3.1112 \(\int \frac {(c+d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=274 \[ -\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f (-d+i c) \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {i c \left (2 c^2+3 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f (c+i d)^{3/2}}-\frac {i (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {(4 d+3 i c) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3} \]

[Out]

-1/8*I*(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^3/f+1/16*I*c*(2*c^2+3*d^2)*arctanh((c+d*t
an(f*x+e))^(1/2)/(c+I*d)^(1/2))/a^3/(c+I*d)^(3/2)/f+1/6*(I*c-d)*(c+d*tan(f*x+e))^(1/2)/f/(a+I*a*tan(f*x+e))^3+
1/24*(3*I*c+4*d)*(c+d*tan(f*x+e))^(1/2)/a/f/(a+I*a*tan(f*x+e))^2-1/16*(2*c^2-I*c*d+2*d^2)*(c+d*tan(f*x+e))^(1/
2)/(I*c-d)/f/(a^3+I*a^3*tan(f*x+e))

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Rubi [A]  time = 1.06, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3558, 3596, 3539, 3537, 63, 208} \[ -\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f (-d+i c) \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {i c \left (2 c^2+3 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f (c+i d)^{3/2}}-\frac {i (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {(4 d+3 i c) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^3,x]

[Out]

((-I/8)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*f) + ((I/16)*c*(2*c^2 + 3*d^2)*A
rcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(a^3*(c + I*d)^(3/2)*f) + ((I*c - d)*Sqrt[c + d*Tan[e + f*x]])
/(6*f*(a + I*a*Tan[e + f*x])^3) + (((3*I)*c + 4*d)*Sqrt[c + d*Tan[e + f*x]])/(24*a*f*(a + I*a*Tan[e + f*x])^2)
- ((2*c^2 - I*c*d + 2*d^2)*Sqrt[c + d*Tan[e + f*x]])/(16*(I*c - d)*f*(a^3 + I*a^3*Tan[e + f*x]))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3539

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3558

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[((b*c - a*d)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1))/(2*a*f*m), x] + Dist[1/(2*a^2*m), Int[(a
+ b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1))
- d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c -
a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && GtQ[n, 1] && (IntegerQ[m] || IntegersQ[2*m,
2*n])

Rule 3596

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((a*A + b*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2
*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

Rubi steps

\begin {align*} \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}-\frac {\int \frac {-\frac {1}{2} a \left (6 c^2-7 i c d+d^2\right )-\frac {1}{2} a (5 c-7 i d) d \tan (e+f x)}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx}{6 a^2}\\ &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}+\frac {\int \frac {\frac {3}{2} a^2 c (c+i d) (4 i c+5 d)+\frac {3}{2} a^2 (i c-d) (3 c-4 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{24 a^4 (i c-d)}\\ &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\int \frac {\frac {3}{2} a^3 (c+i d) \left (4 c^3-2 i c^2 d+5 c d^2-2 i d^3\right )+\frac {3}{2} a^3 (c+i d) d \left (2 c^2-i c d+2 d^2\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{48 a^6 (c+i d)^2}\\ &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(c-i d)^2 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{16 a^3}+\frac {\left (c \left (2 c^2+3 d^2\right )\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{32 a^3 (c+i d)}\\ &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\left (i (c-i d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{16 a^3 f}+\frac {\left (c \left (2 c^2+3 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{32 a^3 (i c-d) f}\\ &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {(c-i d)^2 \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{8 a^3 d f}-\frac {\left (c \left (2 c^2+3 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{16 a^3 (c+i d) d f}\\ &=-\frac {i (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {i c \left (2 c^2+3 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{3/2} f}+\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end {align*}

