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

Optimal. Leaf size=101 \[ -\frac{2 i e^2 (e \sec (c+d x))^{3/2}}{3 a d}+\frac{2 e^3 \sin (c+d x) \sqrt{e \sec (c+d x)}}{a d}-\frac{2 e^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0878165, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3501, 3768, 3771, 2639} \[ -\frac{2 i e^2 (e \sec (c+d x))^{3/2}}{3 a d}+\frac{2 e^3 \sin (c+d x) \sqrt{e \sec (c+d x)}}{a d}-\frac{2 e^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3501

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

Rule 3768

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

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)} \, dx &=-\frac{2 i e^2 (e \sec (c+d x))^{3/2}}{3 a d}+\frac{e^2 \int (e \sec (c+d x))^{3/2} \, dx}{a}\\ &=-\frac{2 i e^2 (e \sec (c+d x))^{3/2}}{3 a d}+\frac{2 e^3 \sqrt{e \sec (c+d x)} \sin (c+d x)}{a d}-\frac{e^4 \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{a}\\ &=-\frac{2 i e^2 (e \sec (c+d x))^{3/2}}{3 a d}+\frac{2 e^3 \sqrt{e \sec (c+d x)} \sin (c+d x)}{a d}-\frac{e^4 \int \sqrt{\cos (c+d x)} \, dx}{a \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 )}{a d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{2 i e^2 (e \sec (c+d x))^{3/2}}{3 a d}+\frac{2 e^3 \sqrt{e \sec (c+d x)} \sin (c+d x)}{a d}\\ \end{align*}

Mathematica [C]  time = 0.721794, size = 102, normalized size = 1.01 \[ \frac{2 i e^3 (\cos (c)+i \sin (c)) (\cos (d x)+i \sin (d x)) \sqrt{e \sec (c+d x)} \left (\sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+i \tan (c+d x)-4\right )}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.233, size = 361, normalized size = 3.6 \begin{align*} -{\frac{2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,ad \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ( 3\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}\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}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) -3\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\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 ) +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 ) +3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+i\sin \left ( dx+c \right ) -3\,\cos \left ( dx+c \right ) \right ) \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{7}{2}}}} \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)),x)

[Out]

-2/3/a/d*(cos(d*x+c)+1)^2*(cos(d*x+c)-1)^2*(3*I*cos(d*x+c)^2*sin(d*x+c)*(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)-3*I*cos(d*x+c)^2*sin(d*x+c)*(1/(cos(d*x+c)+1))^(
1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(I*(cos(d*x+c)-1)/sin(d*x+c),I)+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
)+3*cos(d*x+c)^2+I*sin(d*x+c)-3*cos(d*x+c))*(e/cos(d*x+c))^(7/2)*cos(d*x+c)^2/sin(d*x+c)^5

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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)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{2}{\left (-6 i \, e^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 10 i \, e^{3} e^{\left (i \, d x + i \, c\right )}\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 )} + 3 \,{\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\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 )}}{a d}, x\right )}{3 \,{\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \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)),x, algorithm="fricas")

[Out]

1/3*(sqrt(2)*(-6*I*e^3*e^(3*I*d*x + 3*I*c) - 10*I*e^3*e^(I*d*x + I*c))*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/
2*I*d*x + 1/2*I*c) + 3*(a*d*e^(2*I*d*x + 2*I*c) + a*d)*integral(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*d), x))/(a*d*e^(2*I*d*x + 2*I*c) + a*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)),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}}}{i \, a \tan \left (d x + c\right ) + a}\,{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)),x, algorithm="giac")

[Out]

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