Optimal. Leaf size=270 \[ -\frac{\sqrt{3} a^{2/3} (B+i A) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}-\frac{3 a^{2/3} (B+i A) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac{a^{2/3} (B+i A) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac{a^{2/3} x (A-i B)}{2 \sqrt [3]{2}}-\frac{3 (B+4 i A) (a+i a \tan (c+d x))^{5/3}}{20 a d}+\frac{3 B \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3}}{8 d}-\frac{9 B (a+i a \tan (c+d x))^{2/3}}{8 d} \]
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Rubi [A] time = 0.444395, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3597, 3592, 3527, 3481, 55, 617, 204, 31} \[ -\frac{\sqrt{3} a^{2/3} (B+i A) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}-\frac{3 a^{2/3} (B+i A) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac{a^{2/3} (B+i A) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac{a^{2/3} x (A-i B)}{2 \sqrt [3]{2}}-\frac{3 (B+4 i A) (a+i a \tan (c+d x))^{5/3}}{20 a d}+\frac{3 B \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3}}{8 d}-\frac{9 B (a+i a \tan (c+d x))^{2/3}}{8 d} \]
Antiderivative was successfully verified.
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Rule 3597
Rule 3592
Rule 3527
Rule 3481
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx &=\frac{3 B \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3}}{8 d}+\frac{3 \int \tan (c+d x) (a+i a \tan (c+d x))^{2/3} \left (-2 a B+\frac{2}{3} a (4 A-i B) \tan (c+d x)\right ) \, dx}{8 a}\\ &=\frac{3 B \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3}}{8 d}-\frac{3 (4 i A+B) (a+i a \tan (c+d x))^{5/3}}{20 a d}+\frac{3 \int (a+i a \tan (c+d x))^{2/3} \left (-\frac{2}{3} a (4 A-i B)-2 a B \tan (c+d x)\right ) \, dx}{8 a}\\ &=-\frac{9 B (a+i a \tan (c+d x))^{2/3}}{8 d}+\frac{3 B \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3}}{8 d}-\frac{3 (4 i A+B) (a+i a \tan (c+d x))^{5/3}}{20 a d}+(-A+i B) \int (a+i a \tan (c+d x))^{2/3} \, dx\\ &=-\frac{9 B (a+i a \tan (c+d x))^{2/3}}{8 d}+\frac{3 B \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3}}{8 d}-\frac{3 (4 i A+B) (a+i a \tan (c+d x))^{5/3}}{20 a d}+\frac{(a (i A+B)) \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac{a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}-\frac{a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac{9 B (a+i a \tan (c+d x))^{2/3}}{8 d}+\frac{3 B \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3}}{8 d}-\frac{3 (4 i A+B) (a+i a \tan (c+d x))^{5/3}}{20 a d}+\frac{\left (3 a^{2/3} (i A+B)\right ) \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 )}{2 \sqrt [3]{2} d}-\frac{(3 a (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 )}{2 d}\\ &=\frac{a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}-\frac{a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac{3 a^{2/3} (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac{9 B (a+i a \tan (c+d x))^{2/3}}{8 d}+\frac{3 B \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3}}{8 d}-\frac{3 (4 i A+B) (a+i a \tan (c+d x))^{5/3}}{20 a d}+\frac{\left (3 a^{2/3} (i A+B)\right ) \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 )}{\sqrt [3]{2} d}\\ &=\frac{a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}-\frac{\sqrt{3} a^{2/3} (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 )}{\sqrt [3]{2} d}-\frac{a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac{3 a^{2/3} (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac{9 B (a+i a \tan (c+d x))^{2/3}}{8 d}+\frac{3 B \tan ^2(c+d x) (a+i a \tan (c+d x))^{2/3}}{8 d}-\frac{3 (4 i A+B) (a+i a \tan (c+d x))^{5/3}}{20 a d}\\ \end{align*}
Mathematica [C] time = 2.64225, size = 104, normalized size = 0.39 \[ \frac{3 (a+i a \tan (c+d x))^{2/3} \left (10 (B+i A) \text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )+(8 A-2 i B) \tan (c+d x)-8 i A+5 B \sec ^2(c+d x)-22 B\right )}{40 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 367, normalized size = 1.4 \begin{align*} -{\frac{3\,B}{8\,{a}^{2}d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{8}{3}}}}+{\frac{3\,B}{5\,ad} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{5}{3}}}}-{\frac{{\frac{3\,i}{5}}A}{ad} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{5}{3}}}}-{\frac{3\,B}{2\,d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}}-{\frac{{2}^{{\frac{2}{3}}}B}{2\,d}{a}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ) }-{\frac{{\frac{i}{2}}{2}^{{\frac{2}{3}}}A}{d}{a}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ) }+{\frac{{2}^{{\frac{2}{3}}}B}{4\,d}{a}^{{\frac{2}{3}}}\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 ) }+{\frac{{\frac{i}{4}}{2}^{{\frac{2}{3}}}A}{d}{a}^{{\frac{2}{3}}}\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 ) }-{\frac{\sqrt{3}{2}^{{\frac{2}{3}}}B}{2\,d}{a}^{{\frac{2}{3}}}\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 ) }-{\frac{{\frac{i}{2}}\sqrt{3}{2}^{{\frac{2}{3}}}A}{d}{a}^{{\frac{2}{3}}}\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 ) } \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 = 1.82168, size = 1762, normalized size = 6.53 \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 \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{2}{3}} \left (A + B \tan{\left (c + d x \right )}\right ) \tan ^{2}{\left (c + d 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 (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{2}{3}} \tan \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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