3.436 \(\int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=206 \[ \frac{256 i \sqrt{e \sec (c+d x)}}{585 a^2 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{128 i \sqrt{a+i a \tan (c+d x)}}{585 a^3 d (e \sec (c+d x))^{3/2}}+\frac{32 i}{195 a^2 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}+\frac{16 i}{117 a d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}+\frac{2 i}{13 d (a+i a \tan (c+d x))^{5/2} (e \sec (c+d x))^{3/2}} \]

[Out]

((2*I)/13)/(d*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)) + ((16*I)/117)/(a*d*(e*Sec[c + d*x])^(3/2)*
(a + I*a*Tan[c + d*x])^(3/2)) + ((32*I)/195)/(a^2*d*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (((25
6*I)/585)*Sqrt[e*Sec[c + d*x]])/(a^2*d*e^2*Sqrt[a + I*a*Tan[c + d*x]]) - (((128*I)/585)*Sqrt[a + I*a*Tan[c + d
*x]])/(a^3*d*(e*Sec[c + d*x])^(3/2))

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Rubi [A]  time = 0.398962, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3502, 3497, 3488} \[ \frac{256 i \sqrt{e \sec (c+d x)}}{585 a^2 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{128 i \sqrt{a+i a \tan (c+d x)}}{585 a^3 d (e \sec (c+d x))^{3/2}}+\frac{32 i}{195 a^2 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}+\frac{16 i}{117 a d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}+\frac{2 i}{13 d (a+i a \tan (c+d x))^{5/2} (e \sec (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]

[Out]

((2*I)/13)/(d*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)) + ((16*I)/117)/(a*d*(e*Sec[c + d*x])^(3/2)*
(a + I*a*Tan[c + d*x])^(3/2)) + ((32*I)/195)/(a^2*d*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (((25
6*I)/585)*Sqrt[e*Sec[c + d*x]])/(a^2*d*e^2*Sqrt[a + I*a*Tan[c + d*x]]) - (((128*I)/585)*Sqrt[a + I*a*Tan[c + d
*x]])/(a^3*d*(e*Sec[c + d*x])^(3/2))

Rule 3502

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

Rule 3497

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

Rule 3488

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]

Rubi steps

\begin{align*} \int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2}} \, dx &=\frac{2 i}{13 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2}}+\frac{8 \int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}} \, dx}{13 a}\\ &=\frac{2 i}{13 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2}}+\frac{16 i}{117 a d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac{16 \int \frac{1}{(e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}} \, dx}{39 a^2}\\ &=\frac{2 i}{13 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2}}+\frac{16 i}{117 a d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac{32 i}{195 a^2 d (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}+\frac{64 \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{195 a^3}\\ &=\frac{2 i}{13 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2}}+\frac{16 i}{117 a d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac{32 i}{195 a^2 d (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}-\frac{128 i \sqrt{a+i a \tan (c+d x)}}{585 a^3 d (e \sec (c+d x))^{3/2}}+\frac{128 \int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{585 a^2 e^2}\\ &=\frac{2 i}{13 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{5/2}}+\frac{16 i}{117 a d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac{32 i}{195 a^2 d (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}+\frac{256 i \sqrt{e \sec (c+d x)}}{585 a^2 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{128 i \sqrt{a+i a \tan (c+d x)}}{585 a^3 d (e \sec (c+d x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.548076, size = 107, normalized size = 0.52 \[ \frac{\sec ^4(c+d x) (1040 \sin (2 (c+d x))-120 \sin (4 (c+d x))-1300 i \cos (2 (c+d x))+75 i \cos (4 (c+d x))-351 i)}{2340 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]

[Out]

(Sec[c + d*x]^4*(-351*I - (1300*I)*Cos[2*(c + d*x)] + (75*I)*Cos[4*(c + d*x)] + 1040*Sin[2*(c + d*x)] - 120*Si
n[4*(c + d*x)]))/(2340*a^2*d*(e*Sec[c + d*x])^(3/2)*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])

