3.1118 \(\int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=285 \[ \frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\left (2 i c^3+4 c^2 d-i c d^2+2 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f \sqrt {c+i d}}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {(-d+i c) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}+\frac {(c+i d) (2 d+i c) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2} \]

[Out]

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

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Rubi [A]  time = 1.03, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3558, 3595, 3596, 3539, 3537, 63, 208} \[ \frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\left (4 c^2 d+2 i c^3-i c d^2+2 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f \sqrt {c+i d}}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {(-d+i c) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}+\frac {(c+i d) (2 d+i c) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^3,x]

[Out]

((-I/8)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*f) + (((2*I)*c^3 + 4*c^2*d - I*c
*d^2 + 2*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*Sqrt[c + I*d]*f) + ((c + I*d)*(I*c + 2*
d)*Sqrt[c + d*Tan[e + f*x]])/(8*a*f*(a + I*a*Tan[e + f*x])^2) + (((2*I)*c^2 + 5*c*d - (4*I)*d^2)*Sqrt[c + d*Ta
n[e + f*x]])/(16*f*(a^3 + I*a^3*Tan[e + f*x])) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(6*f*(a + I*a*Tan[e +
f*x])^3)

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 3595

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*b - a*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n)/(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 - 1)*Simp[A*(a*c*m + b*d*n
) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]

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))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx &=\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}-\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-\frac {3}{2} a \left (2 c^2-3 i c d+d^2\right )-\frac {3}{2} a (c-3 i d) d \tan (e+f x)\right )}{(a+i a \tan (e+f x))^2} \, dx}{6 a^2}\\ &=\frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}+\frac {\int \frac {\frac {3}{2} a^2 \left (4 c^3-9 i c^2 d-5 c d^2-2 i d^3\right )+\frac {3}{2} a^2 d \left (3 c^2-7 i c d-6 d^2\right ) \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{24 a^4}\\ &=\frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}-\frac {\int \frac {-\frac {3}{2} a^3 c (i c-d) \left (4 c^2-10 i c d-7 d^2\right )-\frac {3}{2} a^3 (i c-d) d \left (2 c^2-5 i c d-4 d^2\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{48 a^6 (i c-d)}\\ &=\frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}+\frac {(c-i d)^3 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{16 a^3}+\frac {\left (2 c^3-4 i c^2 d-c d^2-2 i d^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{32 a^3}\\ &=\frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}-\frac {(i c+d)^3 \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 (2 i c^3+4 c^2 d-i c d^2+2 d^3\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 f}\\ &=\frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}-\frac {(c-i d)^3 \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 (2 c^3-4 i c^2 d-c d^2-2 i d^3\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 d f}\\ &=-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {\left (2 i c^3+4 c^2 d-i c d^2+2 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 \sqrt {c+i d} f}+\frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}\\ \end {align*}

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Mathematica [A]  time = 2.80, size = 324, normalized size = 1.14 \[ \frac {\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\frac {2}{3} \cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt {c+d \tan (e+f x)} \left (\left (9 i c^2+22 c d-2 i d^2\right ) \sin (2 (e+f x))+\left (13 c^2-14 i c d-6 d^2\right ) \cos (2 (e+f x))+7 c^2+i c d+6 d^2\right )+\frac {2 (\cos (3 e)+i \sin (3 e)) \left (-i \sqrt {-c+i d} \left (2 c^3-4 i c^2 d-c d^2-2 i d^3\right ) \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )-2 \sqrt {-c-i d} (d+i c)^3 \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )\right )}{\sqrt {-c-i d} \sqrt {-c+i d}}\right )}{32 f (a+i a \tan (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*Tan[e + f*x])^(5/2)/(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)*Sqrt[-c + I*d]*(2*c^3 - (4*I)*c^2*d - c*d^2 - (2*I)*d^3)*A
rcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] - 2*Sqrt[-c - I*d]*(I*c + d)^3*ArcTan[Sqrt[c + d*Tan[e + f*x]]/
Sqrt[-c + I*d]])*(Cos[3*e] + I*Sin[3*e]))/(Sqrt[-c - I*d]*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Si
n[3*f*x])*(7*c^2 + I*c*d + 6*d^2 + (13*c^2 - (14*I)*c*d - 6*d^2)*Cos[2*(e + f*x)] + ((9*I)*c^2 + 22*c*d - (2*I
)*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/3))/(32*f*(a + I*a*Tan[e + f*x])^3)

