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

Optimal. Leaf size=130 \[ \frac{2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac{6 a \sin (c+d x) \cos ^3(c+d x)}{5 d (e \cos (c+d x))^{7/2}}-\frac{6 a \cos ^{\frac{7}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac{2 a \sin (c+d x) \cos (c+d x)}{5 d (e \cos (c+d x))^{7/2}} \]

[Out]

(((2*I)/7)*a)/(d*(e*Cos[c + d*x])^(7/2)) - (6*a*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2])/(5*d*(e*Cos[c +
d*x])^(7/2)) + (2*a*Cos[c + d*x]*Sin[c + d*x])/(5*d*(e*Cos[c + d*x])^(7/2)) + (6*a*Cos[c + d*x]^3*Sin[c + d*x]
)/(5*d*(e*Cos[c + d*x])^(7/2))

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Rubi [A]  time = 0.127311, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3515, 3486, 3768, 3771, 2639} \[ \frac{2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac{6 a \sin (c+d x) \cos ^3(c+d x)}{5 d (e \cos (c+d x))^{7/2}}-\frac{6 a \cos ^{\frac{7}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac{2 a \sin (c+d x) \cos (c+d x)}{5 d (e \cos (c+d x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((2*I)/7)*a)/(d*(e*Cos[c + d*x])^(7/2)) - (6*a*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2])/(5*d*(e*Cos[c +
d*x])^(7/2)) + (2*a*Cos[c + d*x]*Sin[c + d*x])/(5*d*(e*Cos[c + d*x])^(7/2)) + (6*a*Cos[c + d*x]^3*Sin[c + d*x]
)/(5*d*(e*Cos[c + d*x])^(7/2))

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 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

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

Mathematica [C]  time = 6.47058, size = 596, normalized size = 4.58 \[ \frac{\cos ^5(c+d x) (a+i a \tan (c+d x)) \left (\left (\frac{2 \sin (c)}{7}+\frac{2}{7} i \cos (c)\right ) \sec ^4(c+d x)+\sec (c) \left (\frac{2 \cos (c)}{5}-\frac{2}{5} i \sin (c)\right ) \sin (d x) \sec ^3(c+d x)+\sec (c) \left (\frac{6 \cos (c)}{5}-\frac{6}{5} i \sin (c)\right ) \sin (d x) \sec (c+d x)+\tan (c) \left (\frac{2 \cos (c)}{5}-\frac{2}{5} i \sin (c)\right ) \sec ^2(c+d x)+\csc (c) \sec (c) \left (\frac{6 \cos (c)}{5}-\frac{6}{5} i \sin (c)\right )\right )}{d (\cos (d x)+i \sin (d x)) (e \cos (c+d x))^{7/2}}-\frac{i \left (\frac{3 \cot (c)}{5}-\frac{3 i}{5}\right ) \cos ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x)) \left (-\frac{2 i \sqrt{2} e^{i d x} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )}{3 \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}-\frac{2 \left (\cos \left (\frac{c}{2}\right )+i \sin \left (\frac{c}{2}\right )\right )^2 \left (2 i \cos (c+d x)+(-\sin (c+d x)-i \cos (c+d x)) \sqrt{\sin (c+d x)-i \cos (c+d x)+1} \sqrt{\sin (c+d x)+i \sin (2 (c+d x))-i \cos (c+d x)+\cos (2 (c+d x))} F\left (\left .\sin ^{-1}\left (\sqrt{\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+\sqrt{\sin (c+d x)-i \cos (c+d x)+1} (\sin (c+d x)+i \cos (c+d x)) \sqrt{\sin (c+d x)+i \sin (2 (c+d x))-i \cos (c+d x)+\cos (2 (c+d x))} E\left (\left .\sin ^{-1}\left (\sqrt{\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )\right )}{\sqrt{\cos (c+d x)}}\right )}{2 d (\cos (d x)+i \sin (d x)) (e \cos (c+d x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/2)*Cos[c + d*x]^(9/2)*((-3*I)/5 + (3*Cot[c])/5)*((((-2*I)/3)*Sqrt[2]*E^(I*d*x)*Sqrt[1 + E^((2*I)*(c + d*x
))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))] -
(2*(Cos[c/2] + I*Sin[c/2])^2*((2*I)*Cos[c + d*x] + EllipticF[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -
1]*((-I)*Cos[c + d*x] - Sin[c + d*x])*Sqrt[1 - I*Cos[c + d*x] + Sin[c + d*x]]*Sqrt[(-I)*Cos[c + d*x] + Cos[2*(
c + d*x)] + Sin[c + d*x] + I*Sin[2*(c + d*x)]] + EllipticE[ArcSin[Sqrt[(-I)*Cos[c + d*x] + Sin[c + d*x]]], -1]
*Sqrt[1 - I*Cos[c + d*x] + Sin[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x])*Sqrt[(-I)*Cos[c + d*x] + Cos[2*(c + d
*x)] + Sin[c + d*x] + I*Sin[2*(c + d*x)]]))/Sqrt[Cos[c + d*x]])*(a + I*a*Tan[c + d*x]))/(d*(e*Cos[c + d*x])^(7
/2)*(Cos[d*x] + I*Sin[d*x])) + (Cos[c + d*x]^5*(Csc[c]*Sec[c]*((6*Cos[c])/5 - ((6*I)/5)*Sin[c]) + Sec[c + d*x]
^4*(((2*I)/7)*Cos[c] + (2*Sin[c])/7) + Sec[c]*Sec[c + d*x]^3*((2*Cos[c])/5 - ((2*I)/5)*Sin[c])*Sin[d*x] + Sec[
c]*Sec[c + d*x]*((6*Cos[c])/5 - ((6*I)/5)*Sin[c])*Sin[d*x] + Sec[c + d*x]^2*((2*Cos[c])/5 - ((2*I)/5)*Sin[c])*
Tan[c])*(a + I*a*Tan[c + d*x]))/(d*(e*Cos[c + d*x])^(7/2)*(Cos[d*x] + I*Sin[d*x]))