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Mathematica [A]  time = 2.78, size = 311, normalized size = 1.14 \[ \frac {\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\frac {2 i (\cos (3 e)+i \sin (3 e)) \left (c \sqrt {-c+i d} \left (2 c^2+3 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )+2 (-c-i d)^{3/2} (c-i d)^2 \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )\right )}{(-c-i d)^{3/2} \sqrt {-c+i d}}+\frac {2 \cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt {c+d \tan (e+f x)} \left (\left (9 i c^2+4 c d+10 i d^2\right ) \sin (2 (e+f x))+\left (13 c^2+4 i c d+6 d^2\right ) \cos (2 (e+f x))+7 c (c+i d)\right )}{3 (c+i d)}\right )}{32 f (a+i a \tan (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^3,x]

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*(((2*I)*(c*Sqrt[-c + I*d]*(2*c^2 + 3*d^2)*ArcTan[Sqrt[c + d*Tan[e +
f*x]]/Sqrt[-c - I*d]] + 2*(-c - I*d)^(3/2)*(c - I*d)^2*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[3
*e] + I*Sin[3*e]))/((-c - I*d)^(3/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Sin[3*f*x])*(7*c*(c + I
*d) + (13*c^2 + (4*I)*c*d + 6*d^2)*Cos[2*(e + f*x)] + ((9*I)*c^2 + 4*c*d + (10*I)*d^2)*Sin[2*(e + f*x)])*Sqrt[
c + d*Tan[e + f*x]])/(3*(c + I*d))))/(32*f*(a + I*a*Tan[e + f*x])^3)

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fricas [B]  time = 0.96, size = 1234, normalized size = 4.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(3/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/96*(3*(-I*a^3*c + a^3*d)*f*sqrt(-(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(a^6*f^2))*e^(6*I*f*x + 6*I*e)*log((2*I
*c^2 + 2*c*d + 2*(a^3*f*e^(2*I*f*x + 2*I*e) + a^3*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*
x + 2*I*e) + 1))*sqrt(-(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(a^6*f^2)) + (2*I*c^2 + 4*c*d - 2*I*d^2)*e^(2*I*f*x
+ 2*I*e))*e^(-2*I*f*x - 2*I*e)/(I*c + d)) + 3*(I*a^3*c - a^3*d)*f*sqrt(-(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(
a^6*f^2))*e^(6*I*f*x + 6*I*e)*log((2*I*c^2 + 2*c*d - 2*(a^3*f*e^(2*I*f*x + 2*I*e) + a^3*f)*sqrt(((c - I*d)*e^(
2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(a^6*f^2)) +
(2*I*c^2 + 4*c*d - 2*I*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(I*c + d)) + 24*(-I*a^3*c + a^3*d)*f*sqr
t((-4*I*c^6 - 12*I*c^4*d^2 - 9*I*c^2*d^4)/((256*I*a^6*c^3 - 768*a^6*c^2*d - 768*I*a^6*c*d^2 + 256*a^6*d^3)*f^2
))*e^(6*I*f*x + 6*I*e)*log(-1/16*(-2*I*c^4 + 2*c^3*d - 3*I*c^2*d^2 + 3*c*d^3 + 16*((a^3*c^2 + 2*I*a^3*c*d - a^
3*d^2)*f*e^(2*I*f*x + 2*I*e) + (a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c +
I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt((-4*I*c^6 - 12*I*c^4*d^2 - 9*I*c^2*d^4)/((256*I*a^6*c^3 - 768*a^6*c^2*d -
768*I*a^6*c*d^2 + 256*a^6*d^3)*f^2)) + (-2*I*c^4 - 3*I*c^2*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((a
^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f)) + 24*(I*a^3*c - a^3*d)*f*sqrt((-4*I*c^6 - 12*I*c^4*d^2 - 9*I*c^2*d^4)/((25
6*I*a^6*c^3 - 768*a^6*c^2*d - 768*I*a^6*c*d^2 + 256*a^6*d^3)*f^2))*e^(6*I*f*x + 6*I*e)*log(-1/16*(-2*I*c^4 + 2
*c^3*d - 3*I*c^2*d^2 + 3*c*d^3 - 16*((a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f*e^(2*I*f*x + 2*I*e) + (a^3*c^2 + 2*I*
a^3*c*d - a^3*d^2)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt((-4*I*c^6
- 12*I*c^4*d^2 - 9*I*c^2*d^4)/((256*I*a^6*c^3 - 768*a^6*c^2*d - 768*I*a^6*c*d^2 + 256*a^6*d^3)*f^2)) + (-2*I*
c^4 - 3*I*c^2*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f)) - (2*c^2 +
4*I*c*d - 2*d^2 + (11*c^2 + 8*d^2)*e^(6*I*f*x + 6*I*e) + (18*c^2 + 7*I*c*d + 8*d^2)*e^(4*I*f*x + 4*I*e) + (9*
c^2 + 11*I*c*d - 2*d^2)*e^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*
e) + 1)))*e^(-6*I*f*x - 6*I*e)/((I*a^3*c - a^3*d)*f)

