3.1117 \(\int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=217 \[ \frac {\sqrt {c+i d} \left (2 i c^2+6 c d-7 i d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 f}+\frac {(c+i d) (5 d+2 i c) \sqrt {c+d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 f}+\frac {(-d+i c) (c+d \tan (e+f x))^{3/2}}{4 f (a+i a \tan (e+f x))^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.64, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3558, 3595, 3539, 3537, 63, 208} \[ \frac {\sqrt {c+i d} \left (2 i c^2+6 c d-7 i d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 f}+\frac {(c+i d) (5 d+2 i c) \sqrt {c+d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 f}+\frac {(-d+i c) (c+d \tan (e+f x))^{3/2}}{4 f (a+i a \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/4)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*f) + (Sqrt[c + I*d]*((2*I)*c^2 +
6*c*d - (7*I)*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*f) + ((c + I*d)*((2*I)*c + 5*d)*Sq
rt[c + d*Tan[e + f*x]])/(8*a^2*f*(1 + I*Tan[e + f*x])) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(4*f*(a + I*a*
Tan[e + f*x])^2)

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 2.01, size = 291, normalized size = 1.34 \[ \frac {\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left (\frac {2 (\cos (2 e)+i \sin (2 e)) \left (-i \sqrt {-c+i d} \left (2 c^3-4 i c^2 d-c d^2-7 i d^3\right ) \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )-2 \sqrt {-c-i d} (d+i c)^3 \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )\right )}{\sqrt {-c-i d} \sqrt {-c+i d}}+2 (c+i d) \cos (e+f x) (\cos (2 f x)-i \sin (2 f x)) \sqrt {c+d \tan (e+f x)} ((-2 c+7 i d) \sin (e+f x)+(5 d+4 i c) \cos (e+f x))\right )}{16 f (a+i a \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*((2*((-I)*Sqrt[-c + I*d]*(2*c^3 - (4*I)*c^2*d - c*d^2 - (7*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[2*e] + I*Sin[2*e]))/(Sqrt[-c - I*d]*Sqrt[-c + I*d]) + 2*(c + I*d)*Cos[e + f*x]*(Cos[2*f*
x] - I*Sin[2*f*x])*(((4*I)*c + 5*d)*Cos[e + f*x] + (-2*c + (7*I)*d)*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]]))/(
16*f*(a + I*a*Tan[e + f*x])^2)

________________________________________________________________________________________

fricas [B]  time = 0.73, size = 1094, normalized size = 5.04 \[ \text {result too large to display} \]

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))^2,x, algorithm="fricas")

[Out]

-1/32*(2*a^2*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^4*f^2))*e^(4*I*f*x + 4
*I*e)*log(1/4*(8*c^3 - 16*I*c^2*d - 8*c*d^2 - (8*I*a^2*f*e^(2*I*f*x + 2*I*e) + 8*I*a^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(-(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^4*f^2)) + 4*(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)) - 2*a^2*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^4*f
^2))*e^(4*I*f*x + 4*I*e)*log(1/4*(8*c^3 - 16*I*c^2*d - 8*c*d^2 - (-8*I*a^2*f*e^(2*I*f*x + 2*I*e) - 8*I*a^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(-(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^4*f^2)) + 4*(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)) - a^2*f*sqrt(-(4*c^5 - 20*I*c^4*d - 40*c^3*d^2 + 20*I*c^2*d^3 - 35
*c*d^4 + 49*I*d^5)/(a^4*f^2))*e^(4*I*f*x + 4*I*e)*log(1/8*(2*I*c^3 + 4*c^2*d - I*c*d^2 + 7*d^3 + (a^2*f*e^(2*I
*f*x + 2*I*e) + a^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*c^5
- 20*I*c^4*d - 40*c^3*d^2 + 20*I*c^2*d^3 - 35*c*d^4 + 49*I*d^5)/(a^4*f^2)) + (2*I*c^3 + 6*c^2*d - 7*I*c*d^2)*e
^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(a^2*f)) + a^2*f*sqrt(-(4*c^5 - 20*I*c^4*d - 40*c^3*d^2 + 20*I*c^2*d^
3 - 35*c*d^4 + 49*I*d^5)/(a^4*f^2))*e^(4*I*f*x + 4*I*e)*log(1/8*(2*I*c^3 + 4*c^2*d - I*c*d^2 + 7*d^3 - (a^2*f*
e^(2*I*f*x + 2*I*e) + a^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*c^5 - 20*I*c^4*d - 40*c^3*d^2 + 20*I*c^2*d^3 - 35*c*d^4 + 49*I*d^5)/(a^4*f^2)) + (2*I*c^3 + 6*c^2*d - 7*I*c*
d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(a^2*f)) - 2*(I*c^2 - 2*c*d - I*d^2 + (3*I*c^2 + 3*c*d + 6*I*d^
2)*e^(4*I*f*x + 4*I*e) + (4*I*c^2 + c*d + 5*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^(-4*I*f*x - 4*I*e)/(a^2*f)

