3.286 \(\int \sec ^7(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\)

Optimal. Leaf size=147 \[ \frac{24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac{64 i a^3 \sec ^7(c+d x)}{429 d (a+i a \tan (c+d x))^{5/2}}+\frac{256 i a^4 \sec ^7(c+d x)}{3003 d (a+i a \tan (c+d x))^{7/2}}+\frac{2 i a \sec ^7(c+d x)}{13 d \sqrt{a+i a \tan (c+d x)}} \]

[Out]

(((256*I)/3003)*a^4*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (((64*I)/429)*a^3*Sec[c + d*x]^7)/(d*(a
+ I*a*Tan[c + d*x])^(5/2)) + (((24*I)/143)*a^2*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (((2*I)/13)
*a*Sec[c + d*x]^7)/(d*Sqrt[a + I*a*Tan[c + d*x]])

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Rubi [A]  time = 0.249842, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3494, 3493} \[ \frac{24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac{64 i a^3 \sec ^7(c+d x)}{429 d (a+i a \tan (c+d x))^{5/2}}+\frac{256 i a^4 \sec ^7(c+d x)}{3003 d (a+i a \tan (c+d x))^{7/2}}+\frac{2 i a \sec ^7(c+d x)}{13 d \sqrt{a+i a \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^7*Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

(((256*I)/3003)*a^4*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (((64*I)/429)*a^3*Sec[c + d*x]^7)/(d*(a
+ I*a*Tan[c + d*x])^(5/2)) + (((24*I)/143)*a^2*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (((2*I)/13)
*a*Sec[c + d*x]^7)/(d*Sqrt[a + I*a*Tan[c + d*x]])

Rule 3494

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] + Dist[(a*(m + 2*n - 2))/(m + n - 1), Int[(
d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0]
&& IGtQ[Simplify[m/2 + n - 1], 0] &&  !IntegerQ[n]

Rule 3493

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] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2
, 0] && EqQ[Simplify[m/2 + n - 1], 0]

Rubi steps

\begin{align*} \int \sec ^7(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx &=\frac{2 i a \sec ^7(c+d x)}{13 d \sqrt{a+i a \tan (c+d x)}}+\frac{1}{13} (12 a) \int \frac{\sec ^7(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i a \sec ^7(c+d x)}{13 d \sqrt{a+i a \tan (c+d x)}}+\frac{1}{143} \left (96 a^2\right ) \int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\\ &=\frac{64 i a^3 \sec ^7(c+d x)}{429 d (a+i a \tan (c+d x))^{5/2}}+\frac{24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i a \sec ^7(c+d x)}{13 d \sqrt{a+i a \tan (c+d x)}}+\frac{1}{429} \left (128 a^3\right ) \int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx\\ &=\frac{256 i a^4 \sec ^7(c+d x)}{3003 d (a+i a \tan (c+d x))^{7/2}}+\frac{64 i a^3 \sec ^7(c+d x)}{429 d (a+i a \tan (c+d x))^{5/2}}+\frac{24 i a^2 \sec ^7(c+d x)}{143 d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i a \sec ^7(c+d x)}{13 d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.586104, size = 95, normalized size = 0.65 \[ \frac{2 \sec ^6(c+d x) \sqrt{a+i a \tan (c+d x)} (7 i (26 \sin (c+d x)+59 \sin (3 (c+d x)))+390 \cos (c+d x)+445 \cos (3 (c+d x))) (\sin (4 (c+d x))+i \cos (4 (c+d x)))}{3003 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^7*Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

(2*Sec[c + d*x]^6*(390*Cos[c + d*x] + 445*Cos[3*(c + d*x)] + (7*I)*(26*Sin[c + d*x] + 59*Sin[3*(c + d*x)]))*(I
*Cos[4*(c + d*x)] + Sin[4*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(3003*d)

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Maple [A]  time = 0.841, size = 141, normalized size = 1. \begin{align*}{\frac{2048\,i \left ( \cos \left ( dx+c \right ) \right ) ^{7}+2048\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) -256\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}+768\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}-80\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+560\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -42\,i\cos \left ( dx+c \right ) +462\,\sin \left ( dx+c \right ) }{3003\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003/d*(1024*I*cos(d*x+c)^7+1024*cos(d*x+c)^6*sin(d*x+c)-128*I*cos(d*x+c)^5+384*sin(d*x+c)*cos(d*x+c)^4-40*I
*cos(d*x+c)^3+280*cos(d*x+c)^2*sin(d*x+c)-21*I*cos(d*x+c)+231*sin(d*x+c))*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x
+c))^(1/2)/cos(d*x+c)^6

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Maxima [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(sec(d*x+c)^7*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 2.30954, size = 467, normalized size = 3.18 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (54912 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 36608 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 13312 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2048 i\right )} e^{\left (i \, d x + i \, c\right )}}{3003 \,{\left (d e^{\left (13 i \, d x + 13 i \, c\right )} + 6 \, d e^{\left (11 i \, d x + 11 i \, c\right )} + 15 \, d e^{\left (9 i \, d x + 9 i \, c\right )} + 20 \, d e^{\left (7 i \, d x + 7 i \, c\right )} + 15 \, d e^{\left (5 i \, d x + 5 i \, c\right )} + 6 \, d e^{\left (3 i \, d x + 3 i \, c\right )} + d e^{\left (i \, d x + i \, c\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3003*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(54912*I*e^(6*I*d*x + 6*I*c) + 36608*I*e^(4*I*d*x + 4*I*c) +
13312*I*e^(2*I*d*x + 2*I*c) + 2048*I)*e^(I*d*x + I*c)/(d*e^(13*I*d*x + 13*I*c) + 6*d*e^(11*I*d*x + 11*I*c) + 1
5*d*e^(9*I*d*x + 9*I*c) + 20*d*e^(7*I*d*x + 7*I*c) + 15*d*e^(5*I*d*x + 5*I*c) + 6*d*e^(3*I*d*x + 3*I*c) + d*e^
(I*d*x + I*c))

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(I*a*tan(d*x + c) + a)*sec(d*x + c)^7, x)