Optimal. Leaf size=131 \[ -\frac{\log \left ((c \cot (a+b x))^{2/3}+c^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac{\log \left (-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}+c^{4/3}\right )}{4 b \sqrt [3]{c}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt{3} c^{2/3}}\right )}{2 b \sqrt [3]{c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0963444, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {3476, 329, 275, 200, 31, 634, 617, 204, 628} \[ -\frac{\log \left ((c \cot (a+b x))^{2/3}+c^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac{\log \left (-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}+c^{4/3}\right )}{4 b \sqrt [3]{c}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt{3} c^{2/3}}\right )}{2 b \sqrt [3]{c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3476
Rule 329
Rule 275
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{c \cot (a+b x)}} \, dx &=-\frac{c \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{x} \left (c^2+x^2\right )} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=-\frac{(3 c) \operatorname{Subst}\left (\int \frac{x}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}\\ &=-\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{c^2+x^3} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{c^{2/3}+x} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}-\frac{\operatorname{Subst}\left (\int \frac{2 c^{2/3}-x}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}\\ &=-\frac{\log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac{\operatorname{Subst}\left (\int \frac{-c^{2/3}+2 x}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{4 b \sqrt [3]{c}}-\frac{\left (3 \sqrt [3]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{4 b}\\ &=-\frac{\log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac{\log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b \sqrt [3]{c}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 (c \cot (a+b x))^{2/3}}{c^{2/3}}\right )}{2 b \sqrt [3]{c}}\\ &=\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 (c \cot (a+b x))^{2/3}}{c^{2/3}}}{\sqrt{3}}\right )}{2 b \sqrt [3]{c}}-\frac{\log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac{\log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b \sqrt [3]{c}}\\ \end{align*}
Mathematica [A] time = 0.150585, size = 98, normalized size = 0.75 \[ \frac{\sqrt [3]{\cot (a+b x)} \left (-2 \log \left (\cot ^{\frac{2}{3}}(a+b x)+1\right )+\log \left (\cot ^{\frac{4}{3}}(a+b x)-\cot ^{\frac{2}{3}}(a+b x)+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \cot ^{\frac{2}{3}}(a+b x)-1}{\sqrt{3}}\right )\right )}{4 b \sqrt [3]{c \cot (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.024, size = 114, normalized size = 0.9 \begin{align*} -{\frac{c}{2\,b}\ln \left ( \left ( c\cot \left ( bx+a \right ) \right ) ^{{\frac{2}{3}}}+\sqrt [3]{{c}^{2}} \right ) \left ({c}^{2} \right ) ^{-{\frac{2}{3}}}}+{\frac{c}{4\,b}\ln \left ( \left ( c\cot \left ( bx+a \right ) \right ) ^{{\frac{4}{3}}}-\sqrt [3]{{c}^{2}} \left ( c\cot \left ( bx+a \right ) \right ) ^{{\frac{2}{3}}}+ \left ({c}^{2} \right ) ^{{\frac{2}{3}}} \right ) \left ({c}^{2} \right ) ^{-{\frac{2}{3}}}}-{\frac{c\sqrt{3}}{2\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{ \left ( c\cot \left ( bx+a \right ) \right ) ^{2/3}}{\sqrt [3]{{c}^{2}}}}-1 \right ) } \right ) \left ({c}^{2} \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.57281, size = 163, normalized size = 1.24 \begin{align*} -\frac{c{\left (\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, \left (\frac{c}{\tan \left (b x + a\right )}\right )^{\frac{2}{3}} -{\left (c^{2}\right )}^{\frac{1}{3}}\right )}}{3 \,{\left (c^{2}\right )}^{\frac{1}{3}}}\right )}{{\left (c^{2}\right )}^{\frac{2}{3}}} - \frac{\log \left (\left (\frac{c}{\tan \left (b x + a\right )}\right )^{\frac{4}{3}} -{\left (c^{2}\right )}^{\frac{1}{3}} \left (\frac{c}{\tan \left (b x + a\right )}\right )^{\frac{2}{3}} +{\left (c^{2}\right )}^{\frac{2}{3}}\right )}{{\left (c^{2}\right )}^{\frac{2}{3}}} + \frac{2 \, \log \left (\left (\frac{c}{\tan \left (b x + a\right )}\right )^{\frac{2}{3}} +{\left (c^{2}\right )}^{\frac{1}{3}}\right )}{{\left (c^{2}\right )}^{\frac{2}{3}}}\right )}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.77121, size = 1639, normalized size = 12.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{c \cot{\left (a + b x \right )}}}\, dx \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 \frac{1}{\left (c \cot \left (b x + a\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]