________________________________________________________________________________________

giac [B]  time = 1.21, size = 516, normalized size = 2.38 \[ \frac {\sqrt {2} {\left (c^{3} - 3 i \, c^{2} d - 3 \, c d^{2} + i \, d^{3}\right )} \arctan \left (\frac {-16 i \, \sqrt {d \tan \left (f x + e\right ) + c} c - 16 i \, \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}}{8 \, {\left (\sqrt {2} \sqrt {c + \sqrt {c^{2} + d^{2}}} c - i \, \sqrt {2} \sqrt {c + \sqrt {c^{2} + d^{2}}} d + \sqrt {2} \sqrt {c^{2} + d^{2}} \sqrt {c + \sqrt {c^{2} + d^{2}}}\right )}}\right )}{4 \, a^{2} \sqrt {c + \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c + \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {{\left (2 i \, c^{3} + 4 \, c^{2} d - i \, c d^{2} + 7 \, d^{3}\right )} \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{2 \, a^{2} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} f {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {2 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{2} d - 2 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{3} d - 5 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d^{2} + i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d^{2} + 7 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{3} - 8 \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{3} - 5 i \, \sqrt {d \tan \left (f x + e\right ) + c} d^{4}}{8 \, {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2} a^{2} f} \]

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))^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.40, size = 978, normalized size = 4.51 \[ \frac {d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{4}}{4 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}+\frac {15 d^{3} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{2}}{8 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}-\frac {7 d^{5} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{8 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}+\frac {5 i d^{6} \sqrt {c +d \tan \left (f x +e \right )}}{8 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}-\frac {i d^{2} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{3}}{8 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}-\frac {d \sqrt {c +d \tan \left (f x +e \right )}\, c^{5}}{4 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}-\frac {d^{3} \sqrt {c +d \tan \left (f x +e \right )}\, c^{3}}{f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}+\frac {9 d^{5} \sqrt {c +d \tan \left (f x +e \right )}\, c}{4 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}-\frac {5 i d^{2} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c^{3}}{8 f \,a^{2} \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}-\frac {15 i d^{4} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c}{8 f \,a^{2} \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}-\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c^{5}}{4 f \,a^{2} \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}-\frac {5 d^{3} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c^{2}}{8 f \,a^{2} \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}+\frac {7 d^{5} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{8 f \,a^{2} \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}+\frac {19 i d^{4} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} c}{8 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}-\frac {3 i d^{2} \sqrt {c +d \tan \left (f x +e \right )}\, c^{4}}{8 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}-\frac {11 i d^{4} \sqrt {c +d \tan \left (f x +e \right )}\, c^{2}}{4 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2} \left (2 i c d +c^{2}-d^{2}\right )}-\frac {i \left (i d -c \right )^{\frac {5}{2}} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{4 f \,a^{2}} \]

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))^2,x)

[Out]

