Optimal. Leaf size=229 \[ -\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\sinh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\sinh ^{-1}(a x)}}+\frac {16 \sqrt {\pi } \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4} \]
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Rubi [A]
time = 0.30, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5779, 5818,
5778, 3388, 2211, 2235, 2236} \begin {gather*} \frac {16 \sqrt {\pi } \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}-\frac {4 x^2}{5 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac {128 x^3 \sqrt {a^2 x^2+1}}{15 a \sqrt {\sinh ^{-1}(a x)}}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac {16 x \sqrt {a^2 x^2+1}}{5 a^3 \sqrt {\sinh ^{-1}(a x)}}-\frac {16 x^4}{15 \sinh ^{-1}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5778
Rule 5779
Rule 5818
Rubi steps
\begin {align*} \int \frac {x^3}{\sinh ^{-1}(a x)^{7/2}} \, dx &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}+\frac {6 \int \frac {x^2}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (8 a) \int \frac {x^4}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \sinh ^{-1}(a x)^{3/2}}+\frac {64}{15} \int \frac {x^3}{\sinh ^{-1}(a x)^{3/2}} \, dx+\frac {8 \int \frac {x}{\sinh ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\sinh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\sinh ^{-1}(a x)}}+\frac {16 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^4}+\frac {128 \text {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{2 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\sinh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\sinh ^{-1}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^4}+\frac {8 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^4}-\frac {64 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\sinh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\sinh ^{-1}(a x)}}+\frac {32 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a^4}-\frac {32 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a^4}-\frac {32 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a^4}+\frac {32 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a^4}+\frac {16 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{5 a^4}+\frac {16 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{5 a^4}\\ &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\sinh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\sinh ^{-1}(a x)}}+\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{5 a^4}+\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{5 a^4}+\frac {64 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}-\frac {64 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}-\frac {64 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x^4}{15 \sinh ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\sinh ^{-1}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\sinh ^{-1}(a x)}}+\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{15 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 210, normalized size = 0.92 \begin {gather*} \frac {4 \sinh ^{-1}(a x) \left (e^{-2 \sinh ^{-1}(a x)} \left (1-4 \sinh ^{-1}(a x)\right )+e^{2 \sinh ^{-1}(a x)} \left (1+4 \sinh ^{-1}(a x)\right )+4 \sqrt {2} \left (-\sinh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-2 \sinh ^{-1}(a x)\right )+4 \sqrt {2} \sinh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \sinh ^{-1}(a x)\right )\right )-4 \sinh ^{-1}(a x) \left (e^{-4 \sinh ^{-1}(a x)} \left (1-8 \sinh ^{-1}(a x)\right )+e^{4 \sinh ^{-1}(a x)} \left (1+8 \sinh ^{-1}(a x)\right )+16 \left (-\sinh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-4 \sinh ^{-1}(a x)\right )+16 \sinh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \sinh ^{-1}(a x)\right )\right )+6 \sinh \left (2 \sinh ^{-1}(a x)\right )-3 \sinh \left (4 \sinh ^{-1}(a x)\right )}{60 a^4 \sinh ^{-1}(a x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 4.28, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\arcsinh \left (a x \right )^{\frac {7}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\operatorname {asinh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{{\mathrm {asinh}\left (a\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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