Optimal. Leaf size=98 \[ -\frac {2 \sqrt {1+a^2 x^2}}{a \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}}-\frac {4 a \sqrt {1+a^2 x^2} \text {Int}\left (\frac {x}{\left (1+a^2 x^2\right )^2 \sqrt {\sinh ^{-1}(a x)}},x\right )}{c \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 \sqrt {1+a^2 x^2}}{a \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}}-\frac {\left (4 a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^2 \sqrt {\sinh ^{-1}(a x)}} \, dx}{c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 4.89, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arcsinh \left (a x \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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