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fricas [B]  time = 1.02, size = 1264, normalized size = 4.44 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/192*(6*a^3*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^6*f^2))*e^(6*I*f*x +
6*I*e)*log(1/8*(16*c^3 - 32*I*c^2*d - 16*c*d^2 - (16*I*a^3*f*e^(2*I*f*x + 2*I*e) + 16*I*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^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3
+ 5*c*d^4 - I*d^5)/(a^6*f^2)) + 8*(2*c^3 - 6*I*c^2*d - 6*c*d^2 + 2*I*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2
*I*e)/(c^2 - 2*I*c*d - d^2)) - 6*a^3*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(
a^6*f^2))*e^(6*I*f*x + 6*I*e)*log(1/8*(16*c^3 - 32*I*c^2*d - 16*c*d^2 - (-16*I*a^3*f*e^(2*I*f*x + 2*I*e) - 16*
I*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^5 - 5*I*c^4*d - 10
*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^6*f^2)) + 8*(2*c^3 - 6*I*c^2*d - 6*c*d^2 + 2*I*d^3)*e^(2*I*f*x +
2*I*e))*e^(-2*I*f*x - 2*I*e)/(c^2 - 2*I*c*d - d^2)) - 3*a^3*f*sqrt(-(4*I*c^6 + 16*c^5*d - 20*I*c^4*d^2 - 15*I
*c^2*d^4 - 4*c*d^5 - 4*I*d^6)/((I*a^6*c - a^6*d)*f^2))*e^(6*I*f*x + 6*I*e)*log(-1/16*(2*c^4 - 2*I*c^3*d + 3*c^
2*d^2 - 3*I*c*d^3 + 2*d^4 - ((I*a^3*c - a^3*d)*f*e^(2*I*f*x + 2*I*e) + (I*a^3*c - a^3*d)*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 + 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d^
4 - 4*c*d^5 - 4*I*d^6)/((I*a^6*c - a^6*d)*f^2)) + (2*c^4 - 4*I*c^3*d - c^2*d^2 - 2*I*c*d^3)*e^(2*I*f*x + 2*I*e
))*e^(-2*I*f*x - 2*I*e)/((I*a^3*c - a^3*d)*f)) + 3*a^3*f*sqrt(-(4*I*c^6 + 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d
^4 - 4*c*d^5 - 4*I*d^6)/((I*a^6*c - a^6*d)*f^2))*e^(6*I*f*x + 6*I*e)*log(-1/16*(2*c^4 - 2*I*c^3*d + 3*c^2*d^2
- 3*I*c*d^3 + 2*d^4 - ((-I*a^3*c + a^3*d)*f*e^(2*I*f*x + 2*I*e) + (-I*a^3*c + a^3*d)*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 + 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d^4 -
4*c*d^5 - 4*I*d^6)/((I*a^6*c - a^6*d)*f^2)) + (2*c^4 - 4*I*c^3*d - c^2*d^2 - 2*I*c*d^3)*e^(2*I*f*x + 2*I*e))*e
^(-2*I*f*x - 2*I*e)/((I*a^3*c - a^3*d)*f)) - 2*(2*I*c^2 - 4*c*d - 2*I*d^2 + (11*I*c^2 + 18*c*d - 4*I*d^2)*e^(6
*I*f*x + 6*I*e) + (18*I*c^2 + 17*c*d + 2*I*d^2)*e^(4*I*f*x + 4*I*e) + (9*I*c^2 - 5*c*d + 4*I*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)/(a^3*
f)

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giac [B]  time = 1.38, size = 602, normalized size = 2.11 \[ -\frac {{\left (2 i \, c^{3} + 4 \, c^{2} d - i \, c d^{2} + 2 \, d^{3}\right )} \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{4 \, a^{3} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} f {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{2 \, a^{3} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {6 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c^{2} d - 12 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{3} d + 6 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{4} d - 15 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c d^{2} + 12 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{2} d^{2} + 3 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{3} d^{2} - 12 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} d^{3} - 20 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d^{3} + 12 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d^{3} + 4 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{4} + 9 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{4}}{48 \, {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{3} a^{3} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.45, size = 1861, normalized size = 6.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^3,x)

[Out]

-1/4/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))^(3/2)*c^6-5/12/f/a^3*d^3/(d*t
an(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))^(3/2)*c^4+5/4/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))^(3/2)*c^2-1/8*I/f/a^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))*c^6+1/8*I/f/a^3*d^6/(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*(I*d-c)^(5/2)*arctan((c+d*tan(f*x+e))^(1/2)/(I*d
-c)^(1/2))+1/4*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))^(5/2)+1/8/f/a^3
*d/(d*tan(f*x+e)-I*d)^3*c^7/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)-5/16/f/a^3*d^3/(d*tan(f*x+e)-
I*d)^3*c^5/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)-5/4/f/a^3*d^5/(d*tan(f*x+e)-I*d)^3*c^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/f/a^3*d^7/(d*tan(f*x+e)-I*d)^3*c/(3*I*c^2*d-I*d^3+c^3-3*c*d
^2)*(c+d*tan(f*x+e))^(1/2)+7/16/f/a^3*d^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))*c+1/8/f/a^3*d/(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))*c^5+5/16/f/a^3*d^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))*c^3-13/16*I/f/a^3*d^6/(d*tan(f*x+e)-I*d)^3*c^2/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e)
)^(1/2)+1/16*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^4-5/3*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))^(3/2)*c^3-5/16*I/f/a^3*d^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))*c^4-5/16*I/f/a^3*d^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))*c^2+7/16*I/f/a^3*d^2/(d
*tan(f*x+e)-I*d)^3*c^6/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)+5/8*I/f/a^3*d^4/(d*tan(f*x+e)-I*d)
^3*c^4/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)+1/6*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))^(3/2)*c+1/16*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^2-1/2*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))
^(3/2)*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))^(5/2)*c^5+5/16/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^3+7/16/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)*c+1/12/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))^(3/2)

