Optimal. Leaf size=163 \[ \frac{3 \tan ^{\frac{5}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left (\frac{5}{3};1,\frac{1}{2};\frac{8}{3};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{10 d \sqrt{a+b \tan (c+d x)}}+\frac{3 \tan ^{\frac{5}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left (\frac{5}{3};1,\frac{1}{2};\frac{8}{3};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{10 d \sqrt{a+b \tan (c+d x)}} \]
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Rubi [A] time = 0.263065, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3575, 912, 130, 511, 510} \[ \frac{3 \tan ^{\frac{5}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left (\frac{5}{3};1,\frac{1}{2};\frac{8}{3};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{10 d \sqrt{a+b \tan (c+d x)}}+\frac{3 \tan ^{\frac{5}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left (\frac{5}{3};1,\frac{1}{2};\frac{8}{3};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{10 d \sqrt{a+b \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3575
Rule 912
Rule 130
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\tan ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^{2/3}}{\sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{i x^{2/3}}{2 (i-x) \sqrt{a+b x}}+\frac{i x^{2/3}}{2 (i+x) \sqrt{a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{i \operatorname{Subst}\left (\int \frac{x^{2/3}}{(i-x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{i \operatorname{Subst}\left (\int \frac{x^{2/3}}{(i+x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{(3 i) \operatorname{Subst}\left (\int \frac{x^4}{\left (i-x^3\right ) \sqrt{a+b x^3}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{x^4}{\left (i+x^3\right ) \sqrt{a+b x^3}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d}\\ &=\frac{\left (3 i \sqrt{1+\frac{b \tan (c+d x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{x^4}{\left (i-x^3\right ) \sqrt{1+\frac{b x^3}{a}}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d \sqrt{a+b \tan (c+d x)}}+\frac{\left (3 i \sqrt{1+\frac{b \tan (c+d x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{x^4}{\left (i+x^3\right ) \sqrt{1+\frac{b x^3}{a}}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d \sqrt{a+b \tan (c+d x)}}\\ &=\frac{3 F_1\left (\frac{5}{3};1,\frac{1}{2};\frac{8}{3};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right ) \tan ^{\frac{5}{3}}(c+d x) \sqrt{1+\frac{b \tan (c+d x)}{a}}}{10 d \sqrt{a+b \tan (c+d x)}}+\frac{3 F_1\left (\frac{5}{3};1,\frac{1}{2};\frac{8}{3};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right ) \tan ^{\frac{5}{3}}(c+d x) \sqrt{1+\frac{b \tan (c+d x)}{a}}}{10 d \sqrt{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [B] time = 59.262, size = 7362, normalized size = 45.17 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.583, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}{\frac{1}{\sqrt{a+b\tan \left ( dx+c \right ) }}}}\, dx \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 (d x + c\right )^{\frac{2}{3}}}{\sqrt{b \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{\frac{2}{3}}{\left (c + d x \right )}}{\sqrt{a + b \tan{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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