1/4/f/a^2*d/(d*tan(f*x+e)-I*d)^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(3/2)*c^4+15/8/f/a^2*d^3/(d*tan(f*x+e)-I*
d)^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(3/2)*c^2-7/8/f/a^2*d^5/(d*tan(f*x+e)-I*d)^2/(-d^2+2*I*c*d+c^2)*(c+d*
tan(f*x+e))^(3/2)+5/8*I/f/a^2*d^6/(d*tan(f*x+e)-I*d)^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(1/2)-1/8*I/f/a^2*d
^2/(d*tan(f*x+e)-I*d)^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(3/2)*c^3-1/4/f/a^2*d/(d*tan(f*x+e)-I*d)^2/(-d^2+2
*I*c*d+c^2)*(c+d*tan(f*x+e))^(1/2)*c^5-1/f/a^2*d^3/(d*tan(f*x+e)-I*d)^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(1
/2)*c^3+9/4/f/a^2*d^5/(d*tan(f*x+e)-I*d)^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(1/2)*c-5/8*I/f/a^2*d^2/(-d^2+2
*I*c*d+c^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^3-15/8*I/f/a^2*d^4/(-d^2+2*I*c*d+c^
2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c-1/4*I/f/a^2/(-d^2+2*I*c*d+c^2)/(-I*d-c)^(1/2
)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^5-5/8/f/a^2*d^3/(-d^2+2*I*c*d+c^2)/(-I*d-c)^(1/2)*arctan((c+
d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^2+7/8/f/a^2*d^5/(-d^2+2*I*c*d+c^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e)
)^(1/2)/(-I*d-c)^(1/2))+19/8*I/f/a^2*d^4/(d*tan(f*x+e)-I*d)^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(3/2)*c-3/8*
I/f/a^2*d^2/(d*tan(f*x+e)-I*d)^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(1/2)*c^4-11/4*I/f/a^2*d^4/(d*tan(f*x+e)-
I*d)^2/(-d^2+2*I*c*d+c^2)*(c+d*tan(f*x+e))^(1/2)*c^2-1/4*I/f/a^2*(I*d-c)^(5/2)*arctan((c+d*tan(f*x+e))^(1/2)/(
I*d-c)^(1/2))

________________________________________________________________________________________

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))^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 9.19, size = 9787, normalized size = 45.10 \[ \text {result too large to display} \]

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)^2,x)

[Out]

