Optimal. Leaf size=117 \[ -\frac{16 a^3}{3 d^2 f \sqrt{d \tan (e+f x)}}+\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac{2 \left (a^3 \tan (e+f x)+a^3\right )}{3 d f (d \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.175519, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3565, 3628, 3532, 208} \[ -\frac{16 a^3}{3 d^2 f \sqrt{d \tan (e+f x)}}+\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac{2 \left (a^3 \tan (e+f x)+a^3\right )}{3 d f (d \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3628
Rule 3532
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{5/2}} \, dx &=-\frac{2 \left (a^3+a^3 \tan (e+f x)\right )}{3 d f (d \tan (e+f x))^{3/2}}+\frac{2 \int \frac{4 a^3 d^2+3 a^3 d^2 \tan (e+f x)+a^3 d^2 \tan ^2(e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{3 d^3}\\ &=-\frac{16 a^3}{3 d^2 f \sqrt{d \tan (e+f x)}}-\frac{2 \left (a^3+a^3 \tan (e+f x)\right )}{3 d f (d \tan (e+f x))^{3/2}}+\frac{2 \int \frac{3 a^3 d^3-3 a^3 d^3 \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{3 d^5}\\ &=-\frac{16 a^3}{3 d^2 f \sqrt{d \tan (e+f x)}}-\frac{2 \left (a^3+a^3 \tan (e+f x)\right )}{3 d f (d \tan (e+f x))^{3/2}}-\frac{\left (12 a^6 d\right ) \operatorname{Subst}\left (\int \frac{1}{-18 a^6 d^6+d x^2} \, dx,x,\frac{3 a^3 d^3+3 a^3 d^3 \tan (e+f x)}{\sqrt{d \tan (e+f x)}}\right )}{f}\\ &=\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{d}+\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac{16 a^3}{3 d^2 f \sqrt{d \tan (e+f x)}}-\frac{2 \left (a^3+a^3 \tan (e+f x)\right )}{3 d f (d \tan (e+f x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 4.11545, size = 272, normalized size = 2.32 \[ \frac{a^3 (\tan (e+f x)+1)^3 \left (8 \sin ^3(e+f x) \tan (e+f x) \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right )-8 \sin (e+f x) \cos ^2(e+f x) \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};-\tan ^2(e+f x)\right )-72 \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\tan ^2(e+f x)\right )-9 \sqrt{2} \cos ^3(e+f x) \tan ^{\frac{5}{2}}(e+f x) \left (2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right )+\log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )-\log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )\right )\right )}{12 f (d \tan (e+f x))^{5/2} (\sin (e+f x)+\cos (e+f x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 388, normalized size = 3.3 \begin{align*}{\frac{{a}^{3}\sqrt{2}}{2\,f{d}^{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{{a}^{3}\sqrt{2}}{f{d}^{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{{a}^{3}\sqrt{2}}{f{d}^{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{{a}^{3}\sqrt{2}}{2\,f{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{1}{\sqrt [4]{{d}^{2}}}}}-{\frac{{a}^{3}\sqrt{2}}{f{d}^{2}}\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{{a}^{3}\sqrt{2}}{f{d}^{2}}\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{2\,{a}^{3}}{3\,fd} \left ( d\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}-6\,{\frac{{a}^{3}}{f{d}^{2}\sqrt{d\tan \left ( fx+e \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 = 1.71563, size = 613, normalized size = 5.24 \begin{align*} \left [\frac{3 \, \sqrt{2} a^{3} \sqrt{d} \log \left (\frac{\tan \left (f x + e\right )^{2} + \frac{2 \, \sqrt{2} \sqrt{d \tan \left (f x + e\right )}{\left (\tan \left (f x + e\right ) + 1\right )}}{\sqrt{d}} + 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{2} - 2 \,{\left (9 \, a^{3} \tan \left (f x + e\right ) + a^{3}\right )} \sqrt{d \tan \left (f x + e\right )}}{3 \, d^{3} f \tan \left (f x + e\right )^{2}}, -\frac{2 \,{\left (3 \, \sqrt{2} a^{3} d \sqrt{-\frac{1}{d}} \arctan \left (\frac{\sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{-\frac{1}{d}}{\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{2} +{\left (9 \, a^{3} \tan \left (f x + e\right ) + a^{3}\right )} \sqrt{d \tan \left (f x + e\right )}\right )}}{3 \, d^{3} f \tan \left (f x + e\right )^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int \frac{1}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}}}\, dx + \int \frac{3 \tan{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}}}\, dx + \int \frac{3 \tan ^{2}{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}}}\, dx + \int \frac{\tan ^{3}{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{5}{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.42373, size = 404, normalized size = 3.45 \begin{align*} \frac{\sqrt{2}{\left (a^{3} d \sqrt{{\left | d \right |}} + a^{3}{\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 )}{2 \, d^{4} f} - \frac{\sqrt{2}{\left (a^{3} d \sqrt{{\left | d \right |}} + a^{3}{\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 )}{2 \, d^{4} f} + \frac{{\left (\sqrt{2} a^{3} d \sqrt{{\left | d \right |}} - \sqrt{2} a^{3}{\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 )}{d^{4} f} + \frac{{\left (\sqrt{2} a^{3} d \sqrt{{\left | d \right |}} - \sqrt{2} a^{3}{\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 )}{d^{4} f} - \frac{2 \,{\left (9 \, a^{3} d \tan \left (f x + e\right ) + a^{3} d\right )}}{3 \, \sqrt{d \tan \left (f x + e\right )} d^{3} f \tan \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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