Optimal. Leaf size=27 \[ \frac {\text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^4}-\frac {3 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5779, 3312, 3298} \[ \frac {\text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^4}-\frac {3 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3298
Rule 3312
Rule 5779
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sinh ^3(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {i \operatorname {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 x}-\frac {i \sinh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}-\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {3 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4}+\frac {\text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 22, normalized size = 0.81 \[ \frac {\text {Shi}\left (3 \sinh ^{-1}(a x)\right )-3 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 23, normalized size = 0.85 \[ -\frac {3 \Shi \left (\arcsinh \left (a x \right )\right )-\Shi \left (3 \arcsinh \left (a x \right )\right )}{4 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^3}{\mathrm {asinh}\left (a\,x\right )\,\sqrt {a^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________