Optimal. Leaf size=40 \[ -\frac {2 \sqrt {1+a^2 x^2}}{a \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {5783}
\begin {gather*} -\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5783
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2}}{a \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 40, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {1+a^2 x^2}}{a \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.23, size = 36, normalized size = 0.90
method | result | size |
default | \(-\frac {2 \sqrt {a^{2} x^{2}+1}}{\sqrt {\arcsinh \left (a x \right )}\, a \sqrt {c \left (a^{2} x^{2}+1\right )}}\) | \(36\) |
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 [A]
time = 0.34, size = 57, normalized size = 1.42 \begin {gather*} -\frac {2 \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{3} c x^{2} + a c\right )} \sqrt {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}} \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}{\sqrt {c \left (a^{2} x^{2} + 1\right )} \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 [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 = -0.02 \begin {gather*} \int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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