Optimal. Leaf size=43 \[ \frac{i (a-i a \tan (c+d x))^4}{8 d \left (a^3+i a^3 \tan (c+d x)\right )^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0431671, antiderivative size = 43, 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 = {3487, 37} \[ \frac{i (a-i a \tan (c+d x))^4}{8 d \left (a^3+i a^3 \tan (c+d x)\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3487
Rule 37
Rubi steps
\begin{align*} \int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{(a-x)^3}{(a+x)^5} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=\frac{i (a-i a \tan (c+d x))^4}{8 d \left (a^3+i a^3 \tan (c+d x)\right )^4}\\ \end{align*}
Mathematica [A] time = 0.0594251, size = 32, normalized size = 0.74 \[ \frac{i \sec ^8(c+d x)}{8 d (a+i a \tan (c+d x))^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.115, size = 63, normalized size = 1.5 \begin{align*}{\frac{1}{d{a}^{8}} \left ( - \left ( \tan \left ( dx+c \right ) -i \right ) ^{-1}-{\frac{3\,i}{ \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+4\, \left ( \tan \left ( dx+c \right ) -i \right ) ^{-3}+{\frac{2\,i}{ \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.14773, size = 217, normalized size = 5.05 \begin{align*} -\frac{35 \, \tan \left (d x + c\right )^{6} - 105 i \, \tan \left (d x + c\right )^{5} - 140 \, \tan \left (d x + c\right )^{4} + 140 i \, \tan \left (d x + c\right )^{3} + 105 \, \tan \left (d x + c\right )^{2} - 35 i \, \tan \left (d x + c\right )}{{\left (35 \, a^{8} \tan \left (d x + c\right )^{7} - 245 i \, a^{8} \tan \left (d x + c\right )^{6} - 735 \, a^{8} \tan \left (d x + c\right )^{5} + 1225 i \, a^{8} \tan \left (d x + c\right )^{4} + 1225 \, a^{8} \tan \left (d x + c\right )^{3} - 735 i \, a^{8} \tan \left (d x + c\right )^{2} - 245 \, a^{8} \tan \left (d x + c\right ) + 35 i \, a^{8}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.52265, size = 49, normalized size = 1.14 \begin{align*} \frac{i \, e^{\left (-8 i \, d x - 8 i \, c\right )}}{8 \, a^{8} 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 [B] time = 1.19459, size = 95, normalized size = 2.21 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]