Optimal. Leaf size=64 \[ \frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{a^2}+\frac {6 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac {6 x}{a}-\frac {3 x \sinh ^{-1}(a x)^2}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5717, 5653, 8} \[ \frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{a^2}+\frac {6 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac {6 x}{a}-\frac {3 x \sinh ^{-1}(a x)^2}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 5653
Rule 5717
Rubi steps
\begin {align*} \int \frac {x \sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a^2}-\frac {3 \int \sinh ^{-1}(a x)^2 \, dx}{a}\\ &=-\frac {3 x \sinh ^{-1}(a x)^2}{a}+\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a^2}+6 \int \frac {x \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {6 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a^2}-\frac {3 x \sinh ^{-1}(a x)^2}{a}+\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a^2}-\frac {6 \int 1 \, dx}{a}\\ &=-\frac {6 x}{a}+\frac {6 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a^2}-\frac {3 x \sinh ^{-1}(a x)^2}{a}+\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 58, normalized size = 0.91 \[ \frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3+6 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)-6 a x-3 a x \sinh ^{-1}(a x)^2}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 92, normalized size = 1.44 \[ -\frac {3 \, a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 6 \, a x - 6 \, \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.42, size = 101, normalized size = 1.58 \[ \frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{a^{2}} - \frac {3 \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 2 \, a {\left (\frac {x}{a} - \frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}}\right )}\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 90, normalized size = 1.41 \[ \frac {\arcsinh \left (a x \right )^{3} x^{2} a^{2}+\arcsinh \left (a x \right )^{3}-3 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x +6 \arcsinh \left (a x \right ) x^{2} a^{2}+6 \arcsinh \left (a x \right )-6 \sqrt {a^{2} x^{2}+1}\, x a}{a^{2} \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 61, normalized size = 0.95 \[ -\frac {3 \, x \operatorname {arsinh}\left (a x\right )^{2}}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{3}}{a^{2}} - \frac {6 \, {\left (x - \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{a}\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,{\mathrm {asinh}\left (a\,x\right )}^3}{\sqrt {a^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.19, size = 61, normalized size = 0.95 \[ \begin {cases} - \frac {3 x \operatorname {asinh}^{2}{\left (a x \right )}}{a} - \frac {6 x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{a^{2}} + \frac {6 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________