3.232 \(\int \frac{1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx\)

Optimal. Leaf size=145 \[ \frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{30 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{77 a d e^4}+\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}} \]

[Out]

(30*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(77*a*d*e^4) + (18*Sin[c + d*x])/(77*a*
d*e*(e*Sec[c + d*x])^(5/2)) + (30*Sin[c + d*x])/(77*a*d*e^3*Sqrt[e*Sec[c + d*x]]) + ((2*I)/11)/(d*(e*Sec[c + d
*x])^(7/2)*(a + I*a*Tan[c + d*x]))

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Rubi [A]  time = 0.109497, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3502, 3769, 3771, 2641} \[ \frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{30 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{77 a d e^4}+\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])),x]

[Out]

(30*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(77*a*d*e^4) + (18*Sin[c + d*x])/(77*a*
d*e*(e*Sec[c + d*x])^(5/2)) + (30*Sin[c + d*x])/(77*a*d*e^3*Sqrt[e*Sec[c + d*x]]) + ((2*I)/11)/(d*(e*Sec[c + d
*x])^(7/2)*(a + I*a*Tan[c + d*x]))

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 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 2641

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

Rubi steps

\begin{align*} \int \frac{1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx &=\frac{2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac{9 \int \frac{1}{(e \sec (c+d x))^{7/2}} \, dx}{11 a}\\ &=\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac{45 \int \frac{1}{(e \sec (c+d x))^{3/2}} \, dx}{77 a e^2}\\ &=\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac{15 \int \sqrt{e \sec (c+d x)} \, dx}{77 a e^4}\\ &=\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac{\left (15 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{77 a e^4}\\ &=\frac{30 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{77 a d e^4}+\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.720903, size = 142, normalized size = 0.98 \[ -\frac{(e \sec (c+d x))^{3/2} \left (78 i \sin (c+d x)+87 i \sin (3 (c+d x))+9 i \sin (5 (c+d x))-148 \cos (c+d x)+34 \cos (3 (c+d x))+2 \cos (5 (c+d x))+240 i \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (c+d x)+i \sin (c+d x))\right )}{616 a d e^5 (\tan (c+d x)-i)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])),x]

[Out]

-((e*Sec[c + d*x])^(3/2)*(-148*Cos[c + d*x] + 34*Cos[3*(c + d*x)] + 2*Cos[5*(c + d*x)] + (240*I)*Sqrt[Cos[c +
d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[c + d*x] + I*Sin[c + d*x]) + (78*I)*Sin[c + d*x] + (87*I)*Sin[3*(c + d*x)
] + (9*I)*Sin[5*(c + d*x)]))/(616*a*d*e^5*(-I + Tan[c + d*x]))

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Maple [A]  time = 0.397, size = 236, normalized size = 1.6 \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 ) ^{3}}{77\,ad{e}^{7} \left ( \sin \left ( dx+c \right ) \right ) ^{4}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{7}{2}}} \left ( 7\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+7\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +15\,i\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}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +15\,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 ) +9\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +15\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c)),x)

[Out]

2/77/a/d*(e/cos(d*x+c))^(7/2)*(cos(d*x+c)-1)^2*(cos(d*x+c)+1)^2*cos(d*x+c)^3*(7*I*cos(d*x+c)^6+7*cos(d*x+c)^5*
sin(d*x+c)+15*I*cos(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)+15*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)/si
n(d*x+c),I)+9*cos(d*x+c)^3*sin(d*x+c)+15*cos(d*x+c)*sin(d*x+c))/e^7/sin(d*x+c)^4

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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(1/(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{{\left (1232 \, a d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )}{\rm integral}\left (-\frac{15 i \, \sqrt{2} \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 )}}{77 \, a d e^{4}}, x\right ) + \sqrt{2} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-11 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 121 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 70 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 226 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 53 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{1232 \, a d e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/1232*(1232*a*d*e^4*e^(6*I*d*x + 6*I*c)*integral(-15/77*I*sqrt(2)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*e^(-1/2*I
*d*x - 1/2*I*c)/(a*d*e^4), x) + sqrt(2)*sqrt(e/(e^(2*I*d*x + 2*I*c) + 1))*(-11*I*e^(10*I*d*x + 10*I*c) - 121*I
*e^(8*I*d*x + 8*I*c) + 70*I*e^(6*I*d*x + 6*I*c) + 226*I*e^(4*I*d*x + 4*I*c) + 53*I*e^(2*I*d*x + 2*I*c) + 7*I)*
e^(1/2*I*d*x + 1/2*I*c))*e^(-6*I*d*x - 6*I*c)/(a*d*e^4)

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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(1/(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{1}{\left (e \sec \left (d x + c\right )\right )^{\frac{7}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*sec(d*x+c))^(7/2)/(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate(1/((e*sec(d*x + c))^(7/2)*(I*a*tan(d*x + c) + a)), x)