Optimal. Leaf size=143 \[ -\frac {2 \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {2 e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}+\frac {2 e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c} \]
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Rubi [A]
time = 0.18, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5773, 5818,
5774, 3388, 2211, 2236, 2235} \begin {gather*} \frac {2 \sqrt {\pi } e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}+\frac {2 \sqrt {\pi } e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}-\frac {4 x}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 \sqrt {c^2 x^2+1}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5773
Rule 5774
Rule 5818
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \, dx &=-\frac {2 \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}+\frac {(2 c) \int \frac {x}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {2 \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {4 \int \frac {1}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{3 b^2}\\ &=-\frac {2 \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {4 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{3 b^3 c}\\ &=-\frac {2 \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {2 \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{3 b^3 c}+\frac {2 \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{3 b^3 c}\\ &=-\frac {2 \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {4 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{3 b^3 c}+\frac {4 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{3 b^3 c}\\ &=-\frac {2 \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}+\frac {2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 181, normalized size = 1.27 \begin {gather*} \frac {e^{-\frac {a+b \sinh ^{-1}(c x)}{b}} \left (-e^{a/b} \left (b+2 a \left (-1+e^{2 \sinh ^{-1}(c x)}\right )-2 b \sinh ^{-1}(c x)+b e^{2 \sinh ^{-1}(c x)} \left (1+2 \sinh ^{-1}(c x)\right )\right )-2 e^{\frac {2 a}{b}+\sinh ^{-1}(c x)} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \left (a+b \sinh ^{-1}(c x)\right ) \Gamma \left (\frac {1}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )-2 b e^{\sinh ^{-1}(c x)} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )\right )}{3 b^2 c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \arcsinh \left (c x \right )\right )^{\frac {5}{2}}}\, dx\]
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 {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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