Optimal. Leaf size=132 \[ -\frac{\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac{\sqrt [3]{b} \log \left (b^{2/3} (b \coth (c+d x))^{2/3}+b^{4/3}+(b \coth (c+d x))^{4/3}\right )}{4 d}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{b^{2/3}+2 (b \coth (c+d x))^{2/3}}{\sqrt{3} b^{2/3}}\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.109114, antiderivative size = 132, 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, 292, 31, 634, 617, 204, 628} \[ -\frac{\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac{\sqrt [3]{b} \log \left (b^{2/3} (b \coth (c+d x))^{2/3}+b^{4/3}+(b \coth (c+d x))^{4/3}\right )}{4 d}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{b^{2/3}+2 (b \coth (c+d x))^{2/3}}{\sqrt{3} b^{2/3}}\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3476
Rule 329
Rule 275
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \sqrt [3]{b \coth (c+d x)} \, dx &=-\frac{b \operatorname{Subst}\left (\int \frac{\sqrt [3]{x}}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=-\frac{(3 b) \operatorname{Subst}\left (\int \frac{x^3}{-b^2+x^6} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}\\ &=-\frac{(3 b) \operatorname{Subst}\left (\int \frac{x}{-b^2+x^3} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 d}\\ &=-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{-b^{2/3}+x} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{-b^{2/3}+x}{b^{4/3}+b^{2/3} x+x^2} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 d}\\ &=-\frac{\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{b^{2/3}+2 x}{b^{4/3}+b^{2/3} x+x^2} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{4 d}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{b^{4/3}+b^{2/3} x+x^2} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{4 d}\\ &=-\frac{\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac{\sqrt [3]{b} \log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 d}+\frac{\left (3 \sqrt [3]{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 (b \coth (c+d x))^{2/3}}{b^{2/3}}\right )}{2 d}\\ &=-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1+\frac{2 (b \coth (c+d x))^{2/3}}{b^{2/3}}}{\sqrt{3}}\right )}{2 d}-\frac{\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac{\sqrt [3]{b} \log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 d}\\ \end{align*}
Mathematica [C] time = 0.040844, size = 38, normalized size = 0.29 \[ \frac{3 (b \coth (c+d x))^{4/3} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\coth ^2(c+d x)\right )}{4 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.015, size = 115, normalized size = 0.9 \begin{align*} -{\frac{b}{2\,d}\ln \left ( \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{{\frac{2}{3}}}-\sqrt [3]{{b}^{2}} \right ){\frac{1}{\sqrt [3]{{b}^{2}}}}}+{\frac{b}{4\,d}\ln \left ( \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{{\frac{4}{3}}}+\sqrt [3]{{b}^{2}} \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{{\frac{2}{3}}}+ \left ({b}^{2} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{b}^{2}}}}}-{\frac{b\sqrt{3}}{2\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{ \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{2/3}}{\sqrt [3]{{b}^{2}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \coth \left (d x + c\right )\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.00133, size = 828, normalized size = 6.27 \begin{align*} -\frac{2 \, \sqrt{3} \left (-b\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3} b - 2 \, \sqrt{3} \left (-b\right )^{\frac{1}{3}} \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{2}{3}}}{3 \, b}\right ) - 2 \, \left (-b\right )^{\frac{1}{3}} \log \left (-\left (-b\right )^{\frac{2}{3}} + \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{2}{3}}\right ) + \left (-b\right )^{\frac{1}{3}} \log \left (\frac{{\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \left (-b\right )^{\frac{2}{3}} \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{2}{3}} -{\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \left (-b\right )^{\frac{1}{3}} +{\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + b\right )} \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{1}{3}}}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1}\right )}{4 \, d} \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 \sqrt [3]{b \coth{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.68703, size = 293, normalized size = 2.22 \begin{align*} -\frac{b{\left (\frac{2 \, \sqrt{3}{\left | b \right |}^{\frac{4}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac{2}{3}} +{\left | b \right |}^{\frac{2}{3}}\right )}}{3 \,{\left | b \right |}^{\frac{2}{3}}}\right )}{b^{2}} - \frac{{\left | b \right |}^{\frac{4}{3}} \log \left (\left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac{2}{3}}{\left | b \right |}^{\frac{2}{3}} +{\left | b \right |}^{\frac{4}{3}} + \frac{{\left (b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac{1}{3}}}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )}{b^{2}} + \frac{2 \,{\left | b \right |}^{\frac{4}{3}} \log \left ({\left | \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac{2}{3}} -{\left | b \right |}^{\frac{2}{3}} \right |}\right )}{b^{2}}\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]