Optimal. Leaf size=172 \[ \frac{\left (3 a^2-12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{8 \sqrt{b} f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(5 a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}-\frac{(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f} \]
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Rubi [A] time = 0.248783, antiderivative size = 172, 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 = {3670, 477, 582, 523, 217, 206, 377, 203} \[ \frac{\left (3 a^2-12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{8 \sqrt{b} f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(5 a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}-\frac{(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 477
Rule 582
Rule 523
Rule 217
Rule 206
Rule 377
Rule 203
Rubi steps
\begin{align*} \int \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (a+b x^2\right )^{3/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (a (4 a-3 b)+(5 a-4 b) b x^2\right )}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=\frac{(5 a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}-\frac{\operatorname{Subst}\left (\int \frac{a (5 a-4 b) b-b \left (3 a^2-12 a b+8 b^2\right ) x^2}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{8 b f}\\ &=\frac{(5 a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}-\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}+\frac{\left (3 a^2-12 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac{(5 a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}-\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{1-(-a+b) x^2} \, dx,x,\frac{\tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f}+\frac{\left (3 a^2-12 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{8 f}\\ &=-\frac{(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f}+\frac{\left (3 a^2-12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{8 \sqrt{b} f}+\frac{(5 a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}\\ \end{align*}
Mathematica [C] time = 6.2563, size = 771, normalized size = 4.48 \[ \frac{\frac{b \left (a^2+4 a b-4 b^2\right ) \sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}}}{\sqrt{2}}\right ),1\right )}{a ((a-b) \cos (2 (e+f x))+a+b)}+\frac{4 b \left (4 a^2-8 a b+4 b^2\right ) \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{(a-b) \cos (2 (e+f x))+a+b}{\cos (2 (e+f x))+1}} \left (\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}}}{\sqrt{2}}\right ),1\right )}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}-\frac{\sin ^4(e+f x) \csc (2 (e+f x)) \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} \Pi \left (-\frac{b}{a-b};\left .\sin ^{-1}\left (\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right )\right |1\right )}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{(a-b) \cos (2 (e+f x))+a+b}}\right )}{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}}{4 f}+\frac{\sqrt{\frac{a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b}{\cos (2 (e+f x))+1}} \left (\frac{1}{8} \sec (e+f x) (5 a \sin (e+f x)-6 b \sin (e+f x))+\frac{1}{4} b \tan (e+f x) \sec ^2(e+f x)\right )}{f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 386, normalized size = 2.2 \begin{align*}{\frac{\tan \left ( fx+e \right ) }{4\,f} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,a\tan \left ( fx+e \right ) }{8\,f}\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}+{\frac{3\,{a}^{2}}{8\,f}\ln \left ( \sqrt{b}\tan \left ( fx+e \right ) +\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}} \right ){\frac{1}{\sqrt{b}}}}-{\frac{b\tan \left ( fx+e \right ) }{2\,f}\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}-{\frac{3\,a}{2\,f}\sqrt{b}\ln \left ( \sqrt{b}\tan \left ( fx+e \right ) +\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}} \right ) }+{\frac{1}{f}{b}^{{\frac{3}{2}}}\ln \left ( \sqrt{b}\tan \left ( fx+e \right ) +\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}} \right ) }-{\frac{1}{f \left ( a-b \right ) }\sqrt{{b}^{4} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ){b}^{2}\tan \left ( fx+e \right ){\frac{1}{\sqrt{{b}^{4} \left ( a-b \right ) }}}{\frac{1}{\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}}} \right ) }+2\,{\frac{a\sqrt{{b}^{4} \left ( a-b \right ) }}{fb \left ( a-b \right ) }\arctan \left ({\frac{ \left ( a-b \right ){b}^{2}\tan \left ( fx+e \right ) }{\sqrt{{b}^{4} \left ( a-b \right ) }\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}} \right ) }-{\frac{{a}^{2}}{f{b}^{2} \left ( a-b \right ) }\sqrt{{b}^{4} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ){b}^{2}\tan \left ( fx+e \right ){\frac{1}{\sqrt{{b}^{4} \left ( a-b \right ) }}}{\frac{1}{\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}}} \right ) } \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{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \tan \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 7.99549, size = 1739, normalized size = 10.11 \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*} \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}} \tan ^{2}{\left (e + f x \right )}\, dx \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{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \tan \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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