Optimal. Leaf size=415 \[ -\frac{\sqrt{3} b \left (a-\sqrt{-b^2}\right )^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt{-b^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-b^2} d}+\frac{\sqrt{3} b \left (a+\sqrt{-b^2}\right )^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt{-b^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-b^2} d}-\frac{3 b \left (a-\sqrt{-b^2}\right )^{2/3} \log \left (\sqrt [3]{a-\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d}+\frac{3 b \left (a+\sqrt{-b^2}\right )^{2/3} \log \left (\sqrt [3]{a+\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d}-\frac{b \left (a-\sqrt{-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}+\frac{b \left (a+\sqrt{-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}-\frac{1}{4} x \left (a-\sqrt{-b^2}\right )^{2/3}-\frac{1}{4} x \left (a+\sqrt{-b^2}\right )^{2/3} \]
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Rubi [A] time = 0.381613, antiderivative size = 415, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3485, 712, 50, 55, 617, 204, 31} \[ -\frac{\sqrt{3} b \left (a-\sqrt{-b^2}\right )^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt{-b^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-b^2} d}+\frac{\sqrt{3} b \left (a+\sqrt{-b^2}\right )^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt{-b^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-b^2} d}-\frac{3 b \left (a-\sqrt{-b^2}\right )^{2/3} \log \left (\sqrt [3]{a-\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d}+\frac{3 b \left (a+\sqrt{-b^2}\right )^{2/3} \log \left (\sqrt [3]{a+\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d}-\frac{b \left (a-\sqrt{-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}+\frac{b \left (a+\sqrt{-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}-\frac{1}{4} x \left (a-\sqrt{-b^2}\right )^{2/3}-\frac{1}{4} x \left (a+\sqrt{-b^2}\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 3485
Rule 712
Rule 50
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int (a+b \tan (c+d x))^{2/3} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^{2/3}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (\frac{\sqrt{-b^2} (a+x)^{2/3}}{2 b^2 \left (\sqrt{-b^2}-x\right )}+\frac{\sqrt{-b^2} (a+x)^{2/3}}{2 b^2 \left (\sqrt{-b^2}+x\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^{2/3}}{\sqrt{-b^2}-x} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt{-b^2} d}-\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^{2/3}}{\sqrt{-b^2}+x} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt{-b^2} d}\\ &=-\frac{\left (b \left (a+\sqrt{-b^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-b^2}-x\right ) \sqrt [3]{a+x}} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt{-b^2} d}+\frac{\left (b^2+a \sqrt{-b^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+x} \left (\sqrt{-b^2}+x\right )} \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=-\frac{1}{4} \left (a-\sqrt{-b^2}\right )^{2/3} x-\frac{1}{4} \left (a+\sqrt{-b^2}\right )^{2/3} x+\frac{\sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 b d}+\frac{b \left (a+\sqrt{-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}-\frac{\left (3 b \left (a+\sqrt{-b^2}\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+\sqrt{-b^2}}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d}+\frac{\left (3 b \left (a+\sqrt{-b^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+\sqrt{-b^2}\right )^{2/3}+\sqrt [3]{a+\sqrt{-b^2}} x+x^2} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d}+\frac{\left (3 \left (b^2+a \sqrt{-b^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a-\sqrt{-b^2}\right )^{2/3}+\sqrt [3]{a-\sqrt{-b^2}} x+x^2} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b d}-\frac{\left (3 \left (b^2+a \sqrt{-b^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a-\sqrt{-b^2}}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b \sqrt [3]{a-\sqrt{-b^2}} d}\\ &=-\frac{1}{4} \left (a-\sqrt{-b^2}\right )^{2/3} x-\frac{1}{4} \left (a+\sqrt{-b^2}\right )^{2/3} x+\frac{\sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 b d}+\frac{b \left (a+\sqrt{-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}+\frac{3 \sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} \log \left (\sqrt [3]{a-\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b d}+\frac{3 b \left (a+\sqrt{-b^2}\right )^{2/3} \log \left (\sqrt [3]{a+\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d}-\frac{\left (3 b \left (a+\sqrt{-b^2}\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt{-b^2}}}\right )}{2 \sqrt{-b^2} d}-\frac{\left (3 \left (b^2+a \sqrt{-b^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt{-b^2}}}\right )}{2 b \sqrt [3]{a-\sqrt{-b^2}} d}\\ &=-\frac{1}{4} \left (a-\sqrt{-b^2}\right )^{2/3} x-\frac{1}{4} \left (a+\sqrt{-b^2}\right )^{2/3} x+\frac{\sqrt{3} \sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt{-b^2}}}}{\sqrt{3}}\right )}{2 b d}+\frac{\sqrt{3} b \left (a+\sqrt{-b^2}\right )^{2/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt{-b^2}}}}{\sqrt{3}}\right )}{2 \sqrt{-b^2} d}+\frac{\sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 b d}+\frac{b \left (a+\sqrt{-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}+\frac{3 \sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} \log \left (\sqrt [3]{a-\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b d}+\frac{3 b \left (a+\sqrt{-b^2}\right )^{2/3} \log \left (\sqrt [3]{a+\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d}\\ \end{align*}
Mathematica [C] time = 0.298816, size = 224, normalized size = 0.54 \[ \frac{\frac{(b+i a) \left (2 \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt{3}}\right )+3 \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a-i b}\right )-\log (\tan (c+d x)+i)\right )}{\sqrt [3]{a-i b}}+\frac{(b-i a) \left (2 \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt{3}}\right )+3 \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a+i b}\right )-\log (-\tan (c+d x)+i)\right )}{\sqrt [3]{a+i b}}}{4 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.014, size = 60, normalized size = 0.1 \begin{align*}{\frac{b}{2\,d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-2\,{{\it \_Z}}^{3}a+{a}^{2}+{b}^{2} \right ) }{\frac{{{\it \_R}}^{4}}{{{\it \_R}}^{5}-{{\it \_R}}^{2}a}\ln \left ( \sqrt [3]{a+b\tan \left ( dx+c \right ) }-{\it \_R} \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 (d x + c\right ) + a\right )}^{\frac{2}{3}}\,{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 \left (a + b \tan{\left (c + d x \right )}\right )^{\frac{2}{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 4.55609, size = 370, normalized size = 0.89 \begin{align*} -\frac{1}{24} \,{\left ({\left (i \, \sqrt{3} + 1\right )} \left (\frac{216 i \, a^{2} - 432 \, a b - 216 i \, b^{2}}{b^{3} d^{3}}\right )^{\frac{1}{3}} \log \left (d\right ) +{\left (-i \, \sqrt{3} + 1\right )} \left (\frac{216 i \, a^{2} - 432 \, a b - 216 i \, b^{2}}{b^{3} d^{3}}\right )^{\frac{1}{3}} \log \left (d\right ) +{\left (i \, \sqrt{3} + 1\right )} \left (\frac{-216 i \, a^{2} - 432 \, a b + 216 i \, b^{2}}{b^{3} d^{3}}\right )^{\frac{1}{3}} \log \left (d\right ) +{\left (-i \, \sqrt{3} + 1\right )} \left (\frac{-216 i \, a^{2} - 432 \, a b + 216 i \, b^{2}}{b^{3} d^{3}}\right )^{\frac{1}{3}} \log \left (d\right ) - 2 \, \left (\frac{-216 i \, a^{2} - 432 \, a b + 216 i \, b^{2}}{b^{3} d^{3}}\right )^{\frac{1}{3}} \log \left (i \, a d + b d -{\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )}^{\frac{1}{3}}{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{1}{3}} d\right ) - 2 \, \left (\frac{216 i \, a^{2} - 432 \, a b - 216 i \, b^{2}}{b^{3} d^{3}}\right )^{\frac{1}{3}} \log \left (-i \, a d + b d -{\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )}^{\frac{1}{3}}{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{1}{3}} d\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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