Optimal. Leaf size=61 \[ -\frac{1}{3} c^2 \left (\frac{1}{c^2 x^2}+1\right )^{3/2}+c^2 \sqrt{\frac{1}{c^2 x^2}+1}+c^2 \tan ^{-1}(c x)-\frac{1}{3 c x^3}+\frac{c}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0945817, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {6342, 266, 43, 325, 203} \[ -\frac{1}{3} c^2 \left (\frac{1}{c^2 x^2}+1\right )^{3/2}+c^2 \sqrt{\frac{1}{c^2 x^2}+1}+c^2 \tan ^{-1}(c x)-\frac{1}{3 c x^3}+\frac{c}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6342
Rule 266
Rule 43
Rule 325
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(c x)}}{x^3 \left (1+c^2 x^2\right )} \, dx &=\frac{\int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^5} \, dx}{c^2}+\frac{\int \frac{1}{x^4 \left (1+c^2 x^2\right )} \, dx}{c}\\ &=-\frac{1}{3 c x^3}-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 c^2}-c \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{1}{3 c x^3}+\frac{c}{x}-\frac{\operatorname{Subst}\left (\int \left (-\frac{c^2}{\sqrt{1+\frac{x}{c^2}}}+c^2 \sqrt{1+\frac{x}{c^2}}\right ) \, dx,x,\frac{1}{x^2}\right )}{2 c^2}+c^3 \int \frac{1}{1+c^2 x^2} \, dx\\ &=c^2 \sqrt{1+\frac{1}{c^2 x^2}}-\frac{1}{3} c^2 \left (1+\frac{1}{c^2 x^2}\right )^{3/2}-\frac{1}{3 c x^3}+\frac{c}{x}+c^2 \tan ^{-1}(c x)\\ \end{align*}
Mathematica [A] time = 0.147343, size = 54, normalized size = 0.89 \[ \frac{\sqrt{\frac{1}{c^2 x^2}+1} \left (2 c^2 x^2-1\right )}{3 x^2}+c^2 \tan ^{-1}(c x)-\frac{1}{3 c x^3}+\frac{c}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.196, size = 193, normalized size = 3.2 \begin{align*}{\frac{{c}^{2}}{3\,{x}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}} \left ( 3\, \left ({\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}} \right ) ^{3/2}{x}^{2}{c}^{2}-3\,\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}{x}^{4}{c}^{2}-3\,\ln \left ( x+\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}} \right ){x}^{3}+3\,\ln \left ( x+\sqrt{-{\frac{ \left ( x{c}^{2}+\sqrt{-{c}^{2}} \right ) \left ( -x{c}^{2}+\sqrt{-{c}^{2}} \right ) }{{c}^{4}}}} \right ){x}^{3}- \left ({\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}} \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}}}}-{\frac{1}{3\,c{x}^{3}}}+{\frac{c}{x}}+{c}^{2}\arctan \left ( cx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.5894, size = 76, normalized size = 1.25 \begin{align*} c^{2} \arctan \left (c x\right ) + \frac{{\left (2 \, c^{2} x^{2} - 1\right )} \sqrt{c^{2} x^{2} + 1}}{3 \, c x^{3}} + \frac{3 \, c^{2} x^{2} - 1}{3 \, c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.52815, size = 155, normalized size = 2.54 \begin{align*} \frac{3 \, c^{3} x^{3} \arctan \left (c x\right ) + 2 \, c^{3} x^{3} + 3 \, c^{2} x^{2} +{\left (2 \, c^{3} x^{3} - c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - 1}{3 \, c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 5.46128, size = 75, normalized size = 1.23 \begin{align*} - 2 c^{5} \left (\frac{\left (1 + \frac{1}{c^{2} x^{2}}\right )^{\frac{3}{2}}}{6 c^{3}} - \frac{\sqrt{1 + \frac{1}{c^{2} x^{2}}}}{2 c^{3}}\right ) - \frac{c^{3} \operatorname{atan}{\left (\frac{1}{x \sqrt{c^{2}}} \right )}}{\sqrt{c^{2}}} + \frac{c}{x} - \frac{1}{3 c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16573, size = 111, normalized size = 1.82 \begin{align*} c^{2} \arctan \left (c x\right ) + \frac{4 \,{\left (3 \,{\left (x{\left | c \right |} - \sqrt{c^{2} x^{2} + 1}\right )}^{2} - 1\right )} c^{2} \mathrm{sgn}\left (x\right )}{3 \,{\left ({\left (x{\left | c \right |} - \sqrt{c^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{3}} + \frac{3 \, c^{2} x^{2} - 1}{3 \, c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]