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giac [B]  time = 0.99, size = 624, normalized size = 2.28 \[ -\frac {2 \, {\left (2 \, c^{3} + 3 \, c d^{2}\right )} \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (-8 i \, a^{3} c f + 8 \, a^{3} d f\right )} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {-6 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c^{2} d + 12 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{3} d - 6 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{4} d - 3 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c d^{2} - 12 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{2} d^{2} + 15 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{3} d^{2} - 6 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} d^{3} + 20 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d^{3} + 6 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d^{3} - 20 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{4} + 9 \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{4} + 6 i \, \sqrt {d \tan \left (f x + e\right ) + c} d^{5}}{{\left (-48 i \, a^{3} c f + 48 \, a^{3} d f\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{3}} - \frac {{\left (-i \, c^{2} - 2 \, c d + i \, d^{2}\right )} \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{2 \, a^{3} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(3/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

-2*(2*c^3 + 3*c*d^2)*arctan(4*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(
-8*c + 8*sqrt(c^2 + d^2)) + I*sqrt(-8*c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2)
)))/((-8*I*a^3*c*f + 8*a^3*d*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) + (-6*I*(d*tan
(f*x + e) + c)^(5/2)*c^2*d + 12*I*(d*tan(f*x + e) + c)^(3/2)*c^3*d - 6*I*sqrt(d*tan(f*x + e) + c)*c^4*d - 3*(d
*tan(f*x + e) + c)^(5/2)*c*d^2 - 12*(d*tan(f*x + e) + c)^(3/2)*c^2*d^2 + 15*sqrt(d*tan(f*x + e) + c)*c^3*d^2 -
6*I*(d*tan(f*x + e) + c)^(5/2)*d^3 + 20*I*(d*tan(f*x + e) + c)^(3/2)*c*d^3 + 6*I*sqrt(d*tan(f*x + e) + c)*c^2
*d^3 - 20*(d*tan(f*x + e) + c)^(3/2)*d^4 + 9*sqrt(d*tan(f*x + e) + c)*c*d^4 + 6*I*sqrt(d*tan(f*x + e) + c)*d^5
)/((-48*I*a^3*c*f + 48*a^3*d*f)*(d*tan(f*x + e) - I*d)^3) - 1/2*(-I*c^2 - 2*c*d + I*d^2)*arctan(4*(sqrt(d*tan(
f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) - I*sqrt(-8*c +
8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/(a^3*sqrt(-8*c + 8*sqrt(c^2 + d^2))*f*
(-I*d/(c - sqrt(c^2 + d^2)) + 1))

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maple [B]  time = 0.45, size = 1266, normalized size = 4.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^(3/2)/(a+I*a*tan(f*x+e))^3,x)

[Out]

