Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{\left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.0382917, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \frac{1}{\left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ \end{align*}
Mathematica [A] time = 11.3646, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.216, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( e{x}^{2}+d \right ) \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c^{3} x^{3} + c x +{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{a b c^{3} e x^{4} +{\left (c^{3} d + c e\right )} a b x^{2} + a b c d +{\left (b^{2} c^{3} e x^{4} +{\left (c^{3} d + c e\right )} b^{2} x^{2} + b^{2} c d +{\left (b^{2} c^{2} e x^{3} + b^{2} c^{2} d 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 x^{3} + a b c^{2} d x\right )} \sqrt{c^{2} x^{2} + 1}} - \int \frac{c^{5} e x^{6} -{\left (c^{5} d - 2 \, c^{3} e\right )} x^{4} -{\left (2 \, c^{3} d - c e\right )} x^{2} +{\left (c^{3} e x^{4} -{\left (c^{3} d - 3 \, c e\right )} x^{2} + c d\right )}{\left (c^{2} x^{2} + 1\right )} - c d +{\left (2 \, c^{4} e x^{5} -{\left (2 \, c^{4} d - 5 \, c^{2} e\right )} x^{3} -{\left (c^{2} d - 2 \, e\right )} x\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{5} e^{2} x^{8} + 2 \,{\left (c^{5} d e + c^{3} e^{2}\right )} a b x^{6} +{\left (c^{5} d^{2} + 4 \, c^{3} d e + c e^{2}\right )} a b x^{4} + a b c d^{2} + 2 \,{\left (c^{3} d^{2} + c d e\right )} a b x^{2} +{\left (a b c^{3} e^{2} x^{6} + 2 \, a b c^{3} d e x^{4} + a b c^{3} d^{2} x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (b^{2} c^{5} e^{2} x^{8} + 2 \,{\left (c^{5} d e + c^{3} e^{2}\right )} b^{2} x^{6} +{\left (c^{5} d^{2} + 4 \, c^{3} d e + c e^{2}\right )} b^{2} x^{4} + b^{2} c d^{2} + 2 \,{\left (c^{3} d^{2} + c d e\right )} b^{2} x^{2} +{\left (b^{2} c^{3} e^{2} x^{6} + 2 \, b^{2} c^{3} d e x^{4} + b^{2} c^{3} d^{2} x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} + 2 \,{\left (b^{2} c^{4} e^{2} x^{7} +{\left (2 \, c^{4} d e + c^{2} e^{2}\right )} b^{2} x^{5} + b^{2} c^{2} d^{2} x +{\left (c^{4} d^{2} + 2 \, c^{2} d e\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^{2} x^{7} +{\left (2 \, c^{4} d e + c^{2} e^{2}\right )} a b x^{5} + a b c^{2} d^{2} x +{\left (c^{4} d^{2} + 2 \, c^{2} d e\right )} a b x^{3}\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{2} e x^{2} + a^{2} d +{\left (b^{2} e x^{2} + b^{2} d\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b e x^{2} + a b d\right )} \operatorname{arsinh}\left (c x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (e x^{2} + d\right )}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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