Optimal. Leaf size=347 \[ -\frac{7 i \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{7 i \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt{3}\right )}{12 a d}+\frac{5 \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{2 \sqrt{3} a d}+\frac{7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}-\frac{5 \log \left (\tan ^{\frac{2}{3}}(c+d x)+1\right )}{6 a d}+\frac{7 i \log \left (\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}-\frac{7 i \log \left (\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}+\frac{5 \log \left (\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right )}{12 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.530917, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {3552, 3529, 3538, 3476, 329, 275, 200, 31, 634, 618, 204, 628, 295, 203} \[ -\frac{7 i \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{7 i \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt{3}\right )}{12 a d}+\frac{5 \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{2 \sqrt{3} a d}+\frac{7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}-\frac{5 \log \left (\tan ^{\frac{2}{3}}(c+d x)+1\right )}{6 a d}+\frac{7 i \log \left (\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}-\frac{7 i \log \left (\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{8 \sqrt{3} a d}+\frac{5 \log \left (\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right )}{12 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3552
Rule 3529
Rule 3538
Rule 3476
Rule 329
Rule 275
Rule 200
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rule 295
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\tan ^{\frac{7}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx &=\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}-\frac{\int \frac{-\frac{10 a}{3}+\frac{7}{3} i a \tan (c+d x)}{\tan ^{\frac{7}{3}}(c+d x)} \, dx}{2 a^2}\\ &=-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}-\frac{\int \frac{\frac{7 i a}{3}+\frac{10}{3} a \tan (c+d x)}{\tan ^{\frac{4}{3}}(c+d x)} \, dx}{2 a^2}\\ &=-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}-\frac{\int \frac{\frac{10 a}{3}-\frac{7}{3} i a \tan (c+d x)}{\sqrt [3]{\tan (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}+\frac{(7 i) \int \tan ^{\frac{2}{3}}(c+d x) \, dx}{6 a}-\frac{5 \int \frac{1}{\sqrt [3]{\tan (c+d x)}} \, dx}{3 a}\\ &=-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}+\frac{(7 i) \operatorname{Subst}\left (\int \frac{x^{2/3}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{6 a d}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{3 a d}\\ &=-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}+\frac{(7 i) \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 a d}-\frac{5 \operatorname{Subst}\left (\int \frac{x}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{a d}\\ &=-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}+\frac{(7 i) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{(7 i) \operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac{(7 i) \operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{1+x^3} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{2 a d}\\ &=\frac{7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}+\frac{(7 i) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{24 a d}+\frac{(7 i) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{24 a d}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{6 a d}-\frac{5 \operatorname{Subst}\left (\int \frac{2-x}{1-x+x^2} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{6 a d}+\frac{(7 i) \operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{8 \sqrt{3} a d}-\frac{(7 i) \operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{8 \sqrt{3} a d}\\ &=\frac{7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac{5 \log \left (1+\tan ^{\frac{2}{3}}(c+d x)\right )}{6 a d}+\frac{7 i \log \left (1-\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}-\frac{7 i \log \left (1+\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}-\frac{(7 i) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}-\frac{(7 i) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{5 \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{12 a d}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{4 a d}\\ &=-\frac{7 i \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{7 i \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac{5 \log \left (1+\tan ^{\frac{2}{3}}(c+d x)\right )}{6 a d}+\frac{7 i \log \left (1-\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}-\frac{7 i \log \left (1+\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}+\frac{5 \log \left (1-\tan ^{\frac{2}{3}}(c+d x)+\tan ^{\frac{4}{3}}(c+d x)\right )}{12 a d}-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \tan ^{\frac{2}{3}}(c+d x)\right )}{2 a d}\\ &=-\frac{7 i \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{7 i \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac{5 \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{2 \sqrt{3} a d}+\frac{7 i \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac{5 \log \left (1+\tan ^{\frac{2}{3}}(c+d x)\right )}{6 a d}+\frac{7 i \log \left (1-\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}-\frac{7 i \log \left (1+\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{8 \sqrt{3} a d}+\frac{5 \log \left (1-\tan ^{\frac{2}{3}}(c+d x)+\tan ^{\frac{4}{3}}(c+d x)\right )}{12 a d}-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{7 i}{2 a d \sqrt [3]{\tan (c+d x)}}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 1.79878, size = 236, normalized size = 0.68 \[ -\frac{i \tan ^{\frac{2}{3}}(c+d x) \csc ^2(c+d x) \sec (c+d x) \left (3 \sqrt [3]{2} e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{2/3} \left (-1+e^{2 i (c+d x)}\right )^2 \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left (1-e^{2 i (c+d x)}\right )\right )+4 \left (-34 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right ) \sin ^2(c+d x) (\cos (c+d x)+i \sin (c+d x))+9 i \sin (c+d x)+9 i \sin (3 (c+d x))-23 \cos (c+d x)+11 \cos (3 (c+d x))\right )\right )}{64 a d (\tan (c+d x)-i)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 290, normalized size = 0.8 \begin{align*}{\frac{{\frac{i}{3}}}{ad}\sqrt [3]{\tan \left ( dx+c \right ) } \left ( -i\sqrt [3]{\tan \left ( dx+c \right ) }+ \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}-1 \right ) ^{-1}}+{\frac{1}{6\,ad} \left ( -i\sqrt [3]{\tan \left ( dx+c \right ) }+ \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}-1 \right ) ^{-1}}+{\frac{17}{24\,ad}\ln \left ( -i\sqrt [3]{\tan \left ( dx+c \right ) }+ \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}-1 \right ) }-{\frac{{\frac{17\,i}{12}}\sqrt{3}}{ad}{\it Artanh} \left ({\frac{\sqrt{3}}{3} \left ( -i+2\,\sqrt [3]{\tan \left ( dx+c \right ) } \right ) } \right ) }+{\frac{1}{8\,ad}\ln \left ( i\sqrt [3]{\tan \left ( dx+c \right ) }+ \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}-1 \right ) }+{\frac{{\frac{i}{4}}\sqrt{3}}{ad}{\it Artanh} \left ({\frac{\sqrt{3}}{3} \left ( i+2\,\sqrt [3]{\tan \left ( dx+c \right ) } \right ) } \right ) }-{\frac{1}{4\,ad}\ln \left ( \sqrt [3]{\tan \left ( dx+c \right ) }-i \right ) }+{\frac{{\frac{i}{6}}}{ad} \left ( \sqrt [3]{\tan \left ( dx+c \right ) }+i \right ) ^{-1}}-{\frac{17}{12\,ad}\ln \left ( \sqrt [3]{\tan \left ( dx+c \right ) }+i \right ) }-{\frac{3}{4\,ad} \left ( \tan \left ( dx+c \right ) \right ) ^{-{\frac{4}{3}}}}+{\frac{3\,i}{ad}{\frac{1}{\sqrt [3]{\tan \left ( dx+c \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 3.01569, size = 2407, normalized size = 6.94 \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 [A] time = 1.17112, size = 325, normalized size = 0.94 \begin{align*} \frac{17 i \, \sqrt{3} \log \left (-\frac{\sqrt{3} - 2 \, \tan \left (d x + c\right )^{\frac{1}{3}} + i}{\sqrt{3} + 2 \, \tan \left (d x + c\right )^{\frac{1}{3}} - i}\right )}{24 \, a d} - \frac{i \, \sqrt{3} \log \left (-\frac{\sqrt{3} - 2 \, \tan \left (d x + c\right )^{\frac{1}{3}} - i}{\sqrt{3} + 2 \, \tan \left (d x + c\right )^{\frac{1}{3}} + i}\right )}{8 \, a d} + \frac{\log \left (\tan \left (d x + c\right )^{\frac{2}{3}} + i \, \tan \left (d x + c\right )^{\frac{1}{3}} - 1\right )}{8 \, a d} + \frac{17 \, \log \left (\tan \left (d x + c\right )^{\frac{2}{3}} - i \, \tan \left (d x + c\right )^{\frac{1}{3}} - 1\right )}{24 \, a d} - \frac{17 \, \log \left (\tan \left (d x + c\right )^{\frac{1}{3}} + i\right )}{12 \, a d} - \frac{\log \left (\tan \left (d x + c\right )^{\frac{1}{3}} - i\right )}{4 \, a d} - \frac{3 \,{\left (-4 i \, \tan \left (d x + c\right ) + 1\right )}}{4 \, a d \tan \left (d x + c\right )^{\frac{4}{3}}} + \frac{i \, \tan \left (d x + c\right )^{\frac{2}{3}}}{2 \, a d{\left (\tan \left (d x + c\right ) - i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]