Optimal. Leaf size=153 \[ \frac{\sqrt{a} \left (3 a^2-10 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{8 b^{5/2} f (a-b)^3}-\frac{a (3 a-7 b) \tan (e+f x)}{8 b^2 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac{a \tan ^3(e+f x)}{4 b f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}-\frac{x}{(a-b)^3} \]
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Rubi [A] time = 0.229099, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3670, 470, 578, 522, 203, 205} \[ \frac{\sqrt{a} \left (3 a^2-10 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{8 b^{5/2} f (a-b)^3}-\frac{a (3 a-7 b) \tan (e+f x)}{8 b^2 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac{a \tan ^3(e+f x)}{4 b f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}-\frac{x}{(a-b)^3} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 470
Rule 578
Rule 522
Rule 203
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{a \tan ^3(e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 a+(3 a-4 b) x^2\right )}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{4 (a-b) b f}\\ &=-\frac{a \tan ^3(e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}-\frac{a (3 a-7 b) \tan (e+f x)}{8 (a-b)^2 b^2 f \left (a+b \tan ^2(e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-a (3 a-7 b)+\left (-3 a^2+7 a b-8 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{8 (a-b)^2 b^2 f}\\ &=-\frac{a \tan ^3(e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}-\frac{a (3 a-7 b) \tan (e+f x)}{8 (a-b)^2 b^2 f \left (a+b \tan ^2(e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{(a-b)^3 f}+\frac{\left (a \left (3 a^2-10 a b+15 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{8 (a-b)^3 b^2 f}\\ &=-\frac{x}{(a-b)^3}+\frac{\sqrt{a} \left (3 a^2-10 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{8 (a-b)^3 b^{5/2} f}-\frac{a \tan ^3(e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}-\frac{a (3 a-7 b) \tan (e+f x)}{8 (a-b)^2 b^2 f \left (a+b \tan ^2(e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 2.17079, size = 142, normalized size = 0.93 \[ \frac{\frac{\sqrt{a} \left (3 a^2-10 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{b^{5/2}}-\frac{a (a-b) \sin (2 (e+f x)) \left (3 \left (a^2-4 a b+3 b^2\right ) \cos (2 (e+f x))+3 a^2-2 a b-9 b^2\right )}{b^2 ((a-b) \cos (2 (e+f x))+a+b)^2}-8 (e+f x)}{8 f (a-b)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 351, normalized size = 2.3 \begin{align*} -{\frac{5\,{a}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{8\,f \left ( a-b \right ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}b}}+{\frac{7\,{a}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{4\,f \left ( a-b \right ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{9\,ab \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{8\,f \left ( a-b \right ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{3\,{a}^{4}\tan \left ( fx+e \right ) }{8\,f \left ( a-b \right ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}{b}^{2}}}+{\frac{5\,{a}^{3}\tan \left ( fx+e \right ) }{4\,f \left ( a-b \right ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}b}}-{\frac{7\,{a}^{2}\tan \left ( fx+e \right ) }{8\,f \left ( a-b \right ) ^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{3\,{a}^{3}}{8\,f \left ( a-b \right ) ^{3}{b}^{2}}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{5\,{a}^{2}}{4\,f \left ( a-b \right ) ^{3}b}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{15\,a}{8\,f \left ( a-b \right ) ^{3}}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{f \left ( a-b \right ) ^{3}}} \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 = 1.7163, size = 1646, normalized size = 10.76 \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(-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 [A] time = 4.32386, size = 290, normalized size = 1.9 \begin{align*} \frac{\frac{{\left (3 \, a^{3} - 10 \, a^{2} b + 15 \, a b^{2}\right )}{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (f x + e\right )}{\sqrt{a b}}\right )\right )}}{{\left (a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right )} \sqrt{a b}} - \frac{8 \,{\left (f x + e\right )}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac{5 \, a^{2} b \tan \left (f x + e\right )^{3} - 9 \, a b^{2} \tan \left (f x + e\right )^{3} + 3 \, a^{3} \tan \left (f x + e\right ) - 7 \, a^{2} b \tan \left (f x + e\right )}{{\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )}{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{2}}}{8 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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