3/16*I/f/a^3*d^4*c/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))-
1/8*I/f/a^3*c^5/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))+1/8
/f/a^3*d/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(5/2)*c^4+1/8/f/a^3*d^3/(d*tan(f*
x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(5/2)*c^2-1/8/f/a^3*d^5/(d*tan(f*x+e)-I*d)^3/(3*I*c
^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(5/2)-1/4/f/a^3*d/(d*tan(f*x+e)-I*d)^3*(c+d*tan(f*x+e))^(3/2)*c^2-5/1
2/f/a^3*d^3/(d*tan(f*x+e)-I*d)^3*(c+d*tan(f*x+e))^(3/2)-1/16*I/f/a^3*d^2*c^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I
*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))-7/16*I/f/a^3*d^6/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d
^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)*c+9/16*I/f/a^3*d^2/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(
c+d*tan(f*x+e))^(1/2)*c^5+1/8/f/a^3*d/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2
)*c^6-7/8/f/a^3*d^3/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)*c^4-3/8/f/a^3*d^
5/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)*c^2+1/8/f/a^3*d^7/(d*tan(f*x+e)-I*
d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)+3/16*I/f/a^3*d^2/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d
^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(5/2)*c^3+1/8*I/f/a^3*(I*d-c)^(3/2)*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/
2))-3/8*I/f/a^3*d^4/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)*c^3+1/4/f/a^3*d*
c^4/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))+3/8/f/a^3*d^3*c
^2/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))+5/16*I/f/a^3*d^4
/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(5/2)*c

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(3/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 9.21, size = 16296, normalized size = 59.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*tan(e + f*x))^(3/2)/(a + a*tan(e + f*x)*1i)^3,x)

[Out]

