Optimal. Leaf size=213 \[ -\frac{\sqrt{3} (B+i A) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2\ 2^{2/3} a^{2/3} d}+\frac{3 (B+i A) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4\ 2^{2/3} a^{2/3} d}+\frac{(B+i A) \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}-\frac{x (A-i B)}{4\ 2^{2/3} a^{2/3}}+\frac{3 (-B+i A)}{4 d (a+i a \tan (c+d x))^{2/3}} \]
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Rubi [A] time = 0.159795, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3526, 3481, 57, 617, 204, 31} \[ -\frac{\sqrt{3} (B+i A) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{2\ 2^{2/3} a^{2/3} d}+\frac{3 (B+i A) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4\ 2^{2/3} a^{2/3} d}+\frac{(B+i A) \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}-\frac{x (A-i B)}{4\ 2^{2/3} a^{2/3}}+\frac{3 (-B+i A)}{4 d (a+i a \tan (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 3526
Rule 3481
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^{2/3}} \, dx &=\frac{3 (i A-B)}{4 d (a+i a \tan (c+d x))^{2/3}}+\frac{(A-i B) \int \sqrt [3]{a+i a \tan (c+d x)} \, dx}{2 a}\\ &=\frac{3 (i A-B)}{4 d (a+i a \tan (c+d x))^{2/3}}-\frac{(i A+B) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{2 d}\\ &=-\frac{(A-i B) x}{4\ 2^{2/3} a^{2/3}}+\frac{(i A+B) \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}+\frac{3 (i A-B)}{4 d (a+i a \tan (c+d x))^{2/3}}-\frac{(3 (i A+B)) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{4\ 2^{2/3} a^{2/3} d}-\frac{(3 (i A+B)) \operatorname{Subst}\left (\int \frac{1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}\\ &=-\frac{(A-i B) x}{4\ 2^{2/3} a^{2/3}}+\frac{(i A+B) \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}+\frac{3 (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4\ 2^{2/3} a^{2/3} d}+\frac{3 (i A-B)}{4 d (a+i a \tan (c+d x))^{2/3}}+\frac{(3 (i A+B)) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{2\ 2^{2/3} a^{2/3} d}\\ &=-\frac{(A-i B) x}{4\ 2^{2/3} a^{2/3}}-\frac{\sqrt{3} (i A+B) \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{2\ 2^{2/3} a^{2/3} d}+\frac{(i A+B) \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}+\frac{3 (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4\ 2^{2/3} a^{2/3} d}+\frac{3 (i A-B)}{4 d (a+i a \tan (c+d x))^{2/3}}\\ \end{align*}
Mathematica [F] time = 0.628965, size = 0, normalized size = 0. \[ \int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^{2/3}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.02, size = 318, normalized size = 1.5 \begin{align*}{\frac{\sqrt [3]{2}B}{4\,d}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ){a}^{-{\frac{2}{3}}}}+{\frac{{\frac{i}{4}}\sqrt [3]{2}A}{d}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ){a}^{-{\frac{2}{3}}}}-{\frac{\sqrt [3]{2}B}{8\,d}\ln \left ( \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}+\sqrt [3]{2}\sqrt [3]{a}\sqrt [3]{a+ia\tan \left ( dx+c \right ) }+{2}^{{\frac{2}{3}}}{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{2}{3}}}}-{\frac{{\frac{i}{8}}\sqrt [3]{2}A}{d}\ln \left ( \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}+\sqrt [3]{2}\sqrt [3]{a}\sqrt [3]{a+ia\tan \left ( dx+c \right ) }+{2}^{{\frac{2}{3}}}{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{2}{3}}}}-{\frac{\sqrt [3]{2}\sqrt{3}B}{4\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ({{2}^{{\frac{2}{3}}}\sqrt [3]{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [3]{a}}}}+1 \right ) } \right ){a}^{-{\frac{2}{3}}}}-{\frac{{\frac{i}{4}}\sqrt [3]{2}\sqrt{3}A}{d}\arctan \left ({\frac{\sqrt{3}}{3} \left ({{2}^{{\frac{2}{3}}}\sqrt [3]{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [3]{a}}}}+1 \right ) } \right ){a}^{-{\frac{2}{3}}}}+{\frac{{\frac{3\,i}{4}}A}{d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}}-{\frac{3\,B}{4\,d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73408, size = 1411, normalized size = 6.62 \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 \frac{A + B \tan{\left (c + d x \right )}}{\left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{2}{3}}}\, 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 \frac{B \tan \left (d x + c\right ) + A}{{\left (i \, a \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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