Optimal. Leaf size=33 \[ \frac{2 i}{5 d (a \cos (c+d x)+i a \sin (c+d x))^{5/2}} \]
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Rubi [A] time = 0.0163566, antiderivative size = 33, 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 = {3071} \[ \frac{2 i}{5 d (a \cos (c+d x)+i a \sin (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3071
Rubi steps
\begin{align*} \int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^{5/2}} \, dx &=\frac{2 i}{5 d (a \cos (c+d x)+i a \sin (c+d x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.036128, size = 32, normalized size = 0.97 \[ \frac{2 i}{5 d (a (\cos (c+d x)+i \sin (c+d x)))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 28, normalized size = 0.9 \begin{align*}{\frac{{\frac{2\,i}{5}}}{d} \left ( a\cos \left ( dx+c \right ) +ia\sin \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.58497, size = 69, normalized size = 2.09 \begin{align*} \frac{2 i \,{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i\right )}^{\frac{5}{2}}}{5 \, a^{\frac{5}{2}} d{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92852, size = 59, normalized size = 1.79 \begin{align*} \frac{2 i \, e^{\left (-\frac{5}{2} i \, d x - \frac{5}{2} i \, c\right )}}{5 \, a^{\frac{5}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.79466, size = 50, normalized size = 1.52 \begin{align*} \frac{2 i}{5 \, d \left (-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i}\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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