Optimal. Leaf size=415 \[ \frac{\sqrt{3} b \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt{-b^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-b^2} d \left (a-\sqrt{-b^2}\right )^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt{-b^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-b^2} d \left (a+\sqrt{-b^2}\right )^{2/3}}-\frac{3 b \log \left (\sqrt [3]{a-\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d \left (a-\sqrt{-b^2}\right )^{2/3}}+\frac{3 b \log \left (\sqrt [3]{a+\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d \left (a+\sqrt{-b^2}\right )^{2/3}}-\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} d \left (a-\sqrt{-b^2}\right )^{2/3}}+\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} d \left (a+\sqrt{-b^2}\right )^{2/3}}-\frac{x}{4 \left (a-\sqrt{-b^2}\right )^{2/3}}-\frac{x}{4 \left (a+\sqrt{-b^2}\right )^{2/3}} \]
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Rubi [A] time = 0.271149, antiderivative size = 415, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3485, 712, 57, 617, 204, 31} \[ \frac{\sqrt{3} b \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt{-b^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-b^2} d \left (a-\sqrt{-b^2}\right )^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt{-b^2}}}+1}{\sqrt{3}}\right )}{2 \sqrt{-b^2} d \left (a+\sqrt{-b^2}\right )^{2/3}}-\frac{3 b \log \left (\sqrt [3]{a-\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d \left (a-\sqrt{-b^2}\right )^{2/3}}+\frac{3 b \log \left (\sqrt [3]{a+\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} d \left (a+\sqrt{-b^2}\right )^{2/3}}-\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} d \left (a-\sqrt{-b^2}\right )^{2/3}}+\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} d \left (a+\sqrt{-b^2}\right )^{2/3}}-\frac{x}{4 \left (a-\sqrt{-b^2}\right )^{2/3}}-\frac{x}{4 \left (a+\sqrt{-b^2}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 3485
Rule 712
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(a+b \tan (c+d x))^{2/3}} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{1}{(a+x)^{2/3} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (\frac{\sqrt{-b^2}}{2 b^2 \left (\sqrt{-b^2}-x\right ) (a+x)^{2/3}}+\frac{\sqrt{-b^2}}{2 b^2 (a+x)^{2/3} \left (\sqrt{-b^2}+x\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{b \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-b^2}-x\right ) (a+x)^{2/3}} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt{-b^2} d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{(a+x)^{2/3} \left (\sqrt{-b^2}+x\right )} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt{-b^2} d}\\ &=-\frac{x}{4 \left (a-\sqrt{-b^2}\right )^{2/3}}-\frac{x}{4 \left (a+\sqrt{-b^2}\right )^{2/3}}-\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} d}+\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} \left (a+\sqrt{-b^2}\right )^{2/3} d}+\frac{(3 b) \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} \left (a-\sqrt{-b^2}\right )^{2/3} d}+\frac{(3 b) \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} \sqrt [3]{a-\sqrt{-b^2}} d}-\frac{(3 b) \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} \left (a+\sqrt{-b^2}\right )^{2/3} d}-\frac{(3 b) \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} \sqrt [3]{a+\sqrt{-b^2}} d}\\ &=-\frac{x}{4 \left (a-\sqrt{-b^2}\right )^{2/3}}-\frac{x}{4 \left (a+\sqrt{-b^2}\right )^{2/3}}-\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} d}+\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} \left (a+\sqrt{-b^2}\right )^{2/3} d}-\frac{3 b \log \left (\sqrt [3]{a-\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} d}+\frac{3 b \log \left (\sqrt [3]{a+\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} \left (a+\sqrt{-b^2}\right )^{2/3} d}-\frac{(3 b) \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} \left (a-\sqrt{-b^2}\right )^{2/3} d}+\frac{(3 b) \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} \left (a+\sqrt{-b^2}\right )^{2/3} d}\\ &=-\frac{x}{4 \left (a-\sqrt{-b^2}\right )^{2/3}}-\frac{x}{4 \left (a+\sqrt{-b^2}\right )^{2/3}}+\frac{\sqrt{3} b \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} \left (a-\sqrt{-b^2}\right )^{2/3} d}-\frac{\sqrt{3} b \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} \left (a+\sqrt{-b^2}\right )^{2/3} d}-\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} d}+\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} \left (a+\sqrt{-b^2}\right )^{2/3} d}-\frac{3 b \log \left (\sqrt [3]{a-\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} \left (a-\sqrt{-b^2}\right )^{2/3} d}+\frac{3 b \log \left (\sqrt [3]{a+\sqrt{-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt{-b^2} \left (a+\sqrt{-b^2}\right )^{2/3} d}\\ \end{align*}
Mathematica [C] time = 0.529837, size = 313, normalized size = 0.75 \[ \frac{i \left (-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt{3}}\right )+\log \left (\sqrt [3]{a-i b} \sqrt [3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a-i b)^{2/3}\right )}{(a-i b)^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt{3}}\right )+\log \left (\sqrt [3]{a+i b} \sqrt [3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a+i b)^{2/3}\right )}{(a+i b)^{2/3}}+\frac{2 \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a-i b}\right )}{(a-i b)^{2/3}}-\frac{2 \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a+i b}\right )}{(a+i b)^{2/3}}\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 57, 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{1}{{{\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 \frac{1}{{\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 \frac{1}{\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 = 2.3313, size = 578, normalized size = 1.39 \begin{align*} -\frac{3}{2} \,{\left ({\left (i \, \sqrt{3} + 1\right )} \left (-\frac{1}{-216 i \, a^{2} b^{3} d^{3} - 432 \, a b^{4} d^{3} + 216 i \, b^{5} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{1}{3}} d{\left (\sqrt{3} + i\right )} - 2 \,{\left (i \, a + b\right )}^{\frac{1}{3}} d\right ) +{\left (i \, \sqrt{3} + 1\right )} \left (-\frac{1}{216 i \, a^{2} b^{3} d^{3} - 432 \, a b^{4} d^{3} - 216 i \, b^{5} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{1}{3}} d{\left (\sqrt{3} + i\right )} + 2 \,{\left (-i \, a + b\right )}^{\frac{1}{3}} d\right ) +{\left (-i \, \sqrt{3} + 1\right )} \left (-\frac{1}{-216 i \, a^{2} b^{3} d^{3} - 432 \, a b^{4} d^{3} + 216 i \, b^{5} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{1}{3}} d{\left (\sqrt{3} - i\right )} + 2 \,{\left (i \, a + b\right )}^{\frac{1}{3}} d\right ) +{\left (-i \, \sqrt{3} + 1\right )} \left (-\frac{1}{216 i \, a^{2} b^{3} d^{3} - 432 \, a b^{4} d^{3} - 216 i \, b^{5} d^{3}}\right )^{\frac{1}{3}} \log \left ({\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{1}{3}} d{\left (\sqrt{3} - i\right )} - 2 \,{\left (-i \, a + b\right )}^{\frac{1}{3}} d\right ) - 2 \, \left (-\frac{1}{-216 i \, a^{2} b^{3} d^{3} - 432 \, a b^{4} d^{3} + 216 i \, b^{5} d^{3}}\right )^{\frac{1}{3}} \log \left (-{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{1}{3}} d + i \,{\left (i \, a + b\right )}^{\frac{1}{3}} d\right ) - 2 \, \left (-\frac{1}{216 i \, a^{2} b^{3} d^{3} - 432 \, a b^{4} d^{3} - 216 i \, b^{5} d^{3}}\right )^{\frac{1}{3}} \log \left (-{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{1}{3}} d - i \,{\left (-i \, a + b\right )}^{\frac{1}{3}} d\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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