(((c + d*tan(e + f*x))^(1/2)*(5*c*d^3 + 10*c^3*d + d^4*10i + c^2*d^2*15i))/(80*a^3*f) - (d*(c + d*tan(e + f*x)
)^(5/2)*(c*d + c^2*2i + d^2*2i)*1i)/(16*(a^3*c*f + a^3*d*f*1i)) + (d*(c^2*6i + d^2*10i)*(c + d*tan(e + f*x))^(
3/2)*1i)/(24*a^3*f))/((c + d*tan(e + f*x))*(c*d*6i + 3*c^2 - 3*d^2) + (c + d*tan(e + f*x))^3 + 3*c*d^2 - c^2*d
*3i - (3*c + d*3i)*(c + d*tan(e + f*x))^2 - c^3 + d^3*1i) - log((((12*c*d^12 - d^13*4i - c^2*d^11*9i + 59*c^3*
d^10 + c^4*d^9*39i + 51*c^5*d^8 + c^6*d^7*64i + c^8*d^5*24i - 8*c^9*d^4 - 4*a^6*c^4*f^2*((216*c^2*d^24 - 16*d^
26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 -
c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 6
4*a^12*c^6*d^2*f^4))^(1/2) - 4*a^6*d^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^2
2 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4
+ 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) - 8*a^6*c^2*d^2*f^
2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^
19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 +
96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2))/(2048*a^6*d^10*f^2 + 6144*a^6*c^2*d^8*f^2 + 6144*a^6*c^4*d
^6*f^2 + 2048*a^6*c^6*d^4*f^2))^(1/2)*(2048*a^9*d^9*f^3 + a^9*c*d^8*f^3*5120i - 32*((12*c*d^12 - d^13*4i - c^2
*d^11*9i + 59*c^3*d^10 + c^4*d^9*39i + 51*c^5*d^8 + c^6*d^7*64i + c^8*d^5*24i - 8*c^9*d^4 - 4*a^6*c^4*f^2*((21
6*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i
- 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^
12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) - 4*a^6*d^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*
176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)
/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2)
- 8*a^6*c^2*d^2*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209
*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*
a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2))/(2048*a^6*d^10*f^2 + 6144*a^6*c^2*d^8*f^
2 + 6144*a^6*c^4*d^6*f^2 + 2048*a^6*c^6*d^4*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(a^6*c^2*d^3*f^2*4096i - 20
48*a^6*c*d^4*f^2 + 2048*a^6*c^3*d^2*f^2)*(a^6*d^2*f^2 - a^6*c^2*f^2 + a^6*c*d*f^2*2i) + a^9*c^3*d^6*f^3*8192i
- 4096*a^9*c^4*d^5*f^3 + a^9*c^5*d^4*f^3*3072i - 2048*a^9*c^6*d^3*f^3) + 32*(c + d*tan(e + f*x))^(1/2)*(a^6*d^
2*f^2 - a^6*c^2*f^2 + a^6*c*d*f^2*2i)*(c*d^7*8i + 4*d^8 - 5*c^2*d^6 + c^3*d^5*16i - 16*c^4*d^4 + c^5*d^3*8i -
8*c^6*d^2))*((12*c*d^12 - d^13*4i - c^2*d^11*9i + 59*c^3*d^10 + c^4*d^9*39i + 51*c^5*d^8 + c^6*d^7*64i + c^8*d
^5*24i - 8*c^9*d^4 - 4*a^6*c^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*
d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^1
2*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) - 4*a^6*d^4*f^2*((216*c^2*
d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111
*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4
*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) - 8*a^6*c^2*d^2*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*17
6i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(
16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2))/(
2048*a^6*d^10*f^2 + 6144*a^6*c^2*d^8*f^2 + 6144*a^6*c^4*d^6*f^2 + 2048*a^6*c^6*d^4*f^2))^(1/2) - a^3*c^2*d^10*
f*42i + 28*a^3*c^3*d^9*f - a^3*c^4*d^8*f*64i + 108*a^3*c^5*d^7*f + a^3*c^6*d^6*f*30i + 44*a^3*c^7*d^5*f + a^3*
c^8*d^4*f*36i - 8*a^3*c^9*d^3*f - 12*a^3*c*d^11*f)*((12*c*d^12 - d^13*4i - c^2*d^11*9i + 59*c^3*d^10 + c^4*d^9
*39i + 51*c^5*d^8 + c^6*d^7*64i + c^8*d^5*24i - 8*c^9*d^4 - 4*a^6*c^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96
i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i +
9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2
*f^4))^(1/2) - 4*a^6*d^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*3
30i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*
f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) - 8*a^6*c^2*d^2*f^2*((216*c^2*d^
24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c
^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d
^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2))/(2048*a^6*d^10*f^2 + 6144*a^6*c^2*d^8*f^2 + 6144*a^6*c^4*d^6*f^2 + 2048*
a^6*c^6*d^4*f^2))^(1/2) - log((((12*c*d^12 - d^13*4i - c^2*d^11*9i + 59*c^3*d^10 + c^4*d^9*39i + 51*c^5*d^8 +
c^6*d^7*64i + c^8*d^5*24i - 8*c^9*d^4 + 4*a^6*c^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i +
111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^
12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) + 4*a^6
*d^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 +
c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^
6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) + 8*a^6*c^2*d^2*f^2*((216*c^2*d^24 - 16*d^26 - c*d^2
5*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*5
4i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6
*d^2*f^4))^(1/2))/(2048*a^6*d^10*f^2 + 6144*a^6*c^2*d^8*f^2 + 6144*a^6*c^4*d^6*f^2 + 2048*a^6*c^6*d^4*f^2))^(1
/2)*(2048*a^9*d^9*f^3 + a^9*c*d^8*f^3*5120i - 32*((12*c*d^12 - d^13*4i - c^2*d^11*9i + 59*c^3*d^10 + c^4*d^9*3
