3.179 \(\int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\)

Optimal. Leaf size=68 \[ \frac{i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}+\frac{i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8} \]

[Out]

((I/9)*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^8) + ((I/63)*Sec[c + d*x]^7)/(a*d*(a + I*a*Tan[c + d*x])^7)

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Rubi [A]  time = 0.0803466, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3502, 3488} \[ \frac{i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}+\frac{i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^8,x]

[Out]

((I/9)*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^8) + ((I/63)*Sec[c + d*x]^7)/(a*d*(a + I*a*Tan[c + d*x])^7)

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 3488

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

Rubi steps

\begin{align*} \int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac{\int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{9 a}\\ &=\frac{i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac{i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}\\ \end{align*}

Mathematica [A]  time = 0.101362, size = 40, normalized size = 0.59 \[ -\frac{(\tan (c+d x)-8 i) \sec ^7(c+d x)}{63 a^8 d (\tan (c+d x)-i)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^8,x]

[Out]

-(Sec[c + d*x]^7*(-8*I + Tan[c + d*x]))/(63*a^8*d*(-I + Tan[c + d*x])^8)

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Maple [B]  time = 0.108, size = 156, normalized size = 2.3 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ( -{\frac{86}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+136\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-5}+{\frac{7\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}+{\frac{128}{9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{9}}}+{\frac{{\frac{496\,i}{3}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}-{\frac{928}{7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-1}-{\frac{64\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}-{\frac{76\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}} \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))^8,x)

[Out]

2/d/a^8*(-86/3/(tan(1/2*d*x+1/2*c)-I)^3+136/(tan(1/2*d*x+1/2*c)-I)^5+7*I/(tan(1/2*d*x+1/2*c)-I)^2+128/9/(tan(1
/2*d*x+1/2*c)-I)^9+496/3*I/(tan(1/2*d*x+1/2*c)-I)^6-928/7/(tan(1/2*d*x+1/2*c)-I)^7+1/(tan(1/2*d*x+1/2*c)-I)-64
*I/(tan(1/2*d*x+1/2*c)-I)^8-76*I/(tan(1/2*d*x+1/2*c)-I)^4)

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Maxima [A]  time = 1.21565, size = 72, normalized size = 1.06 \begin{align*} \frac{7 i \, \cos \left (9 \, d x + 9 \, c\right ) + 9 i \, \cos \left (7 \, d x + 7 \, c\right ) + 7 \, \sin \left (9 \, d x + 9 \, c\right ) + 9 \, \sin \left (7 \, d x + 7 \, c\right )}{126 \, a^{8} d} \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))^8,x, algorithm="maxima")

[Out]

1/126*(7*I*cos(9*d*x + 9*c) + 9*I*cos(7*d*x + 7*c) + 7*sin(9*d*x + 9*c) + 9*sin(7*d*x + 7*c))/(a^8*d)

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Fricas [A]  time = 2.44531, size = 92, normalized size = 1.35 \begin{align*} \frac{{\left (9 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{126 \, a^{8} d} \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))^8,x, algorithm="fricas")

[Out]

1/126*(9*I*e^(2*I*d*x + 2*I*c) + 7*I)*e^(-9*I*d*x - 9*I*c)/(a^8*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(sec(d*x+c)**7/(a+I*a*tan(d*x+c))**8,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.22857, size = 169, normalized size = 2.49 \begin{align*} \frac{2 \,{\left (63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 63 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 483 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 315 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 693 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 189 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 225 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 9 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8\right )}}{63 \, a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{9}} \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))^8,x, algorithm="giac")

[Out]

2/63*(63*tan(1/2*d*x + 1/2*c)^8 - 63*I*tan(1/2*d*x + 1/2*c)^7 - 483*tan(1/2*d*x + 1/2*c)^6 + 315*I*tan(1/2*d*x
+ 1/2*c)^5 + 693*tan(1/2*d*x + 1/2*c)^4 - 189*I*tan(1/2*d*x + 1/2*c)^3 - 225*tan(1/2*d*x + 1/2*c)^2 + 9*I*tan
(1/2*d*x + 1/2*c) + 8)/(a^8*d*(tan(1/2*d*x + 1/2*c) - I)^9)