Optimal. Leaf size=125 \[ -\frac{i (a+i a \tan (e+f x))^m \sqrt{\frac{c+d \tan (e+f x)}{c+i d}} F_1\left (m;\frac{5}{2},1;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d},\frac{1}{2} (i \tan (e+f x)+1)\right )}{2 f m (c+i d)^2 \sqrt{c+d \tan (e+f x)}} \]
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Rubi [A] time = 0.157546, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3564, 137, 136} \[ -\frac{i (a+i a \tan (e+f x))^m \sqrt{\frac{c+d \tan (e+f x)}{c+i d}} F_1\left (m;\frac{5}{2},1;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d},\frac{1}{2} (i \tan (e+f x)+1)\right )}{2 f m (c+i d)^2 \sqrt{c+d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3564
Rule 137
Rule 136
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{(a+x)^{-1+m}}{\left (-a^2+a x\right ) \left (c-\frac{i d x}{a}\right )^{5/2}} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac{\left (i a^2 \sqrt{\frac{c+d \tan (e+f x)}{c+i d}}\right ) \operatorname{Subst}\left (\int \frac{(a+x)^{-1+m}}{\left (-a^2+a x\right ) \left (\frac{c}{c+i d}-\frac{i d x}{a (c+i d)}\right )^{5/2}} \, dx,x,i a \tan (e+f x)\right )}{(c+i d)^2 f \sqrt{c+d \tan (e+f x)}}\\ &=-\frac{i F_1\left (m;\frac{5}{2},1;1+m;-\frac{d (1+i \tan (e+f x))}{i c-d},\frac{1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m \sqrt{\frac{c+d \tan (e+f x)}{c+i d}}}{2 (c+i d)^2 f m \sqrt{c+d \tan (e+f x)}}\\ \end{align*}
Mathematica [F] time = 17.7164, size = 0, normalized size = 0. \[ \int \frac{(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.325, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{m} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (\frac{2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{m} \sqrt{\frac{{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}{\left (-i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, e^{\left (4 i \, f x + 4 i \, e\right )} - 3 i \, e^{\left (2 i \, f x + 2 i \, e\right )} - i\right )}}{-i \, c^{3} + 3 \, c^{2} d + 3 i \, c d^{2} - d^{3} +{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-3 i \, c^{3} - 3 \, c^{2} d - 3 i \, c d^{2} - 3 \, d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-3 i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} + 3 \, d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}, x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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