Optimal. Leaf size=164 \[ \frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{8 a^3 f}+\frac{5 d^2 \sqrt{d \tan (e+f x)}}{8 a^3 f (\tan (e+f x)+1)}-\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{2 \sqrt{2} a^3 f}-\frac{d^2 \sqrt{d \tan (e+f x)}}{4 a f (a \tan (e+f x)+a)^2} \]
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Rubi [A] time = 0.559563, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3565, 3649, 3654, 3532, 208, 3634, 63, 205} \[ \frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{8 a^3 f}+\frac{5 d^2 \sqrt{d \tan (e+f x)}}{8 a^3 f (\tan (e+f x)+1)}-\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{2 \sqrt{2} a^3 f}-\frac{d^2 \sqrt{d \tan (e+f x)}}{4 a f (a \tan (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3649
Rule 3654
Rule 3532
Rule 208
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{(d \tan (e+f x))^{5/2}}{(a+a \tan (e+f x))^3} \, dx &=-\frac{d^2 \sqrt{d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}+\frac{\int \frac{\frac{a^2 d^3}{2}-2 a^2 d^3 \tan (e+f x)+\frac{5}{2} a^2 d^3 \tan ^2(e+f x)}{\sqrt{d \tan (e+f x)} (a+a \tan (e+f x))^2} \, dx}{4 a^3}\\ &=\frac{5 d^2 \sqrt{d \tan (e+f x)}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 \sqrt{d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}+\frac{\int \frac{-\frac{3}{2} a^4 d^4+\frac{5}{2} a^4 d^4 \tan ^2(e+f x)}{\sqrt{d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{8 a^6 d}\\ &=\frac{5 d^2 \sqrt{d \tan (e+f x)}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 \sqrt{d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}+\frac{\int \frac{-4 a^5 d^4+4 a^5 d^4 \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{16 a^8 d}+\frac{d^3 \int \frac{1+\tan ^2(e+f x)}{\sqrt{d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{16 a^2}\\ &=\frac{5 d^2 \sqrt{d \tan (e+f x)}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 \sqrt{d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}+\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{d x} (a+a x)} \, dx,x,\tan (e+f x)\right )}{16 a^2 f}-\frac{\left (2 a^2 d^7\right ) \operatorname{Subst}\left (\int \frac{1}{-32 a^{10} d^8+d x^2} \, dx,x,\frac{-4 a^5 d^4-4 a^5 d^4 \tan (e+f x)}{\sqrt{d \tan (e+f x)}}\right )}{f}\\ &=-\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d}+\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{2 \sqrt{2} a^3 f}+\frac{5 d^2 \sqrt{d \tan (e+f x)}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 \sqrt{d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{a+\frac{a x^2}{d}} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{8 a^2 f}\\ &=\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{8 a^3 f}-\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d}+\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{2 \sqrt{2} a^3 f}+\frac{5 d^2 \sqrt{d \tan (e+f x)}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 \sqrt{d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}\\ \end{align*}
Mathematica [A] time = 2.91742, size = 192, normalized size = 1.17 \[ \frac{\sec (e+f x) (d \tan (e+f x))^{5/2} (\sin (e+f x)+\cos (e+f x))^3 \left (\frac{\csc ^4(e+f x) (5 \sin (2 (e+f x))+3 \cos (2 (e+f x))+3)}{(\cot (e+f x)+1)^2}+\frac{2 \csc (e+f x) \sec (e+f x) \left (\tan ^{-1}\left (\sqrt{\tan (e+f x)}\right )+\sqrt{2} \left (\log \left (-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right )-\log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )\right )\right )}{\tan ^{\frac{3}{2}}(e+f x)}\right )}{16 a^3 f (\tan (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 440, normalized size = 2.7 \begin{align*} -{\frac{{d}^{2}\sqrt{2}}{16\,f{a}^{3}}\sqrt [4]{{d}^{2}}\ln \left ({ \left ( d\tan \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ) }-{\frac{{d}^{2}\sqrt{2}}{8\,f{a}^{3}}\sqrt [4]{{d}^{2}}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{{d}^{2}\sqrt{2}}{8\,f{a}^{3}}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{{d}^{3}\sqrt{2}}{16\,f{a}^{3}}\ln \left ({ \left ( d\tan \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}+{\frac{{d}^{3}\sqrt{2}}{8\,f{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}-{\frac{{d}^{3}\sqrt{2}}{8\,f{a}^{3}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}+{\frac{5\,{d}^{3}}{8\,f{a}^{3} \left ( d\tan \left ( fx+e \right ) +d \right ) ^{2}} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{4}}{8\,f{a}^{3} \left ( d\tan \left ( fx+e \right ) +d \right ) ^{2}}\sqrt{d\tan \left ( fx+e \right ) }}+{\frac{1}{8\,f{a}^{3}}{d}^{{\frac{5}{2}}}\arctan \left ({\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{d}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89411, size = 1154, normalized size = 7.04 \begin{align*} \left [\frac{4 \,{\left (\sqrt{2} d^{2} \tan \left (f x + e\right )^{2} + 2 \, \sqrt{2} d^{2} \tan \left (f x + e\right ) + \sqrt{2} d^{2}\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{d \tan \left (f x + e\right )}{\left (\sqrt{2} \tan \left (f x + e\right ) + \sqrt{2}\right )} \sqrt{-d}}{2 \, d \tan \left (f x + e\right )}\right ) +{\left (d^{2} \tan \left (f x + e\right )^{2} + 2 \, d^{2} \tan \left (f x + e\right ) + d^{2}\right )} \sqrt{-d} \log \left (\frac{d \tan \left (f x + e\right ) + 2 \, \sqrt{d \tan \left (f x + e\right )} \sqrt{-d} - d}{\tan \left (f x + e\right ) + 1}\right ) + 2 \,{\left (5 \, d^{2} \tan \left (f x + e\right ) + 3 \, d^{2}\right )} \sqrt{d \tan \left (f x + e\right )}}{16 \,{\left (a^{3} f \tan \left (f x + e\right )^{2} + 2 \, a^{3} f \tan \left (f x + e\right ) + a^{3} f\right )}}, \frac{{\left (d^{2} \tan \left (f x + e\right )^{2} + 2 \, d^{2} \tan \left (f x + e\right ) + d^{2}\right )} \sqrt{d} \arctan \left (\frac{\sqrt{d \tan \left (f x + e\right )}}{\sqrt{d}}\right ) +{\left (\sqrt{2} d^{2} \tan \left (f x + e\right )^{2} + 2 \, \sqrt{2} d^{2} \tan \left (f x + e\right ) + \sqrt{2} d^{2}\right )} \sqrt{d} \log \left (\frac{d \tan \left (f x + e\right )^{2} - 2 \, \sqrt{d \tan \left (f x + e\right )}{\left (\sqrt{2} \tan \left (f x + e\right ) + \sqrt{2}\right )} \sqrt{d} + 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) +{\left (5 \, d^{2} \tan \left (f x + e\right ) + 3 \, d^{2}\right )} \sqrt{d \tan \left (f x + e\right )}}{8 \,{\left (a^{3} f \tan \left (f x + e\right )^{2} + 2 \, a^{3} f \tan \left (f x + e\right ) + a^{3} f\right )}}\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 [B] time = 1.27793, size = 439, normalized size = 2.68 \begin{align*} -\frac{1}{16} \, d^{4}{\left (\frac{2 \, \sqrt{2}{\left (d \sqrt{{\left | d \right |}} -{\left | d \right |}^{\frac{3}{2}}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} + 2 \, \sqrt{d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{a^{3} d^{3} f} + \frac{2 \, \sqrt{2}{\left (d \sqrt{{\left | d \right |}} -{\left | d \right |}^{\frac{3}{2}}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} - 2 \, \sqrt{d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{a^{3} d^{3} f} - \frac{2 \, \arctan \left (\frac{\sqrt{d \tan \left (f x + e\right )}}{\sqrt{d}}\right )}{a^{3} d^{\frac{3}{2}} f} + \frac{\sqrt{2}{\left (d \sqrt{{\left | d \right |}} +{\left | d \right |}^{\frac{3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) + \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{a^{3} d^{3} f} - \frac{\sqrt{2}{\left (d \sqrt{{\left | d \right |}} +{\left | d \right |}^{\frac{3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) - \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{a^{3} d^{3} f} - \frac{2 \,{\left (5 \, \sqrt{d \tan \left (f x + e\right )} d \tan \left (f x + e\right ) + 3 \, \sqrt{d \tan \left (f x + e\right )} d\right )}}{{\left (d \tan \left (f x + e\right ) + d\right )}^{2} a^{3} d f}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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