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

Optimal. Leaf size=190 \[ \frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{7/2}}{15 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{7/2}}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \sin (c+d x) \cos (c+d x) (e \cos (c+d x))^{7/2}}{15 a^2 d}+\frac{6 \tan (c+d x) (e \cos (c+d x))^{7/2}}{35 a^2 d}+\frac{2 \tan (c+d x) \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{7 a^2 d} \]

[Out]

(2*(e*Cos[c + d*x])^(7/2)*EllipticF[(c + d*x)/2, 2])/(7*a^2*d*Cos[c + d*x]^(7/2)) + (2*Cos[c + d*x]*(e*Cos[c +
d*x])^(7/2)*Sin[c + d*x])/(15*a^2*d) + (6*(e*Cos[c + d*x])^(7/2)*Tan[c + d*x])/(35*a^2*d) + (2*(e*Cos[c + d*x
])^(7/2)*Sec[c + d*x]^2*Tan[c + d*x])/(7*a^2*d) + (((4*I)/15)*Cos[c + d*x]^2*(e*Cos[c + d*x])^(7/2))/(d*(a^2 +
I*a^2*Tan[c + d*x]))

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Rubi [A]  time = 0.220811, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3515, 3500, 3769, 3771, 2641} \[ \frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{7/2}}{15 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (e \cos (c+d x))^{7/2}}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \sin (c+d x) \cos (c+d x) (e \cos (c+d x))^{7/2}}{15 a^2 d}+\frac{6 \tan (c+d x) (e \cos (c+d x))^{7/2}}{35 a^2 d}+\frac{2 \tan (c+d x) \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{7 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*Cos[c + d*x])^(7/2)*EllipticF[(c + d*x)/2, 2])/(7*a^2*d*Cos[c + d*x]^(7/2)) + (2*Cos[c + d*x]*(e*Cos[c +
d*x])^(7/2)*Sin[c + d*x])/(15*a^2*d) + (6*(e*Cos[c + d*x])^(7/2)*Tan[c + d*x])/(35*a^2*d) + (2*(e*Cos[c + d*x
])^(7/2)*Sec[c + d*x]^2*Tan[c + d*x])/(7*a^2*d) + (((4*I)/15)*Cos[c + d*x]^2*(e*Cos[c + d*x])^(7/2))/(d*(a^2 +
I*a^2*Tan[c + d*x]))

Rule 3515

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(d*Co
s[e + f*x])^m*(d*Sec[e + f*x])^m, Int[(a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e,
f, m, n}, x] &&  !IntegerQ[m]

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 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{(e \cos (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx &=\left ((e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^2} \, dx\\ &=\frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{7/2}}{15 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (11 e^2 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{1}{(e \sec (c+d x))^{11/2}} \, dx}{15 a^2}\\ &=\frac{2 \cos (c+d x) (e \cos (c+d x))^{7/2} \sin (c+d x)}{15 a^2 d}+\frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{7/2}}{15 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (3 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{1}{(e \sec (c+d x))^{7/2}} \, dx}{5 a^2}\\ &=\frac{2 \cos (c+d x) (e \cos (c+d x))^{7/2} \sin (c+d x)}{15 a^2 d}+\frac{6 (e \cos (c+d x))^{7/2} \tan (c+d x)}{35 a^2 d}+\frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{7/2}}{15 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (3 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{1}{(e \sec (c+d x))^{3/2}} \, dx}{7 a^2 e^2}\\ &=\frac{2 \cos (c+d x) (e \cos (c+d x))^{7/2} \sin (c+d x)}{15 a^2 d}+\frac{6 (e \cos (c+d x))^{7/2} \tan (c+d x)}{35 a^2 d}+\frac{2 (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{7 a^2 d}+\frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{7/2}}{15 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left ((e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \sqrt{e \sec (c+d x)} \, dx}{7 a^2 e^4}\\ &=\frac{2 \cos (c+d x) (e \cos (c+d x))^{7/2} \sin (c+d x)}{15 a^2 d}+\frac{6 (e \cos (c+d x))^{7/2} \tan (c+d x)}{35 a^2 d}+\frac{2 (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{7 a^2 d}+\frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{7/2}}{15 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{(e \cos (c+d x))^{7/2} \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{7 a^2 \cos ^{\frac{7}{2}}(c+d x)}\\ &=\frac{2 (e \cos (c+d x))^{7/2} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \cos (c+d x) (e \cos (c+d x))^{7/2} \sin (c+d x)}{15 a^2 d}+\frac{6 (e \cos (c+d x))^{7/2} \tan (c+d x)}{35 a^2 d}+\frac{2 (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{7 a^2 d}+\frac{4 i \cos ^2(c+d x) (e \cos (c+d x))^{7/2}}{15 d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.23597, size = 156, normalized size = 0.82 \[ \frac{e^3 \sqrt{e \cos (c+d x)} \left (\sqrt{\cos (c+d x)} (134 \sin (c+d x)-117 \sin (3 (c+d x))-11 \sin (5 (c+d x))-296 i \cos (c+d x)+68 i \cos (3 (c+d x))+4 i \cos (5 (c+d x)))-240 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))\right )}{840 a^2 d \cos ^{\frac{5}{2}}(c+d x) (\tan (c+d x)-i)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^3*Sqrt[e*Cos[c + d*x]]*(-240*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + Sqrt[Cos[c
+ d*x]]*((-296*I)*Cos[c + d*x] + (68*I)*Cos[3*(c + d*x)] + (4*I)*Cos[5*(c + d*x)] + 134*Sin[c + d*x] - 117*Si
n[3*(c + d*x)] - 11*Sin[5*(c + d*x)])))/(840*a^2*d*Cos[c + d*x]^(5/2)*(-I + Tan[c + d*x])^2)

