Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.03, 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 = {} \[ \int \frac {1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \frac {1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 5.21, size = 0, normalized size = 0.00 \[ \int \frac {1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} e^{2} x^{2} + 2 \, a^{2} d e x + a^{2} d^{2} + {\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b e^{2} x^{2} + 2 \, a b d e x + a b d^{2}\right )} \operatorname {arsinh}\left (c x\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (e x + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x +d \right )^{2} \left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx \]
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 {c^{3} x^{3} + c x + {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a b c^{3} e^{2} x^{4} + 2 \, a b c^{3} d e x^{3} + 2 \, a b c d e x + a b c d^{2} + {\left (c^{3} d^{2} + c e^{2}\right )} a b x^{2} + {\left (b^{2} c^{3} e^{2} x^{4} + 2 \, b^{2} c^{3} d e x^{3} + 2 \, b^{2} c d e x + b^{2} c d^{2} + {\left (c^{3} d^{2} + c e^{2}\right )} b^{2} x^{2} + {\left (b^{2} c^{2} e^{2} x^{3} + 2 \, b^{2} c^{2} d e x^{2} + b^{2} c^{2} d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a b c^{2} e^{2} x^{3} + 2 \, a b c^{2} d e x^{2} + a b c^{2} d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}} - \int \frac {c^{5} e x^{5} - c^{5} d x^{4} + 2 \, c^{3} e x^{3} - 2 \, c^{3} d x^{2} + c e x + {\left (c^{3} e x^{3} - c^{3} d x^{2} + 3 \, c e x + c d\right )} {\left (c^{2} x^{2} + 1\right )} - c d + {\left (2 \, c^{4} e x^{4} - 2 \, c^{4} d x^{3} + 5 \, c^{2} e x^{2} - c^{2} d x + 2 \, e\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{5} e^{3} x^{7} + 3 \, a b c^{5} d e^{2} x^{6} + {\left (3 \, c^{5} d^{2} e + 2 \, c^{3} e^{3}\right )} a b x^{5} + 3 \, a b c d^{2} e x + {\left (c^{5} d^{3} + 6 \, c^{3} d e^{2}\right )} a b x^{4} + a b c d^{3} + {\left (6 \, c^{3} d^{2} e + c e^{3}\right )} a b x^{3} + {\left (2 \, c^{3} d^{3} + 3 \, c d e^{2}\right )} a b x^{2} + {\left (a b c^{3} e^{3} x^{5} + 3 \, a b c^{3} d e^{2} x^{4} + 3 \, a b c^{3} d^{2} e x^{3} + a b c^{3} d^{3} x^{2}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (b^{2} c^{5} e^{3} x^{7} + 3 \, b^{2} c^{5} d e^{2} x^{6} + {\left (3 \, c^{5} d^{2} e + 2 \, c^{3} e^{3}\right )} b^{2} x^{5} + 3 \, b^{2} c d^{2} e x + {\left (c^{5} d^{3} + 6 \, c^{3} d e^{2}\right )} b^{2} x^{4} + b^{2} c d^{3} + {\left (6 \, c^{3} d^{2} e + c e^{3}\right )} b^{2} x^{3} + {\left (2 \, c^{3} d^{3} + 3 \, c d e^{2}\right )} b^{2} x^{2} + {\left (b^{2} c^{3} e^{3} x^{5} + 3 \, b^{2} c^{3} d e^{2} x^{4} + 3 \, b^{2} c^{3} d^{2} e x^{3} + b^{2} c^{3} d^{3} x^{2}\right )} {\left (c^{2} x^{2} + 1\right )} + 2 \, {\left (b^{2} c^{4} e^{3} x^{6} + 3 \, b^{2} c^{4} d e^{2} x^{5} + 3 \, b^{2} c^{2} d^{2} e x^{2} + b^{2} c^{2} d^{3} x + {\left (3 \, c^{4} d^{2} e + c^{2} e^{3}\right )} b^{2} x^{4} + {\left (c^{4} d^{3} + 3 \, c^{2} d e^{2}\right )} b^{2} x^{3}\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} e^{3} x^{6} + 3 \, a b c^{4} d e^{2} x^{5} + 3 \, a b c^{2} d^{2} e x^{2} + a b c^{2} d^{3} x + {\left (3 \, c^{4} d^{2} e + c^{2} e^{3}\right )} a b x^{4} + {\left (c^{4} d^{3} + 3 \, c^{2} d e^{2}\right )} a b x^{3}\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 = -0.05 \[ \int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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