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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))^(5/2)/(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 = 9.34, size = 9472, normalized size = 33.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*tan(e + f*x))^(5/2)/(a + a*tan(e + f*x)*1i)^3,x)

[Out]

- atan((((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) - 65536*a^12*c*d^2*f^4*(
c + d*tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8
*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*
c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25
*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024
- d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/2
56)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d
^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^
4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*
c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)
*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1
024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 32*a^6*f^
2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*
d^2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2
*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5
)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (28
5*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c
^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/
2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*1i - ((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^
6*c^3*d^3*f^2) + 65536*a^12*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4
*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*
d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((
5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/12
8)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)
/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*
d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 -
(35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2
))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512
+ (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024
+ (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*
f^2*(d^6 + c^2*d^4)))^(1/2) - 32*a^6*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*
80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^
6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((
55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*
c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) -
((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6
)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*1i)/(((2*a^3*f*(1536*a^6*c*d^
5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) - 65536*a^12*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(20*
c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10
- (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f
^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/5
12 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/102
4 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^
6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40
i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4
+ 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512
- (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)
/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d
^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 32*a^6*f^2*(c + d*tan(e + f*x))^(1/2)*(c*
d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*(-(20*c*d^10 + c^2*d^9*55i
- 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 -
18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6
+ 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/12
8 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1
024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4
)))^(1/2) + ((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) + 65536*a^12*c*d^2*f
^4*(c + d*tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i
+ 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 +
10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 -
(25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/
1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^
4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c
^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^
7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((
((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/
128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^
8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) - 32*a^
6*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*
c^6*d^2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6
*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6
*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 -
(285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (2
25*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))
^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 4*a^3*f*(c*d^10*30i + 8*d^11 - 25*c^2*d^9 + c^3*d^8*55i - 153*
c^4*d^7 - c^5*d^6*165i + 96*c^6*d^5 + c^7*d^4*30i - 4*c^8*d^3)))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4
*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*
d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((
5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/12
8)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)
/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*2i - atan
((((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) - 65536*a^12*c*d^2*f^4*(c + d*
tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d
^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^
7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d
^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^1
6/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a
^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i
- 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^
6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15
)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(
a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 +
(11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 32*a^6*f^2*(c +
d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*
(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*
c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/
(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*
d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^1
2)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2
048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*1i - ((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*
d^3*f^2) + 65536*a^12*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*4
0i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(
a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^
15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)
/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024
+ (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 +
c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c
^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 -
4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55
*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155
*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d
^6 + c^2*d^4)))^(1/2) - 32*a^6*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i +
80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^
6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2
*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^
13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c
^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 -
(c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*1i)/(((2*a^3*f*(1536*a^6*c*d^5*f^2
+ a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) - 65536*a^12*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(20*c*d^10
+ c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35
*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2
- 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (
55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (1
55*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*
(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*
c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c
^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*
c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024
- d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/25
6)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 32*a^6*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*20
i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*(-(20*c*d^10 + c^2*d^9*55i - 35*
c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5
*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*
c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5
*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 -
(665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1
/2) + ((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) + 65536*a^12*c*d^2*f^4*(c
+ d*tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c
^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^
4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c
^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 -
d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256
)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7
*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)
/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*
d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1
i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/102
4 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) - 32*a^6*f^2*
(c + d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^
2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(
((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*
1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*
c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4
*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2)
)/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 4*a^3*f*(c*d^10*30i + 8*d^11 - 25*c^2*d^9 + c^3*d^8*55i - 153*c^4*d^
7 - c^5*d^6*165i + 96*c^6*d^5 + c^7*d^4*30i - 4*c^8*d^3)))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*4
0i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(
a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^
15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)
/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024
+ (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*2i - (((c + d*t
an(e + f*x))^(3/2)*(10*c*d^3 + 6*c^3*d - d^4*2i - c^2*d^2*6i))/(24*a^3*f) - ((c + d*tan(e + f*x))^(1/2)*(c*d^4
*15i + 10*c^4*d + 20*c^2*d^3 + c^3*d^2*5i))/(80*a^3*f) + (d*(c + d*tan(e + f*x))^(5/2)*(5*c*d + c^2*2i - d^2*4
i)*1i)/(16*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*3
i - (3*c + d*3i)*(c + d*tan(e + f*x))^2 - c^3 + d^3*1i)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**(5/2)/(a+I*a*tan(f*x+e))**3,x)

[Out]

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