- atan((((a^2*f*(a^4*d^6*f^2*640i - 256*a^4*c*d^5*f^2 + a^4*c^2*d^4*f^2*640i - 256*a^4*c^3*d^3*f^2) - 4096*a^8
*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(d^11*45i - 15*c*d^10 + c^2*d^9*105i - 95*c^3*d^8 + c^4*d^7*20i - 72*c
^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (
295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^
2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*
f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d
^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(d^11*45i - 15*c*d^10 + c^2*d^9*1
05i - 95*c^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*(((
(105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i
)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (
11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2)
- (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2)
+ 8*a^4*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*10i + 53*d^8 - 5*c^2*d^6 - c^3*d^5*60i + 80*c^4*d^4 + c^5*d^3*4
0i - 8*c^6*d^2))*(-(d^11*45i - 15*c*d^10 + c^2*d^9*105i - 95*c^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i
+ 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5
*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4
*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 +
105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*
f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2)*1i - ((a^2*f*(a^4*d^6*f^2*640i - 256*a^4*c*d^5*f^2 + a^4*
c^2*d^4*f^2*640i - 256*a^4*c^3*d^3*f^2) + 4096*a^8*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(d^11*45i - 15*c*d^1
0 + c^2*d^9*105i - 95*c^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256
*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^
11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^
8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*
1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2
*d^4)))^(1/2))*(-(d^11*45i - 15*c*d^10 + c^2*d^9*105i - 95*c^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i +
8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c
^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d
^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 10
5*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^
2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2) - 8*a^4*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*10i + 53*d^8
- 5*c^2*d^6 - c^3*d^5*60i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*(-(d^11*45i - 15*c*d^10 + c^2*d^9*105i - 95
*c^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3
*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f
^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*
d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c
*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2)*1i)/(((
a^2*f*(a^4*d^6*f^2*640i - 256*a^4*c*d^5*f^2 + a^4*c^2*d^4*f^2*640i - 256*a^4*c^3*d^3*f^2) - 4096*a^8*c*d^2*f^4
*(c + d*tan(e + f*x))^(1/2)*(-(d^11*45i - 15*c*d^10 + c^2*d^9*105i - 95*c^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c
^6*d^5*40i + 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^
11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16
- (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + ((
(45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7
*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(d^11*45i - 15*c*d^10 + c^2*d^9*105i - 95*c
^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d
^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4
) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^
6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d
^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2) + 8*a^4*f
^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*10i + 53*d^8 - 5*c^2*d^6 - c^3*d^5*60i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6
*d^2))*(-(d^11*45i - 15*c*d^10 + c^2*d^9*105i - 95*c^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^
4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/
16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32
+ (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^
9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(
1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2) + ((a^2*f*(a^4*d^6*f^2*640i - 256*a^4*c*d^5*f^2 + a^4*c^2*d^4*f^2*6
40i - 256*a^4*c^3*d^3*f^2) + 4096*a^8*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(d^11*45i - 15*c*d^10 + c^2*d^9*1
05i - 95*c^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*(((
(105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i
)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (
11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2)
- (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2)
)*(-(d^11*45i - 15*c*d^10 + c^2*d^9*105i - 95*c^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + a
^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 -
(105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (21
5*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 2
0*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))
/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2) - 8*a^4*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*10i + 53*d^8 - 5*c^2*d^6 -
c^3*d^5*60i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*(-(d^11*45i - 15*c*d^10 + c^2*d^9*105i - 95*c^3*d^8 + c^
4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (
35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^
16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^1
2*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^
3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2) - 2*a^2*f*(240*c^2*d
^9 - 63*d^11 - c*d^10*215i + c^3*d^8*20i + 193*c^4*d^7 + c^5*d^6*205i - 106*c^6*d^5 - c^7*d^4*30i + 4*c^8*d^3)
))*(-(d^11*45i - 15*c*d^10 + c^2*d^9*105i - 95*c^3*d^8 + c^4*d^7*20i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 +
a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 -
(105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (2
15*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 +
20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2)
)/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2)*2i - atan((((a^2*f*(a^4*d^6*f^2*640i - 256*a^4*c*d^5*f^2 + a^4*c^2*d^4*
f^2*640i - 256*a^4*c^3*d^3*f^2) - 4096*a^8*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*((15*c*d^10 - d^11*45i - c^2*d
^9*105i + 95*c^3*d^8 - c^4*d^7*20i + 72*c^5*d^6 + c^6*d^5*40i - 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)
*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8
)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64
- (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*
f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(
1/2))*((15*c*d^10 - d^11*45i - c^2*d^9*105i + 95*c^3*d^8 - c^4*d^7*20i + 72*c^5*d^6 + c^6*d^5*40i - 8*c^7*d^4
+ a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16
