Optimal. Leaf size=222 \[ -\frac{\sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{d^{3/2} f}+\frac{\sqrt{2} a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right )}{d^{3/2} f}-\frac{a^2 \log \left (\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} d^{3/2} f}+\frac{a^2 \log \left (\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} d^{3/2} f}-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}} \]
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Rubi [A] time = 0.199365, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3542, 12, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{d^{3/2} f}+\frac{\sqrt{2} a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right )}{d^{3/2} f}-\frac{a^2 \log \left (\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} d^{3/2} f}+\frac{a^2 \log \left (\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} d^{3/2} f}-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3542
Rule 12
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx &=-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}}+\frac{\int \frac{2 a^2 d}{\sqrt{d \tan (e+f x)}} \, dx}{d^2}\\ &=-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}}+\frac{\left (2 a^2\right ) \int \frac{1}{\sqrt{d \tan (e+f x)}} \, dx}{d}\\ &=-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (d^2+x^2\right )} \, dx,x,d \tan (e+f x)\right )}{f}\\ &=-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}}+\frac{\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{d-x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{d f}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{d+x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{d f}\\ &=-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}}-\frac{a^2 \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}+2 x}{-d-\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{\sqrt{2} d^{3/2} f}-\frac{a^2 \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}-2 x}{-d+\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{\sqrt{2} d^{3/2} f}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{d f}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{d f}\\ &=-\frac{a^2 \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} d^{3/2} f}+\frac{a^2 \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} d^{3/2} f}-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}}+\frac{\left (\sqrt{2} a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{d^{3/2} f}-\frac{\left (\sqrt{2} a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{d^{3/2} f}\\ &=-\frac{\sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{d^{3/2} f}+\frac{\sqrt{2} a^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{d^{3/2} f}-\frac{a^2 \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} d^{3/2} f}+\frac{a^2 \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} d^{3/2} f}-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.83606, size = 232, normalized size = 1.05 \[ -\frac{a^2 (\tan (e+f x)+1)^2 \left (-4 \sin ^2(e+f x) \tan (e+f x) \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right )+6 \sin (2 (e+f x)) \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\tan ^2(e+f x)\right )+3 \sqrt{2} \cos ^2(e+f x) \tan ^{\frac{3}{2}}(e+f x) \left (2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right )+\log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )-\log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )\right )\right )}{6 f (d \tan (e+f x))^{3/2} (\sin (e+f x)+\cos (e+f x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 195, normalized size = 0.9 \begin{align*} -2\,{\frac{{a}^{2}}{df\sqrt{d\tan \left ( fx+e \right ) }}}+{\frac{{a}^{2}\sqrt{2}}{2\,f{d}^{2}}\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}^{2}\sqrt{2}}{f{d}^{2}}\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}^{2}\sqrt{2}}{f{d}^{2}}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \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 [B] time = 2.0335, size = 1825, normalized size = 8.22 \begin{align*} \text{result too large to display} \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^{2} \left (\int \frac{1}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{2 \tan{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{\tan ^{2}{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27856, size = 304, normalized size = 1.37 \begin{align*} \frac{\frac{2 \, \sqrt{2} a^{2} \sqrt{{\left | d \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 f} + \frac{2 \, \sqrt{2} a^{2} \sqrt{{\left | d \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 f} + \frac{\sqrt{2} a^{2} \sqrt{{\left | d \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 )}{d f} - \frac{\sqrt{2} a^{2} \sqrt{{\left | d \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 )}{d f} - \frac{4 \, a^{2}}{\sqrt{d \tan \left (f x + e\right )} f}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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