9i + 51*c^5*d^8 + c^6*d^7*64i + c^8*d^5*24i - 8*c^9*d^4 + 4*a^6*c^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i
+ c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9
*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f
^4))^(1/2) + 4*a^6*d^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330
i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^
4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) + 8*a^6*c^2*d^2*f^2*((216*c^2*d^24
- 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8
*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4
*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2))/(2048*a^6*d^10*f^2 + 6144*a^6*c^2*d^8*f^2 + 6144*a^6*c^4*d^6*f^2 + 2048*a^
6*c^6*d^4*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(a^6*c^2*d^3*f^2*4096i - 2048*a^6*c*d^4*f^2 + 2048*a^6*c^3*d^
2*f^2)*(a^6*d^2*f^2 - a^6*c^2*f^2 + a^6*c*d*f^2*2i) + a^9*c^3*d^6*f^3*8192i - 4096*a^9*c^4*d^5*f^3 + a^9*c^5*d
^4*f^3*3072i - 2048*a^9*c^6*d^3*f^3) + 32*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2*f^2 - a^6*c^2*f^2 + a^6*c*d*f^2*
2i)*(c*d^7*8i + 4*d^8 - 5*c^2*d^6 + c^3*d^5*16i - 16*c^4*d^4 + c^5*d^3*8i - 8*c^6*d^2))*((12*c*d^12 - d^13*4i
- c^2*d^11*9i + 59*c^3*d^10 + c^4*d^9*39i + 51*c^5*d^8 + c^6*d^7*64i + c^8*d^5*24i - 8*c^9*d^4 + 4*a^6*c^4*f^2
*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^1
9*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 +
96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) + 4*a^6*d^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*
d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*
d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(
1/2) + 8*a^6*c^2*d^2*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i
- 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4
+ 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2))/(2048*a^6*d^10*f^2 + 6144*a^6*c^2*d
^8*f^2 + 6144*a^6*c^4*d^6*f^2 + 2048*a^6*c^6*d^4*f^2))^(1/2) - a^3*c^2*d^10*f*42i + 28*a^3*c^3*d^9*f - a^3*c^4
*d^8*f*64i + 108*a^3*c^5*d^7*f + a^3*c^6*d^6*f*30i + 44*a^3*c^7*d^5*f + a^3*c^8*d^4*f*36i - 8*a^3*c^9*d^3*f -
12*a^3*c*d^11*f)*((12*c*d^12 - d^13*4i - c^2*d^11*9i + 59*c^3*d^10 + c^4*d^9*39i + 51*c^5*d^8 + c^6*d^7*64i +
c^8*d^5*24i - 8*c^9*d^4 + 4*a^6*c^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 +
c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 1
6*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) + 4*a^6*d^4*f^2*((216
*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i
- 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^1
2*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) + 8*a^6*c^2*d^2*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^
23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^
16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/
2))/(2048*a^6*d^10*f^2 + 6144*a^6*c^2*d^8*f^2 + 6144*a^6*c^4*d^6*f^2 + 2048*a^6*c^6*d^4*f^2))^(1/2) + log((((c
*d^10*4i + 4*d^11 + 21*c^2*d^9 + c^3*d^8*21i + 24*c^4*d^7 + c^5*d^6*24i + 8*c^6*d^5 + c^7*d^4*8i - a^6*c^2*f^2
*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) +
(((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2
*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111
*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*
c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a
^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i + a^6*d^2*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*
c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d
^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^
14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(
a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512
+ (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i + 8*a^6
*c*d*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^
2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2
*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/102
4 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)
- (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4
*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*d^2*f^2*1i - a^6*c^2*f^2*1i + 2*
a^6*c*d*f^2)))^(1/2)*(2048*a^9*d^9*f^3 + a^9*c*d^8*f^3*5120i + a^9*c^3*d^6*f^3*8192i - 4096*a^9*c^4*d^5*f^3 +
a^9*c^5*d^4*f^3*3072i - 2048*a^9*c^6*d^3*f^3 + 32*(c + d*tan(e + f*x))^(1/2)*(a^6*c^2*d^3*f^2*4096i - 2048*a^6
*c*d^4*f^2 + 2048*a^6*c^3*d^2*f^2)*(a^6*d^2*f^2 - a^6*c^2*f^2 + a^6*c*d*f^2*2i)*((c*d^10*4i + 4*d^11 + 21*c^2*
d^9 + c^3*d^8*21i + 24*c^4*d^7 + c^5*d^6*24i + 8*c^6*d^5 + c^7*d^4*8i - a^6*c^2*f^2*(((3*c*d^12 + (59*c^3*d^10
)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*
d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 +
256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^
6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^1
3)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^
2*f^4)))^(1/2)*4i + a^6*d^2*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6
*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^
