Optimal. Leaf size=126 \[ \frac{1}{6} \left (a+b \tan ^4(x)\right )^{3/2}+\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}-\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )-\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right ) \]
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Rubi [A] time = 0.208331, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {3670, 1248, 735, 815, 844, 217, 206, 725} \[ \frac{1}{6} \left (a+b \tan ^4(x)\right )^{3/2}+\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}-\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )-\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right ) \]
Antiderivative was successfully verified.
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Rule 3670
Rule 1248
Rule 735
Rule 815
Rule 844
Rule 217
Rule 206
Rule 725
Rubi steps
\begin{align*} \int \tan (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \frac{x \left (a+b x^4\right )^{3/2}}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{1+x} \, dx,x,\tan ^2(x)\right )\\ &=\frac{1}{6} \left (a+b \tan ^4(x)\right )^{3/2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a-b x) \sqrt{a+b x^2}}{1+x} \, dx,x,\tan ^2(x)\right )\\ &=\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}+\frac{1}{6} \left (a+b \tan ^4(x)\right )^{3/2}+\frac{\operatorname{Subst}\left (\int \frac{a b (2 a+b)-b^2 (3 a+2 b) x}{(1+x) \sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )}{4 b}\\ &=\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}+\frac{1}{6} \left (a+b \tan ^4(x)\right )^{3/2}+\frac{1}{2} (a+b)^2 \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )-\frac{1}{4} (b (3 a+2 b)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )\\ &=\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}+\frac{1}{6} \left (a+b \tan ^4(x)\right )^{3/2}-\frac{1}{2} (a+b)^2 \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 )-\frac{1}{4} (b (3 a+2 b)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )\\ &=-\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )-\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )+\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}+\frac{1}{6} \left (a+b \tan ^4(x)\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 4.39079, size = 166, normalized size = 1.32 \[ \frac{1}{12} \left (\sqrt{a+b \tan ^4(x)} \left (8 a+2 b \tan ^4(x)-3 b \tan ^2(x)+6 b\right )-6 (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )-6 \sqrt{b} (a+b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )-\frac{3 \sqrt{a} \sqrt{b} \sqrt{a+b \tan ^4(x)} \sinh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a}}\right )}{\sqrt{\frac{b \tan ^4(x)}{a}+1}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 313, normalized size = 2.5 \begin{align*}{\frac{b \left ( \tan \left ( x \right ) \right ) ^{4}}{6}\sqrt{a+b \left ( \tan \left ( x \right ) \right ) ^{4}}}+{\frac{2\,a}{3}\sqrt{a+b \left ( \tan \left ( x \right ) \right ) ^{4}}}-{\frac{b \left ( \tan \left ( x \right ) \right ) ^{2}}{4}\sqrt{a+b \left ( \tan \left ( x \right ) \right ) ^{4}}}-{\frac{3\,a}{4}\sqrt{b}\ln \left ( \sqrt{b} \left ( \tan \left ( x \right ) \right ) ^{2}+\sqrt{a+b \left ( \tan \left ( x \right ) \right ) ^{4}} \right ) }+{\frac{b}{2}\sqrt{a+b \left ( \tan \left ( x \right ) \right ) ^{4}}}-{\frac{1}{2}{b}^{{\frac{3}{2}}}\ln \left ( \sqrt{b} \left ( \tan \left ( x \right ) \right ) ^{2}+\sqrt{a+b \left ( \tan \left ( x \right ) \right ) ^{4}} \right ) }-{\frac{{a}^{2}}{2}\ln \left ({\frac{1}{1+ \left ( \tan \left ( x \right ) \right ) ^{2}} \left ( 2\,a+2\,b-2\,b \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) +2\,\sqrt{a+b}\sqrt{b \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{2}-2\,b \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) +a+b} \right ) } \right ){\frac{1}{\sqrt{a+b}}}}-{ab\ln \left ({\frac{1}{1+ \left ( \tan \left ( x \right ) \right ) ^{2}} \left ( 2\,a+2\,b-2\,b \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) +2\,\sqrt{a+b}\sqrt{b \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{2}-2\,b \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) +a+b} \right ) } \right ){\frac{1}{\sqrt{a+b}}}}-{\frac{{b}^{2}}{2}\ln \left ({\frac{1}{1+ \left ( \tan \left ( x \right ) \right ) ^{2}} \left ( 2\,a+2\,b-2\,b \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) +2\,\sqrt{a+b}\sqrt{b \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{2}-2\,b \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) +a+b} \right ) } \right ){\frac{1}{\sqrt{a+b}}}} \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 (x\right )^{4} + a\right )}^{\frac{3}{2}} \tan \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.23001, size = 1609, normalized size = 12.77 \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 ^{4}{\left (x \right )}\right )^{\frac{3}{2}} \tan{\left (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 (x\right )^{4} + a\right )}^{\frac{3}{2}} \tan \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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