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Maple [B]  time = 3.086, size = 387, normalized size = 2. \begin{align*}{\frac{2\,{e}^{4}}{105\,{a}^{2}d} \left ( 14\,i\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -3584\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{16}+1568\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}+12544\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{14}\cos \left ( 1/2\,dx+c/2 \right ) -25088\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11}-19264\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{12}\cos \left ( 1/2\,dx+c/2 \right ) -224\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}+16800\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}\cos \left ( 1/2\,dx+c/2 \right ) +25088\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{13}-9104\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+3584\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{17}+3128\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -6272\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}-700\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -14336\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{15}-15\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +90\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +15680\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9} \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105/a^2/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^4*(14*I*sin(1/2*d*x+1/2*c)-3584*cos(1/2*d*x
+1/2*c)*sin(1/2*d*x+1/2*c)^16+1568*I*sin(1/2*d*x+1/2*c)^5+12544*sin(1/2*d*x+1/2*c)^14*cos(1/2*d*x+1/2*c)-25088
*I*sin(1/2*d*x+1/2*c)^11-19264*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)-224*I*sin(1/2*d*x+1/2*c)^3+16800*sin(1
/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+25088*I*sin(1/2*d*x+1/2*c)^13-9104*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8
+3584*I*sin(1/2*d*x+1/2*c)^17+3128*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-6272*I*sin(1/2*d*x+1/2*c)^7-700*sin
(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-14336*I*sin(1/2*d*x+1/2*c)^15-15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*
d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+90*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+15680*I
*sin(1/2*d*x+1/2*c)^9)/d

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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*cos(d*x+c))^(7/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (1680 \, a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )}{\rm integral}\left (-\frac{2 i \, \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{3} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{7 \,{\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}}, x\right ) + \sqrt{\frac{1}{2}}{\left (-15 i \, e^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 185 i \, e^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 430 i \, e^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 162 i \, e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 49 i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, e^{3}\right )} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{1680 \, a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(1680*a^2*d*e^(7*I*d*x + 7*I*c)*integral(-2/7*I*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*e^3*e^(-1/2*I
*d*x - 1/2*I*c)/(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d), x) + sqrt(1/2)*(-15*I*e^3*e^(10*I*d*x + 10*I*c) - 185*I*e
^3*e^(8*I*d*x + 8*I*c) + 430*I*e^3*e^(6*I*d*x + 6*I*c) + 162*I*e^3*e^(4*I*d*x + 4*I*c) + 49*I*e^3*e^(2*I*d*x +
2*I*c) + 7*I*e^3)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*e^(-1/2*I*d*x - 1/2*I*c))*e^(-7*I*d*x - 7*I*c)/(a^2*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*cos(d*x+c))**(7/2)/(a+I*a*tan(d*x+c))**2,x)

[Out]

Exception raised: AttributeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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