4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1
024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4
+ 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*
d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i + 8*a^6*c*d*f^2*(((3*c*d^12 + (59*
c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 -
(9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(2
56*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3
*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27
*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^1
2*c^2*d^2*f^4)))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*d^2*f^2*1i - a^6*c^2*f^2*1i + 2*a^6*c*d*f^2)))^(1/2)) - 32*
(c + d*tan(e + f*x))^(1/2)*(a^6*d^2*f^2 - a^6*c^2*f^2 + a^6*c*d*f^2*2i)*(c*d^7*8i + 4*d^8 - 5*c^2*d^6 + c^3*d^
5*16i - 16*c^4*d^4 + c^5*d^3*8i - 8*c^6*d^2))*((c*d^10*4i + 4*d^11 + 21*c^2*d^9 + c^3*d^8*21i + 24*c^4*d^7 + c
^5*d^6*24i + 8*c^6*d^5 + c^7*d^4*8i - a^6*c^2*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/
(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^
8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024
- (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*
f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d
^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i + a^6*d^2*f^2*((
(3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + ((
(39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^
2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^
8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5
*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12
*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i + 8*a^6*c*d*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c
^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^
7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^1
4)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a
^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512
+ (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2))/(2048*(d^6
+ c^2*d^4)*(a^6*d^2*f^2*1i - a^6*c^2*f^2*1i + 2*a^6*c*d*f^2)))^(1/2) - a^3*c^2*d^10*f*42i + 28*a^3*c^3*d^9*f
- a^3*c^4*d^8*f*64i + 108*a^3*c^5*d^7*f + a^3*c^6*d^6*f*30i + 44*a^3*c^7*d^5*f + a^3*c^8*d^4*f*36i - 8*a^3*c^9
*d^3*f - 12*a^3*c*d^11*f)*((c*d^10*4i + 4*d^11 + 21*c^2*d^9 + c^3*d^8*21i + 24*c^4*d^7 + c^5*d^6*24i + 8*c^6*d
^5 + c^7*d^4*8i - a^6*c^2*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d
^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*
f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/102
4 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 +
2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^
5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i + a^6*d^2*f^2*(((3*c*d^12 + (59*c^3*
d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*
c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d
^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^1
0*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3
*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^
2*d^2*f^4)))^(1/2)*4i + 8*a^6*c*d*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2
+ a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(
a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d
^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d
^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3
*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*d^2
*f^2*1i - a^6*c^2*f^2*1i + 2*a^6*c*d*f^2)))^(1/2) + log((((c*d^10*4i + 4*d^11 + 21*c^2*d^9 + c^3*d^8*21i + 24*
c^4*d^7 + c^5*d^6*24i + 8*c^6*d^5 + c^7*d^4*8i + a^6*c^2*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 -
2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6
*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*
d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)
/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/5
12 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i - a^6
*d^2*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^
2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2
*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/102
4 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)
- (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4
*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i - 8*a^6*c*d*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d
^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13
+ 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(
((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d
^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^
7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2))
/(2048*(d^6 + c^2*d^4)*(a^6*d^2*f^2*1i - a^6*c^2*f^2*1i + 2*a^6*c*d*f^2)))^(1/2)*(2048*a^9*d^9*f^3 + a^9*c*d^8
*f^3*5120i + a^9*c^3*d^6*f^3*8192i - 4096*a^9*c^4*d^5*f^3 + a^9*c^5*d^4*f^3*3072i - 2048*a^9*c^6*d^3*f^3 + 32*
(c + d*tan(e + f*x))^(1/2)*(a^6*c^2*d^3*f^2*4096i - 2048*a^6*c*d^4*f^2 + 2048*a^6*c^3*d^2*f^2)*(a^6*d^2*f^2 -
