3.219 \(\int \frac{(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx\)

Optimal. Leaf size=125 \[ \frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}-\frac{2 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}} \]

[Out]

(-2*a^4*EllipticE[(c + d*x)/2, 2])/(15*d*e^4*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (((4*I)/9)*a*(a + I*a*
Tan[c + d*x])^3)/(d*(e*Sec[c + d*x])^(9/2)) + (((4*I)/15)*(a^4 + I*a^4*Tan[c + d*x]))/(d*e^2*(e*Sec[c + d*x])^
(5/2))

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Rubi [A]  time = 0.131385, antiderivative size = 125, 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 = {3496, 3771, 2639} \[ \frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}-\frac{2 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(9/2),x]

[Out]

(-2*a^4*EllipticE[(c + d*x)/2, 2])/(15*d*e^4*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (((4*I)/9)*a*(a + I*a*
Tan[c + d*x])^3)/(d*(e*Sec[c + d*x])^(9/2)) + (((4*I)/15)*(a^4 + I*a^4*Tan[c + d*x]))/(d*e^2*(e*Sec[c + d*x])^
(5/2))

Rule 3496

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(2*b*(
d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*m), x] - Dist[(b^2*(m + 2*n - 2))/(d^2*m), Int[(d*Sec[e + f
*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n,
1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILt
Q[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && 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{(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{9/2}} \, dx &=-\frac{4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}-\frac{a^2 \int \frac{(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{5/2}} \, dx}{3 e^2}\\ &=-\frac{4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}+\frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}-\frac{a^4 \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{15 e^4}\\ &=-\frac{4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}+\frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}-\frac{a^4 \int \sqrt{\cos (c+d x)} \, dx}{15 e^4 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=-\frac{2 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}+\frac{4 i \left (a^4+i a^4 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}\\ \end{align*}

Mathematica [C]  time = 3.667, size = 108, normalized size = 0.86 \[ -\frac{i a^4 e^{i (c+d x)} \left (-2 \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+7 e^{2 i (c+d x)}+5 e^{4 i (c+d x)}+2\right ) \sqrt{e \sec (c+d x)}}{45 d e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(9/2),x]

[Out]

((-I/45)*a^4*E^(I*(c + d*x))*(2 + 7*E^((2*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x)) - 2*Sqrt[1 + E^((2*I)*(c + d*x
))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sqrt[e*Sec[c + d*x]])/(d*e^5)

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Maple [B]  time = 0.285, size = 370, normalized size = 3. \begin{align*}{\frac{2\,{a}^{4}}{45\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}} \left ( -40\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+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 ) -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 EllipticF} \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 ) -40\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}+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}}}-3\,i\sin \left ( dx+c \right ){\it EllipticF} \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}}}+36\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+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 ) \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(9/2),x)

[Out]

2/45*a^4/d*(-40*I*sin(d*x+c)*cos(d*x+c)^5+3*I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*Ellip
ticE(I*(cos(d*x+c)-1)/sin(d*x+c),I)*cos(d*x+c)*sin(d*x+c)-3*I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)
+1))^(1/2)*EllipticF(I*(cos(d*x+c)-1)/sin(d*x+c),I)*cos(d*x+c)*sin(d*x+c)-40*cos(d*x+c)^6+3*I*sin(d*x+c)*Ellip
ticE(I*(cos(d*x+c)-1)/sin(d*x+c),I)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-3*I*sin(d*x+c)*
EllipticF(I*(cos(d*x+c)-1)/sin(d*x+c),I)*(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+56*cos(d*x+c)^4-13*cos(d*x+c)^2-3*cos(d*x+c))/sin(d*x+c)/cos(d*x+c)^5/(e/cos(d*x+c))^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{\left (e \sec \left (d x + c\right )\right )^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(9/2),x, algorithm="maxima")

[Out]

integrate((I*a*tan(d*x + c) + a)^4/(e*sec(d*x + c))^(9/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(9/2),x, algorithm="fricas")

[Out]

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

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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((a+I*a*tan(d*x+c))**4/(e*sec(d*x+c))**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{\left (e \sec \left (d x + c\right )\right )^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))^4/(e*sec(d*x+c))^(9/2),x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^4/(e*sec(d*x + c))^(9/2), x)