Optimal. Leaf size=189 \[ -\frac{31 d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{8 a^3 f}+\frac{27 d^4 \sqrt{d \tan (e+f x)}}{8 a^3 f}-\frac{9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (\tan (e+f x)+1)}+\frac{d^{9/2} \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{2 \sqrt{2} a^3 f}-\frac{d^2 (d \tan (e+f x))^{5/2}}{4 a f (a \tan (e+f x)+a)^2} \]
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Rubi [A] time = 0.755399, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {3565, 3645, 3647, 3654, 3532, 208, 3634, 63, 205} \[ -\frac{31 d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{8 a^3 f}+\frac{27 d^4 \sqrt{d \tan (e+f x)}}{8 a^3 f}-\frac{9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (\tan (e+f x)+1)}+\frac{d^{9/2} \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{2 \sqrt{2} a^3 f}-\frac{d^2 (d \tan (e+f x))^{5/2}}{4 a f (a \tan (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3645
Rule 3647
Rule 3654
Rule 3532
Rule 208
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{(d \tan (e+f x))^{9/2}}{(a+a \tan (e+f x))^3} \, dx &=-\frac{d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}+\frac{\int \frac{(d \tan (e+f x))^{3/2} \left (\frac{5 a^2 d^3}{2}-2 a^2 d^3 \tan (e+f x)+\frac{9}{2} a^2 d^3 \tan ^2(e+f x)\right )}{(a+a \tan (e+f x))^2} \, dx}{4 a^3}\\ &=-\frac{9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}+\frac{\int \frac{\sqrt{d \tan (e+f x)} \left (\frac{27 a^4 d^4}{2}-4 a^4 d^4 \tan (e+f x)+\frac{27}{2} a^4 d^4 \tan ^2(e+f x)\right )}{a+a \tan (e+f x)} \, dx}{8 a^6}\\ &=\frac{27 d^4 \sqrt{d \tan (e+f x)}}{8 a^3 f}-\frac{9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}+\frac{\int \frac{-\frac{27}{4} a^5 d^5-\frac{35}{4} a^5 d^5 \tan ^2(e+f x)}{\sqrt{d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{4 a^7}\\ &=\frac{27 d^4 \sqrt{d \tan (e+f x)}}{8 a^3 f}-\frac{9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}+\frac{\int \frac{2 a^6 d^5-2 a^6 d^5 \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{8 a^9}-\frac{\left (31 d^5\right ) \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{27 d^4 \sqrt{d \tan (e+f x)}}{8 a^3 f}-\frac{9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}-\frac{\left (31 d^5\right ) \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 (a^3 d^{10}\right ) \operatorname{Subst}\left (\int \frac{1}{-8 a^{12} d^{10}+d x^2} \, dx,x,\frac{2 a^6 d^5+2 a^6 d^5 \tan (e+f x)}{\sqrt{d \tan (e+f x)}}\right )}{f}\\ &=\frac{d^{9/2} \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{27 d^4 \sqrt{d \tan (e+f x)}}{8 a^3 f}-\frac{9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}-\frac{\left (31 d^4\right ) \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{31 d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{8 a^3 f}+\frac{d^{9/2} \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{27 d^4 \sqrt{d \tan (e+f x)}}{8 a^3 f}-\frac{9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac{d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}\\ \end{align*}
Mathematica [A] time = 6.26722, size = 346, normalized size = 1.83 \[ \frac{\cot (e+f x) \csc ^3(e+f x) (d \tan (e+f x))^{9/2} (\sin (e+f x)+\cos (e+f x))^3 \left (-\frac{11 \sin (e+f x)}{8 (\sin (e+f x)+\cos (e+f x))}-\frac{1}{8 (\sin (e+f x)+\cos (e+f x))^2}+\frac{7}{2}\right )}{f (a \tan (e+f x)+a)^3}+\frac{\sec ^3(e+f x) (d \tan (e+f x))^{9/2} (\sin (e+f x)+\cos (e+f x))^3 \left (\frac{2 \sqrt{2} \cos (2 (e+f x)) \csc (e+f x) \sec ^3(e+f x) \left (\log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )-\log \left (-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right )\right )}{(1-\tan (e+f x)) \left (\tan ^2(e+f x)+1\right ) (\cot (e+f x)+1)}-\frac{62 \tan ^{-1}\left (\sqrt{\tan (e+f x)}\right ) (\tan (e+f x)+1) \csc (e+f x) \sec ^3(e+f x)}{\left (\tan ^2(e+f x)+1\right )^2 (\cot (e+f x)+1)}\right )}{16 f \tan ^{\frac{9}{2}}(e+f x) (a \tan (e+f x)+a)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 461, normalized size = 2.4 \begin{align*} 2\,{\frac{{d}^{4}\sqrt{d\tan \left ( fx+e \right ) }}{{a}^{3}f}}+{\frac{{d}^{4}\sqrt{2}}{16\,{a}^{3}f}\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}^{4}\sqrt{2}}{8\,{a}^{3}f}\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}^{4}\sqrt{2}}{8\,{a}^{3}f}\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}^{5}\sqrt{2}}{16\,{a}^{3}f}\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}^{5}\sqrt{2}}{8\,{a}^{3}f}\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}^{5}\sqrt{2}}{8\,{a}^{3}f}\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{13\,{d}^{5}}{8\,{a}^{3}f \left ( d\tan \left ( fx+e \right ) +d \right ) ^{2}} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{11\,{d}^{6}}{8\,{a}^{3}f \left ( d\tan \left ( fx+e \right ) +d \right ) ^{2}}\sqrt{d\tan \left ( fx+e \right ) }}-{\frac{31}{8\,{a}^{3}f}{d}^{{\frac{9}{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 = 1.95026, size = 1235, normalized size = 6.53 \begin{align*} \left [-\frac{4 \,{\left (\sqrt{2} d^{4} \tan \left (f x + e\right )^{2} + 2 \, \sqrt{2} d^{4} \tan \left (f x + e\right ) + \sqrt{2} d^{4}\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 ) - 31 \,{\left (d^{4} \tan \left (f x + e\right )^{2} + 2 \, d^{4} \tan \left (f x + e\right ) + d^{4}\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 (16 \, d^{4} \tan \left (f x + e\right )^{2} + 45 \, d^{4} \tan \left (f x + e\right ) + 27 \, d^{4}\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{31 \,{\left (d^{4} \tan \left (f x + e\right )^{2} + 2 \, d^{4} \tan \left (f x + e\right ) + d^{4}\right )} \sqrt{d} \arctan \left (\frac{\sqrt{d \tan \left (f x + e\right )}}{\sqrt{d}}\right ) -{\left (\sqrt{2} d^{4} \tan \left (f x + e\right )^{2} + 2 \, \sqrt{2} d^{4} \tan \left (f x + e\right ) + \sqrt{2} d^{4}\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 (16 \, d^{4} \tan \left (f x + e\right )^{2} + 45 \, d^{4} \tan \left (f x + e\right ) + 27 \, d^{4}\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 = 1.39279, size = 466, normalized size = 2.47 \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 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 f} - \frac{62 \, \sqrt{d} \arctan \left (\frac{\sqrt{d \tan \left (f x + e\right )}}{\sqrt{d}}\right )}{a^{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 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 f} + \frac{32 \, \sqrt{d \tan \left (f x + e\right )}}{a^{3} f} + \frac{2 \,{\left (13 \, \sqrt{d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right ) + 11 \, \sqrt{d \tan \left (f x + e\right )} d^{2}\right )}}{{\left (d \tan \left (f x + e\right ) + d\right )}^{2} a^{3} f}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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