Optimal. Leaf size=68 \[ \frac{i \sec ^3(c+d x)}{15 a d (a+i a \tan (c+d x))^3}+\frac{i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0793596, 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} \[ \frac{i \sec ^3(c+d x)}{15 a d (a+i a \tan (c+d x))^3}+\frac{i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=\frac{i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4}+\frac{\int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{5 a}\\ &=\frac{i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4}+\frac{i \sec ^3(c+d x)}{15 a d (a+i a \tan (c+d x))^3}\\ \end{align*}
Mathematica [A] time = 0.0796416, size = 40, normalized size = 0.59 \[ -\frac{(\tan (c+d x)-4 i) \sec ^3(c+d x)}{15 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.089, size = 90, normalized size = 1.3 \begin{align*} 2\,{\frac{1}{{a}^{4}d} \left ( 8/5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-5}+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-1}-{\frac{4\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{3\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}-14/3\, \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]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.99218, size = 72, normalized size = 1.06 \begin{align*} \frac{3 i \, \cos \left (5 \, d x + 5 \, c\right ) + 5 i \, \cos \left (3 \, d x + 3 \, c\right ) + 3 \, \sin \left (5 \, d x + 5 \, c\right ) + 5 \, \sin \left (3 \, d x + 3 \, c\right )}{30 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.32098, size = 90, normalized size = 1.32 \begin{align*} \frac{{\left (5 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{30 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17768, size = 99, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 15 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 25 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 5 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4\right )}}{15 \, a^{4} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]