Optimal. Leaf size=251 \[ \frac{\sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}-\frac{3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{i a^{4/3} x}{2^{2/3}}+\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{6 (a+i a \tan (c+d x))^{7/3}}{35 a d}-\frac{9 (a+i a \tan (c+d x))^{4/3}}{20 d}-\frac{3 a \sqrt [3]{a+i a \tan (c+d x)}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.34185, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {3560, 3592, 3527, 3478, 3481, 57, 617, 204, 31} \[ \frac{\sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{d}-\frac{3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{i a^{4/3} x}{2^{2/3}}+\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{6 (a+i a \tan (c+d x))^{7/3}}{35 a d}-\frac{9 (a+i a \tan (c+d x))^{4/3}}{20 d}-\frac{3 a \sqrt [3]{a+i a \tan (c+d x)}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3560
Rule 3592
Rule 3527
Rule 3478
Rule 3481
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \tan ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx &=\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{3 \int \tan (c+d x) (a+i a \tan (c+d x))^{4/3} \left (2 a+\frac{4}{3} i a \tan (c+d x)\right ) \, dx}{10 a}\\ &=\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{6 (a+i a \tan (c+d x))^{7/3}}{35 a d}-\frac{3 \int (a+i a \tan (c+d x))^{4/3} \left (-\frac{4 i a}{3}+2 a \tan (c+d x)\right ) \, dx}{10 a}\\ &=-\frac{9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{6 (a+i a \tan (c+d x))^{7/3}}{35 a d}+i \int (a+i a \tan (c+d x))^{4/3} \, dx\\ &=-\frac{3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{6 (a+i a \tan (c+d x))^{7/3}}{35 a d}+(2 i a) \int \sqrt [3]{a+i a \tan (c+d x)} \, dx\\ &=-\frac{3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{6 (a+i a \tan (c+d x))^{7/3}}{35 a d}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^{4/3} x}{2^{2/3}}-\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{6 (a+i a \tan (c+d x))^{7/3}}{35 a d}+\frac{\left (3 a^{4/3}\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^{2/3} d}+\frac{\left (3 a^{5/3}\right ) \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 )}{\sqrt [3]{2} d}\\ &=-\frac{i a^{4/3} x}{2^{2/3}}-\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{6 (a+i a \tan (c+d x))^{7/3}}{35 a d}-\frac{\left (3 \sqrt [3]{2} a^{4/3}\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 )}{d}\\ &=-\frac{i a^{4/3} x}{2^{2/3}}+\frac{\sqrt [3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{d}-\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac{3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac{9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{6 (a+i a \tan (c+d x))^{7/3}}{35 a d}\\ \end{align*}
Mathematica [F] time = 180.004, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.018, size = 217, normalized size = 0.9 \begin{align*} -{\frac{3}{10\,{a}^{2}d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{10}{3}}}}+{\frac{3}{7\,ad} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{7}{3}}}}-{\frac{3}{4\,d} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}}}-3\,{\frac{a\sqrt [3]{a+ia\tan \left ( dx+c \right ) }}{d}}-{\frac{\sqrt [3]{2}}{d}{a}^{{\frac{4}{3}}}\ln \left ( \sqrt [3]{a+ia\tan \left ( dx+c \right ) }-\sqrt [3]{2}\sqrt [3]{a} \right ) }+{\frac{\sqrt [3]{2}}{2\,d}{a}^{{\frac{4}{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}\sqrt{3}}{d}{a}^{{\frac{4}{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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.82993, size = 1466, normalized size = 5.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{4}{3}} \tan \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]