Optimal. Leaf size=61 \[ -\frac{4 c \text{Unintegrable}\left (\frac{x}{\left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b}-\frac{1}{b c \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.109984, 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 = {} \[ \int \frac{1}{\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac{1}{b c \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}-\frac{(4 c) \int \frac{x}{\left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b}\\ \end{align*}
Mathematica [A] time = 2.77519, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.182, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}} \left ({c}^{2}{x}^{2}+1 \right ) ^{-{\frac{5}{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 x + \sqrt{c^{2} x^{2} + 1}}{{\left (a b c^{4} x^{3} + a b c^{2} x\right )}{\left (c^{2} x^{2} + 1\right )} +{\left ({\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (b^{2} c^{5} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (a b c^{5} x^{4} + 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt{c^{2} x^{2} + 1}} - \int \frac{4 \, c^{4} x^{4} + 3 \, c^{2} x^{2} +{\left (4 \, c^{2} x^{2} + 1\right )}{\left (c^{2} x^{2} + 1\right )} + 4 \,{\left (2 \, c^{3} x^{3} + c x\right )} \sqrt{c^{2} x^{2} + 1} - 1}{{\left (a b c^{6} x^{6} + 2 \, a b c^{4} x^{4} + a b c^{2} x^{2}\right )}{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 2 \,{\left (a b c^{7} x^{7} + 3 \, a b c^{5} x^{5} + 3 \, a b c^{3} x^{3} + a b c x\right )}{\left (c^{2} x^{2} + 1\right )} +{\left ({\left (b^{2} c^{6} x^{6} + 2 \, b^{2} c^{4} x^{4} + b^{2} c^{2} x^{2}\right )}{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 2 \,{\left (b^{2} c^{7} x^{7} + 3 \, b^{2} c^{5} x^{5} + 3 \, b^{2} c^{3} x^{3} + b^{2} c x\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (b^{2} c^{8} x^{8} + 4 \, b^{2} c^{6} x^{6} + 6 \, b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (a b c^{8} x^{8} + 4 \, a b c^{6} x^{6} + 6 \, a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} + a b\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{\sqrt{c^{2} x^{2} + 1}}{a^{2} c^{6} x^{6} + 3 \, a^{2} c^{4} x^{4} + 3 \, a^{2} c^{2} x^{2} +{\left (b^{2} c^{6} x^{6} + 3 \, b^{2} c^{4} x^{4} + 3 \, b^{2} c^{2} x^{2} + b^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + a^{2} + 2 \,{\left (a b c^{6} x^{6} + 3 \, a b c^{4} x^{4} + 3 \, a b c^{2} x^{2} + a b\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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2} \left (c^{2} x^{2} + 1\right )^{\frac{5}{2}}}\, dx \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 (c^{2} x^{2} + 1\right )}^{\frac{5}{2}}{\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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