Optimal. Leaf size=30 \[ \text {Int}\left (\frac {\left (c^2 x^2+1\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.14, 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 = {} \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 14.51, size = 0, normalized size = 0.00 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b x^{3} \operatorname {arsinh}\left (c x\right ) + a^{2} x^{3}}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx \]
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 \[ -\frac {{\left (c^{4} x^{4} + 2 \, c^{2} x^{2} + 1\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (c^{5} x^{5} + 2 \, c^{3} x^{3} + c x\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{3} x^{5} + \sqrt {c^{2} x^{2} + 1} a b c^{2} x^{4} + a b c x^{3} + {\left (b^{2} c^{3} x^{5} + \sqrt {c^{2} x^{2} + 1} b^{2} c^{2} x^{4} + b^{2} c x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )} + \int \frac {{\left (c^{5} x^{5} - 3 \, c^{3} x^{3} - 4 \, c x\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + {\left (2 \, c^{6} x^{6} - 3 \, c^{4} x^{4} - 8 \, c^{2} x^{2} - 3\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (c^{7} x^{7} - 3 \, c^{3} x^{3} - 2 \, c x\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{5} x^{8} + {\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{6} + 2 \, a b c^{3} x^{6} + a b c x^{4} + {\left (b^{2} c^{5} x^{8} + {\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{6} + 2 \, b^{2} c^{3} x^{6} + b^{2} c x^{4} + 2 \, {\left (b^{2} c^{4} x^{7} + b^{2} c^{2} x^{5}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{4} x^{7} + a b c^{2} x^{5}\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (c^2\,x^2+1\right )}^{3/2}}{x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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