Optimal. Leaf size=94 \[ \frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15 \sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}-\frac {15 \sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5772, 5798,
5819, 3389, 2211, 2235, 2236} \begin {gather*} -\frac {5 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{2 a}+\frac {15 \sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}-\frac {15 \sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5772
Rule 5798
Rule 5819
Rubi steps
\begin {align*} \int \sinh ^{-1}(a x)^{5/2} \, dx &=x \sinh ^{-1}(a x)^{5/2}-\frac {1}{2} (5 a) \int \frac {x \sinh ^{-1}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15}{4} \int \sqrt {\sinh ^{-1}(a x)} \, dx\\ &=\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}-\frac {1}{8} (15 a) \int \frac {x}{\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \, dx\\ &=\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a}\\ &=\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a}-\frac {15 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a}\\ &=\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a}-\frac {15 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a}\\ &=\frac {15}{4} x \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{2 a}+x \sinh ^{-1}(a x)^{5/2}+\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{16 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 45, normalized size = 0.48 \begin {gather*} -\frac {\frac {\sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-\sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}+\Gamma \left (\frac {7}{2},\sinh ^{-1}(a x)\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 2.53, size = 78, normalized size = 0.83
method | result | size |
default | \(-\frac {-16 \arcsinh \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, a x +40 \arcsinh \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}-60 \sqrt {\arcsinh \left (a x \right )}\, \sqrt {\pi }\, a x -15 \pi \erf \left (\sqrt {\arcsinh \left (a x \right )}\right )+15 \pi \erfi \left (\sqrt {\arcsinh \left (a x \right )}\right )}{16 \sqrt {\pi }\, a}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________