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Maple [B]  time = 8.012, size = 396, normalized size = 3.1 \begin{align*} -{\frac{2\,a}{35\,{e}^{3}d} \left ( 168\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-336\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-252\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+504\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +126\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-280\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -21\,{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +5\,i\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) \left ( 8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-12\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+6\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) ^{-1} \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((a+I*a*tan(d*x+c))/(e*cos(d*x+c))^(7/2),x)

[Out]

-2/35/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)/(-2*sin(1/2
*d*x+1/2*c)^2*e+e)^(1/2)/e^3*(168*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(
1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^6-336*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-252*(2*sin(1/2*d*x+1/
2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^4+504*si
n(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+126*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipt
icE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2-280*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-21*EllipticE(
cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)+56*sin(1/2*d*x+1/2*c
)^2*cos(1/2*d*x+1/2*c)+5*I*sin(1/2*d*x+1/2*c))*a/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (-84 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 308 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 92 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 28 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\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 )} + 35 \,{\left (d e^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )}{\rm integral}\left (\frac{6 i \, \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} a e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{5 \,{\left (d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )}}, x\right )}{35 \,{\left (d e^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(sqrt(1/2)*(-84*I*a*e^(8*I*d*x + 8*I*c) - 308*I*a*e^(6*I*d*x + 6*I*c) - 92*I*a*e^(4*I*d*x + 4*I*c) - 28*I
*a*e^(2*I*d*x + 2*I*c))*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*e^(-1/2*I*d*x - 1/2*I*c) + 35*(d*e^4*e^(8*I*d*x + 8*I*
c) + 4*d*e^4*e^(6*I*d*x + 6*I*c) + 6*d*e^4*e^(4*I*d*x + 4*I*c) + 4*d*e^4*e^(2*I*d*x + 2*I*c) + d*e^4)*integral
(6/5*I*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*a*e^(1/2*I*d*x + 1/2*I*c)/(d*e^4*e^(2*I*d*x + 2*I*c) + d*e^4)
, x))/(d*e^4*e^(8*I*d*x + 8*I*c) + 4*d*e^4*e^(6*I*d*x + 6*I*c) + 6*d*e^4*e^(4*I*d*x + 4*I*c) + 4*d*e^4*e^(2*I*
d*x + 2*I*c) + d*e^4)

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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))/(e*cos(d*x+c))**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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