Optimal. Leaf size=109 \[ -\frac{(b-2 a) \tan ^2(x)+3 a}{6 a (a+b)^2 \sqrt{a+b \tan ^4(x)}}-\frac{1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}} \]
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Rubi [A] time = 0.227757, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {3670, 1252, 823, 12, 725, 206} \[ -\frac{(b-2 a) \tan ^2(x)+3 a}{6 a (a+b)^2 \sqrt{a+b \tan ^4(x)}}-\frac{1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 1252
Rule 823
Rule 12
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan ^3(x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{x^3}{\left (1+x^2\right ) \left (a+b x^4\right )^{5/2}} \, dx,x,\tan (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(1+x) \left (a+b x^2\right )^{5/2}} \, dx,x,\tan ^2(x)\right )\\ &=-\frac{1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{a b-2 a b x}{(1+x) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan ^2(x)\right )}{6 a b (a+b)}\\ &=-\frac{1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac{3 a-(2 a-b) \tan ^2(x)}{6 a (a+b)^2 \sqrt{a+b \tan ^4(x)}}+\frac{\operatorname{Subst}\left (\int -\frac{3 a^2 b^2}{(1+x) \sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )}{6 a^2 b^2 (a+b)^2}\\ &=-\frac{1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac{3 a-(2 a-b) \tan ^2(x)}{6 a (a+b)^2 \sqrt{a+b \tan ^4(x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )}{2 (a+b)^2}\\ &=-\frac{1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac{3 a-(2 a-b) \tan ^2(x)}{6 a (a+b)^2 \sqrt{a+b \tan ^4(x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b-x^2} \, dx,x,\frac{a-b \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )}{2 (a+b)^2}\\ &=\frac{\tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac{1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac{3 a-(2 a-b) \tan ^2(x)}{6 a (a+b)^2 \sqrt{a+b \tan ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.7698, size = 104, normalized size = 0.95 \[ \frac{1}{6} \left (\frac{3 a^2 \tan ^2(x)+b (2 a-b) \tan ^6(x)-3 a b \tan ^4(x)-a (4 a+b)}{a (a+b)^2 \left (a+b \tan ^4(x)\right )^{3/2}}+\frac{3 \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )}{(a+b)^{5/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.093, size = 654, normalized size = 6. \begin{align*} \text{result too large to display} \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 \frac{\tan \left (x\right )^{3}}{{\left (b \tan \left (x\right )^{4} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.04866, size = 1273, normalized size = 11.68 \begin{align*} \left [\frac{3 \,{\left (a b^{2} \tan \left (x\right )^{8} + 2 \, a^{2} b \tan \left (x\right )^{4} + a^{3}\right )} \sqrt{a + b} \log \left (\frac{{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} - 2 \, \sqrt{b \tan \left (x\right )^{4} + a}{\left (b \tan \left (x\right )^{2} - a\right )} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + 2 \,{\left ({\left (2 \, a^{2} b + a b^{2} - b^{3}\right )} \tan \left (x\right )^{6} - 3 \,{\left (a^{2} b + a b^{2}\right )} \tan \left (x\right )^{4} - 4 \, a^{3} - 5 \, a^{2} b - a b^{2} + 3 \,{\left (a^{3} + a^{2} b\right )} \tan \left (x\right )^{2}\right )} \sqrt{b \tan \left (x\right )^{4} + a}}{12 \,{\left ({\left (a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right )} \tan \left (x\right )^{8} + a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3} + 2 \,{\left (a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right )} \tan \left (x\right )^{4}\right )}}, \frac{3 \,{\left (a b^{2} \tan \left (x\right )^{8} + 2 \, a^{2} b \tan \left (x\right )^{4} + a^{3}\right )} \sqrt{-a - b} \arctan \left (\frac{\sqrt{b \tan \left (x\right )^{4} + a}{\left (b \tan \left (x\right )^{2} - a\right )} \sqrt{-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) +{\left ({\left (2 \, a^{2} b + a b^{2} - b^{3}\right )} \tan \left (x\right )^{6} - 3 \,{\left (a^{2} b + a b^{2}\right )} \tan \left (x\right )^{4} - 4 \, a^{3} - 5 \, a^{2} b - a b^{2} + 3 \,{\left (a^{3} + a^{2} b\right )} \tan \left (x\right )^{2}\right )} \sqrt{b \tan \left (x\right )^{4} + a}}{6 \,{\left ({\left (a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right )} \tan \left (x\right )^{8} + a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3} + 2 \,{\left (a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right )} \tan \left (x\right )^{4}\right )}}\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*} \int \frac{\tan ^{3}{\left (x \right )}}{\left (a + b \tan ^{4}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.249, size = 806, normalized size = 7.39 \begin{align*} \frac{{\left ({\left (\frac{{\left (2 \, a^{7} b^{2} + 11 \, a^{6} b^{3} + 24 \, a^{5} b^{4} + 25 \, a^{4} b^{5} + 10 \, a^{3} b^{6} - 3 \, a^{2} b^{7} - 4 \, a b^{8} - b^{9}\right )} \tan \left (x\right )^{2}}{a^{9} b + 8 \, a^{8} b^{2} + 28 \, a^{7} b^{3} + 56 \, a^{6} b^{4} + 70 \, a^{5} b^{5} + 56 \, a^{4} b^{6} + 28 \, a^{3} b^{7} + 8 \, a^{2} b^{8} + a b^{9}} - \frac{3 \,{\left (a^{7} b^{2} + 6 \, a^{6} b^{3} + 15 \, a^{5} b^{4} + 20 \, a^{4} b^{5} + 15 \, a^{3} b^{6} + 6 \, a^{2} b^{7} + a b^{8}\right )}}{a^{9} b + 8 \, a^{8} b^{2} + 28 \, a^{7} b^{3} + 56 \, a^{6} b^{4} + 70 \, a^{5} b^{5} + 56 \, a^{4} b^{6} + 28 \, a^{3} b^{7} + 8 \, a^{2} b^{8} + a b^{9}}\right )} \tan \left (x\right )^{2} + \frac{3 \,{\left (a^{8} b + 6 \, a^{7} b^{2} + 15 \, a^{6} b^{3} + 20 \, a^{5} b^{4} + 15 \, a^{4} b^{5} + 6 \, a^{3} b^{6} + a^{2} b^{7}\right )}}{a^{9} b + 8 \, a^{8} b^{2} + 28 \, a^{7} b^{3} + 56 \, a^{6} b^{4} + 70 \, a^{5} b^{5} + 56 \, a^{4} b^{6} + 28 \, a^{3} b^{7} + 8 \, a^{2} b^{8} + a b^{9}}\right )} \tan \left (x\right )^{2} - \frac{4 \, a^{8} b + 25 \, a^{7} b^{2} + 66 \, a^{6} b^{3} + 95 \, a^{5} b^{4} + 80 \, a^{4} b^{5} + 39 \, a^{3} b^{6} + 10 \, a^{2} b^{7} + a b^{8}}{a^{9} b + 8 \, a^{8} b^{2} + 28 \, a^{7} b^{3} + 56 \, a^{6} b^{4} + 70 \, a^{5} b^{5} + 56 \, a^{4} b^{6} + 28 \, a^{3} b^{7} + 8 \, a^{2} b^{8} + a b^{9}}}{6 \,{\left (b \tan \left (x\right )^{4} + a\right )}^{\frac{3}{2}}} + \frac{\arctan \left (\frac{\sqrt{b} \tan \left (x\right )^{2} - \sqrt{b \tan \left (x\right )^{4} + a} + \sqrt{b}}{\sqrt{-a - b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{-a - b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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