3.143 \(\int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=32 \[ \frac{i \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^3} \]

[Out]

((I/3)*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^3)

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Rubi [A]  time = 0.0372742, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {3488} \[ \frac{i \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^3,x]

[Out]

((I/3)*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^3)

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 ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=\frac{i \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^3}\\ \end{align*}

Mathematica [A]  time = 0.046857, size = 32, normalized size = 1. \[ \frac{i \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^3,x]

[Out]

((I/3)*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^3)

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Maple [A]  time = 0.081, size = 57, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{d{a}^{3}} \left ({\frac{2\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-1}-4/3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-3} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3/(a+I*a*tan(d*x+c))^3,x)

[Out]

2/d/a^3*(2*I/(tan(1/2*d*x+1/2*c)-I)^2+1/(tan(1/2*d*x+1/2*c)-I)-4/3/(tan(1/2*d*x+1/2*c)-I)^3)

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Maxima [A]  time = 0.989576, size = 39, normalized size = 1.22 \begin{align*} \frac{i \, \cos \left (3 \, d x + 3 \, c\right ) + \sin \left (3 \, d x + 3 \, c\right )}{3 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/3*(I*cos(3*d*x + 3*c) + sin(3*d*x + 3*c))/(a^3*d)

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Fricas [A]  time = 2.22272, size = 49, normalized size = 1.53 \begin{align*} \frac{i \, e^{\left (-3 i \, d x - 3 i \, c\right )}}{3 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/3*I*e^(-3*I*d*x - 3*I*c)/(a^3*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)**3/(a+I*a*tan(d*x+c))**3,x)

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.201, size = 49, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}}{3 \, a^{3} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

2/3*(3*tan(1/2*d*x + 1/2*c)^2 - 1)/(a^3*d*(tan(1/2*d*x + 1/2*c) - I)^3)