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Maple [A]  time = 0.315, size = 159, normalized size = 0.8 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( 180\,i \left ( \cos \left ( dx+c \right ) \right ) ^{7}+180\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) -55\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}+35\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+8\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+48\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +64\,i\cos \left ( dx+c \right ) +128\,\sin \left ( dx+c \right ) \right ) }{585\,d{a}^{3}{e}^{3}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/585/d/a^3*(e/cos(d*x+c))^(3/2)*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)^2*(180*I*cos(d*x+c)
^7+180*cos(d*x+c)^6*sin(d*x+c)-55*I*cos(d*x+c)^5+35*sin(d*x+c)*cos(d*x+c)^4+8*I*cos(d*x+c)^3+48*cos(d*x+c)^2*s
in(d*x+c)+64*I*cos(d*x+c)+128*sin(d*x+c))/e^3

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Maxima [A]  time = 2.05163, size = 305, normalized size = 1.48 \begin{align*} \frac{45 i \, \cos \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ) + 260 i \, \cos \left (\frac{9}{13} \, \arctan \left (\sin \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ), \cos \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right )\right )\right ) + 702 i \, \cos \left (\frac{5}{13} \, \arctan \left (\sin \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ), \cos \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right )\right )\right ) - 195 i \, \cos \left (\frac{3}{13} \, \arctan \left (\sin \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ), \cos \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right )\right )\right ) + 2340 i \, \cos \left (\frac{1}{13} \, \arctan \left (\sin \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ), \cos \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right )\right )\right ) + 45 \, \sin \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ) + 260 \, \sin \left (\frac{9}{13} \, \arctan \left (\sin \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ), \cos \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right )\right )\right ) + 702 \, \sin \left (\frac{5}{13} \, \arctan \left (\sin \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ), \cos \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right )\right )\right ) + 195 \, \sin \left (\frac{3}{13} \, \arctan \left (\sin \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ), \cos \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right )\right )\right ) + 2340 \, \sin \left (\frac{1}{13} \, \arctan \left (\sin \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right ), \cos \left (\frac{13}{2} \, d x + \frac{13}{2} \, c\right )\right )\right )}{4680 \, a^{\frac{5}{2}} d e^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*sec(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

1/4680*(45*I*cos(13/2*d*x + 13/2*c) + 260*I*cos(9/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c)))
+ 702*I*cos(5/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))) - 195*I*cos(3/13*arctan2(sin(13/2*d*
x + 13/2*c), cos(13/2*d*x + 13/2*c))) + 2340*I*cos(1/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c)
)) + 45*sin(13/2*d*x + 13/2*c) + 260*sin(9/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))) + 702*s
in(5/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))) + 195*sin(3/13*arctan2(sin(13/2*d*x + 13/2*c)
, cos(13/2*d*x + 13/2*c))) + 2340*sin(1/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))))/(a^(5/2)*
d*e^(3/2))

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Fricas [A]  time = 2.16607, size = 360, normalized size = 1.75 \begin{align*} \frac{\sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-195 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 2145 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 3042 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 962 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 305 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 45 i\right )} e^{\left (-\frac{13}{2} i \, d x - \frac{13}{2} i \, c\right )}}{4680 \, a^{3} d e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*sec(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/4680*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*(-195*I*e^(10*I*d*x + 10*I*c) + 214
5*I*e^(8*I*d*x + 8*I*c) + 3042*I*e^(6*I*d*x + 6*I*c) + 962*I*e^(4*I*d*x + 4*I*c) + 305*I*e^(2*I*d*x + 2*I*c) +
45*I)*e^(-13/2*I*d*x - 13/2*I*c)/(a^3*d*e^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*sec(d*x+c))**(3/2)/(a+I*a*tan(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*sec(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((e*sec(d*x + c))^(3/2)*(I*a*tan(d*x + c) + a)^(5/2)), x)