a^6*c^2*f^2 + a^6*c*d*f^2*2i)*((c*d^10*4i + 4*d^11 + 21*c^2*d^9 + c^3*d^8*21i + 24*c^4*d^7 + c^5*d^6*24i + 8*c
^6*d^5 + c^7*d^4*8i + a^6*c^2*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a
^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*
c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)
/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f
^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^1
1*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i - a^6*d^2*f^2*(((3*c*d^12 + (59*
c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 -
(9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(2
56*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3
*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27
*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^1
2*c^2*d^2*f^4)))^(1/2)*4i - 8*a^6*c*d*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4
*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1
i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c
^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^
12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128
- (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6
*d^2*f^2*1i - a^6*c^2*f^2*1i + 2*a^6*c*d*f^2)))^(1/2)) - 32*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2*f^2 - a^6*c^2*
f^2 + a^6*c*d*f^2*2i)*(c*d^7*8i + 4*d^8 - 5*c^2*d^6 + c^3*d^5*16i - 16*c^4*d^4 + c^5*d^3*8i - 8*c^6*d^2))*((c*
d^10*4i + 4*d^11 + 21*c^2*d^9 + c^3*d^8*21i + 24*c^4*d^7 + c^5*d^6*24i + 8*c^6*d^5 + c^7*d^4*8i + a^6*c^2*f^2*
(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) +
(((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*
d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*
c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c
^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^
12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i - a^6*d^2*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c
^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^
7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^1
4)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a
^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512
+ (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i - 8*a^6*
c*d*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2
*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*
a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024
+ (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) -
(((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*
f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*d^2*f^2*1i - a^6*c^2*f^2*1i + 2*a
^6*c*d*f^2)))^(1/2) - a^3*c^2*d^10*f*42i + 28*a^3*c^3*d^9*f - a^3*c^4*d^8*f*64i + 108*a^3*c^5*d^7*f + a^3*c^6*
d^6*f*30i + 44*a^3*c^7*d^5*f + a^3*c^8*d^4*f*36i - 8*a^3*c^9*d^3*f - 12*a^3*c*d^11*f)*((c*d^10*4i + 4*d^11 + 2
1*c^2*d^9 + c^3*d^8*21i + 24*c^4*d^7 + c^5*d^6*24i + 8*c^6*d^5 + c^7*d^4*8i + a^6*c^2*f^2*(((3*c*d^12 + (59*c^
3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (
9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256
*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c
^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c
^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*
c^2*d^2*f^4)))^(1/2)*4i - a^6*d^2*f^2*(((3*c*d^12 + (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2
+ a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(
a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2 + 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d
^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d
^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256 - (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3
*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4)))^(1/2)*4i - 8*a^6*c*d*f^2*(((3*c*d^12
+ (59*c^3*d^10)/4 + (51*c^5*d^8)/4 - 2*c^9*d^4)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2) + (((39*c^4*d^
9)/4 - (9*c^2*d^11)/4 - d^13 + 16*c^6*d^7 + 6*c^8*d^5)*1i)/(a^6*c^4*f^2 + a^6*d^4*f^2 + 2*a^6*c^2*d^2*f^2))^2
+ 4*(256*d^6 + 256*c^2*d^4)*(((9*c^2*d^14)/1024 - (123*c^4*d^12)/1024 - (41*c^6*d^10)/1024 + (111*c^8*d^8)/102
4 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^4*f^4 + a^12*d^4*f^4 + 2*a^12*c^2*d^2*f^4) - (((27*c^5*d^11)/256
- (27*c^3*d^13)/512 + (81*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^4*f^4 + a^12*d^4*f^4 +
2*a^12*c^2*d^2*f^4)))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*d^2*f^2*1i - a^6*c^2*f^2*1i + 2*a^6*c*d*f^2)))^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {i \left (\int \frac {c \sqrt {c + d \tan {\left (e + f x \right )}}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx + \int \frac {d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**(3/2)/(a+I*a*tan(f*x+e))**3,x)

[Out]

I*(Integral(c*sqrt(c + d*tan(e + f*x))/(tan(e + f*x)**3 - 3*I*tan(e + f*x)**2 - 3*tan(e + f*x) + I), x) + Inte
gral(d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)/(tan(e + f*x)**3 - 3*I*tan(e + f*x)**2 - 3*tan(e + f*x) + I), x))
/a**3

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