- (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 +
(215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9
+ 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/
2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2) + 8*a^4*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*10i + 53*d^8 - 5*c^2*d^
6 - c^3*d^5*60i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*((15*c*d^10 - d^11*45i - c^2*d^9*105i + 95*c^3*d^8 -
c^4*d^7*20i + 72*c^5*d^6 + c^6*d^5*40i - 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 -
(35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*
d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c
^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*
c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2)*1i - ((a^2*f*(a^4*
d^6*f^2*640i - 256*a^4*c*d^5*f^2 + a^4*c^2*d^4*f^2*640i - 256*a^4*c^3*d^3*f^2) + 4096*a^8*c*d^2*f^4*(c + d*tan
(e + f*x))^(1/2)*((15*c*d^10 - d^11*45i - c^2*d^9*105i + 95*c^3*d^8 - c^4*d^7*20i + 72*c^5*d^6 + c^6*d^5*40i -
8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*
c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*
d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 1
05*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f
^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2))*((15*c*d^10 - d^11*45i - c^2*d^9*105i + 95*c^3*d^8 - c^4*
d^7*20i + 72*c^5*d^6 + c^6*d^5*40i - 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35
*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16
)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*
d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*
d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2) - 8*a^4*f^2*(c + d*tan
(e + f*x))^(1/2)*(c*d^7*10i + 53*d^8 - 5*c^2*d^6 - c^3*d^5*60i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*((15*c
*d^10 - d^11*45i - c^2*d^9*105i + 95*c^3*d^8 - c^4*d^7*20i + 72*c^5*d^6 + c^6*d^5*40i - 8*c^7*d^4 + a^4*f^2*(4
*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*
d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^1
0)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7
- 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4
*f^2*(d^6 + c^2*d^4)))^(1/2)*1i)/(((a^2*f*(a^4*d^6*f^2*640i - 256*a^4*c*d^5*f^2 + a^4*c^2*d^4*f^2*640i - 256*a
^4*c^3*d^3*f^2) - 4096*a^8*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*((15*c*d^10 - d^11*45i - c^2*d^9*105i + 95*c^3
*d^8 - c^4*d^7*20i + 72*c^5*d^6 + c^6*d^5*40i - 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^1
3)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4)
- ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)
/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^1
0 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2))*((15*c*d^1
0 - d^11*45i - c^2*d^9*105i + 95*c^3*d^8 - c^4*d^7*20i + 72*c^5*d^6 + c^6*d^5*40i - 8*c^7*d^4 + a^4*f^2*(4*(25
6*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)
/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/1
6 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 4
0*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2
*(d^6 + c^2*d^4)))^(1/2) + 8*a^4*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*10i + 53*d^8 - 5*c^2*d^6 - c^3*d^5*60i
+ 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*((15*c*d^10 - d^11*45i - c^2*d^9*105i + 95*c^3*d^8 - c^4*d^7*20i + 72
*c^5*d^6 + c^6*d^5*40i - 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 +
(295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*
c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^
8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5
*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2) + ((a^2*f*(a^4*d^6*f^2*640i - 256*
a^4*c*d^5*f^2 + a^4*c^2*d^4*f^2*640i - 256*a^4*c^3*d^3*f^2) + 4096*a^8*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*((
15*c*d^10 - d^11*45i - c^2*d^9*105i + 95*c^3*d^8 - c^4*d^7*20i + 72*c^5*d^6 + c^6*d^5*40i - 8*c^7*d^4 + a^4*f^
2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*
c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6
*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4
*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512
*a^4*f^2*(d^6 + c^2*d^4)))^(1/2))*((15*c*d^10 - d^11*45i - c^2*d^9*105i + 95*c^3*d^8 - c^4*d^7*20i + 72*c^5*d^
6 + c^6*d^5*40i - 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c
^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^1
4)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4))
+ (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 -
8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2) - 8*a^4*f^2*(c + d*tan(e + f*x))^(1/2)*(c
*d^7*10i + 53*d^8 - 5*c^2*d^6 - c^3*d^5*60i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*((15*c*d^10 - d^11*45i -
c^2*d^9*105i + 95*c^3*d^8 - c^4*d^7*20i + 72*c^5*d^6 + c^6*d^5*40i - 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2
*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d
^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^
8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4)) + (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/
(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4
)))^(1/2) - 2*a^2*f*(240*c^2*d^9 - 63*d^11 - c*d^10*215i + c^3*d^8*20i + 193*c^4*d^7 + c^5*d^6*205i - 106*c^6*
d^5 - c^7*d^4*30i + 4*c^8*d^3)))*((15*c*d^10 - d^11*45i - c^2*d^9*105i + 95*c^3*d^8 - c^4*d^7*20i + 72*c^5*d^6
+ c^6*d^5*40i - 8*c^7*d^4 + a^4*f^2*(4*(256*d^6 + 256*c^2*d^4)*((((105*c^3*d^13)/16 - (35*c*d^15)/8 + (295*c^
5*d^11)/16 + (5*c^7*d^9)/16 - (105*c^9*d^7)/16 + (5*c^11*d^5)/8)*1i)/(a^8*f^4) - ((49*d^16)/64 - (149*c^2*d^14
)/16 - (235*c^4*d^12)/32 + (215*c^6*d^10)/16 + (505*c^8*d^8)/64 - (11*c^10*d^6)/4 + (c^12*d^4)/16)/(a^8*f^4))
+ (((45*d^11 + 105*c^2*d^9 + 20*c^4*d^7 - 40*c^6*d^5)*1i)/(a^4*f^2) - (15*c*d^10 + 95*c^3*d^8 + 72*c^5*d^6 - 8
*c^7*d^4)/(a^4*f^2))^2)^(1/2))/(512*a^4*f^2*(d^6 + c^2*d^4)))^(1/2)*2i - (((c + d*tan(e + f*x))^(1/2)*(24*c*d^
3 + 6*c^3*d + d^4*15i - c^2*d^2*3i))/(24*a^2*f) - ((c + d*tan(e + f*x))^(3/2)*(2*c^2*d - c*d^2*5i + 7*d^3))/(8
*a^2*f))/(c*d*2i - (2*c + d*2i)*(c + d*tan(e + f*x)) + (c + d*tan(e + f*x))^2 + c^2 - d^2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {c^{2} \sqrt {c + d \tan {\left (e + f x \right )}}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx + \int \frac {d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx + \int \frac {2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx}{a^{2}} \]

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))**2,x)

[Out]

-(Integral(c**2*sqrt(c + d*tan(e + f*x))/(tan(e + f*x)**2 - 2*I*tan(e + f*x) - 1), x) + Integral(d**2*sqrt(c +
d*tan(e + f*x))*tan(e + f*x)**2/(tan(e + f*x)**2 - 2*I*tan(e + f*x) - 1), x) + Integral(2*c*d*sqrt(c + d*tan(
e + f*x))*tan(e + f*x)/(tan(e + f*x)**2 - 2*I*tan(e + f*x) - 1), x))/a**2

________________________________________________________________________________________