Optimal. Leaf size=62 \[ \frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x^2 \sqrt {\frac {1}{c^2 x^2}+1}}{6 c}-\frac {b \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{6 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6284, 266, 51, 63, 208} \[ \frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x^2 \sqrt {\frac {1}{c^2 x^2}+1}}{6 c}-\frac {b \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{6 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 208
Rule 266
Rule 6284
Rubi steps
\begin {align*} \int x^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b \int \frac {x}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{3 c}\\ &=\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{6 c}\\ &=\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{12 c^3}\\ &=\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b \operatorname {Subst}\left (\int \frac {1}{-c^2+c^2 x^2} \, dx,x,\sqrt {1+\frac {1}{c^2 x^2}}\right )}{6 c}\\ &=\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{6 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 85, normalized size = 1.37 \[ \frac {a x^3}{3}+\frac {b x^2 \sqrt {\frac {c^2 x^2+1}{c^2 x^2}}}{6 c}-\frac {b \log \left (x \left (\sqrt {\frac {c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{6 c^3}+\frac {1}{3} b x^3 \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.58, size = 186, normalized size = 3.00 \[ \frac {2 \, a c^{3} x^{3} + b c^{2} x^{2} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, b c^{3} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 2 \, b c^{3} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + b \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + 2 \, {\left (b c^{3} x^{3} - b c^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{6 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 88, normalized size = 1.42 \[ \frac {\frac {a \,c^{3} x^{3}}{3}+b \left (\frac {c^{3} x^{3} \mathrm {arccsch}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (c x \sqrt {c^{2} x^{2}+1}-\arcsinh \left (c x \right )\right )}{6 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 96, normalized size = 1.55 \[ \frac {1}{3} \, a x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________