3.259 \(\int \frac{(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=132 \[ -\frac{4 i e^4}{15 d \left (a^4+i a^4 \tan (c+d x)\right ) \sqrt{e \sec (c+d x)}}-\frac{2 e^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^4 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2 (e \sec (c+d x))^{3/2}}{9 a d (a+i a \tan (c+d x))^3} \]

[Out]

(-2*e^4*EllipticE[(c + d*x)/2, 2])/(15*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((4*I)/9)*e^2*(e*Sec[
c + d*x])^(3/2))/(a*d*(a + I*a*Tan[c + d*x])^3) - (((4*I)/15)*e^4)/(d*Sqrt[e*Sec[c + d*x]]*(a^4 + I*a^4*Tan[c
+ d*x]))

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Rubi [A]  time = 0.137749, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {3500, 3771, 2639} \[ -\frac{4 i e^4}{15 d \left (a^4+i a^4 \tan (c+d x)\right ) \sqrt{e \sec (c+d x)}}-\frac{2 e^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^4 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2 (e \sec (c+d x))^{3/2}}{9 a d (a+i a \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^4,x]

[Out]

(-2*e^4*EllipticE[(c + d*x)/2, 2])/(15*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((4*I)/9)*e^2*(e*Sec[
c + d*x])^(3/2))/(a*d*(a + I*a*Tan[c + d*x])^3) - (((4*I)/15)*e^4)/(d*Sqrt[e*Sec[c + d*x]]*(a^4 + I*a^4*Tan[c
+ d*x]))

Rule 3500

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^4} \, dx &=\frac{4 i e^2 (e \sec (c+d x))^{3/2}}{9 a d (a+i a \tan (c+d x))^3}-\frac{e^2 \int \frac{(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx}{3 a^2}\\ &=\frac{4 i e^2 (e \sec (c+d x))^{3/2}}{9 a d (a+i a \tan (c+d x))^3}-\frac{4 i e^4}{15 d \sqrt{e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{e^4 \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{15 a^4}\\ &=\frac{4 i e^2 (e \sec (c+d x))^{3/2}}{9 a d (a+i a \tan (c+d x))^3}-\frac{4 i e^4}{15 d \sqrt{e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{e^4 \int \sqrt{\cos (c+d x)} \, dx}{15 a^4 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=-\frac{2 e^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^4 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2 (e \sec (c+d x))^{3/2}}{9 a d (a+i a \tan (c+d x))^3}-\frac{4 i e^4}{15 d \sqrt{e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 0.739934, size = 149, normalized size = 1.13 \[ \frac{e^3 e^{-i d x} \sec ^4(c+d x) \sqrt{e \sec (c+d x)} (\sin (c+2 d x)-i \cos (c+2 d x)) \left (6 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )+3 i \sin (2 (c+d x))-7 \cos (2 (c+d x))-7\right )}{45 a^4 d (\tan (c+d x)-i)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^4,x]

[Out]

(e^3*Sec[c + d*x]^4*Sqrt[e*Sec[c + d*x]]*(-7 - 7*Cos[2*(c + d*x)] + 6*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c
+ d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + (3*I)*Sin[2*(c + d*x)])*((-I)*Cos[c + 2*d*
x] + Sin[c + 2*d*x]))/(45*a^4*d*E^(I*d*x)*(-I + Tan[c + d*x])^4)

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Maple [B]  time = 0.269, size = 370, normalized size = 2.8 \begin{align*} -{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{45\,{a}^{4}d\sin \left ( dx+c \right ) } \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{7}{2}}} \left ( -40\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+40\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}+3\,i{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-3\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +36\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3\,i{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-3\,i\sin \left ( dx+c \right ){\it EllipticE} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-56\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+13\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,\cos \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^4,x)

[Out]

-2/45/a^4/d*(e/cos(d*x+c))^(7/2)*cos(d*x+c)^3*(-40*I*sin(d*x+c)*cos(d*x+c)^5+40*cos(d*x+c)^6+3*I*EllipticF(I*(
cos(d*x+c)-1)/sin(d*x+c),I)*sin(d*x+c)*cos(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-3
*I*EllipticE(I*(cos(d*x+c)-1)/sin(d*x+c),I)*cos(d*x+c)*sin(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)+36*I*sin(d*x+c)*cos(d*x+c)^3+3*I*EllipticF(I*(cos(d*x+c)-1)/sin(d*x+c),I)*sin(d*x+c)*(1/(cos(d*
x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-3*I*EllipticE(I*(cos(d*x+c)-1)/sin(d*x+c),I)*sin(d*x+c)*(1/(c
os(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-56*cos(d*x+c)^4+13*cos(d*x+c)^2+3*cos(d*x+c))/sin(d*x+c)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (45 \, a^{4} d e^{\left (5 i \, d x + 5 i \, c\right )}{\rm integral}\left (\frac{i \, \sqrt{2} e^{3} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{15 \, a^{4} d}, x\right ) + \sqrt{2}{\left (-6 i \, e^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 4 i \, e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i \, e^{3}\right )} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{45 \, a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/45*(45*a^4*d*e^(5*I*d*x + 5*I*c)*integral(1/15*I*sqrt(2)*e^3*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x
+ 1/2*I*c)/(a^4*d), x) + sqrt(2)*(-6*I*e^3*e^(6*I*d*x + 6*I*c) - 4*I*e^3*e^(4*I*d*x + 4*I*c) + 7*I*e^3*e^(2*I*
d*x + 2*I*c) + 5*I*e^3)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c))*e^(-5*I*d*x - 5*I*c)/(a^4*d
)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))**(7/2)/(a+I*a*tan(d*x+c))**4,x)

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \sec \left (d x + c\right )\right )^{\frac{7}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((e*sec(d*x + c))^(7/2)/(I*a*tan(d*x + c) + a)^4, x)