Optimal. Leaf size=174 \[ -\frac{2 d \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}},x\right )}{b c}+\frac{\sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{2 d \left (c^2 x^2+1\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}} \]
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Rubi [A] time = 0.796411, 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 = {} \[ \int \frac{d+c^2 d x^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{d+c^2 d x^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{(2 d) \int \frac{\sqrt{1+c^2 x^2}}{x^2 \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}+\frac{(4 c d) \int \frac{\sqrt{1+c^2 x^2}}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(4 d) \operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \left (\frac{c^2}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}\right ) \, dx}{b c}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(4 d) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}+\frac{\cosh (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}-\frac{(2 c d) \int \frac{1}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(2 d) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}+\frac{d \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(2 d) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2}+\frac{(2 d) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}+\frac{d e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{(2 d) \int \frac{1}{x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ \end{align*}
Mathematica [A] time = 4.07211, size = 0, normalized size = 0. \[ \int \frac{d+c^2 d x^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.184, size = 0, normalized size = 0. \begin{align*} \int{\frac{{c}^{2}d{x}^{2}+d}{x} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{-{\frac{3}{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*} \int \frac{c^{2} d x^{2} + d}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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*} d \left (\int \frac{c^{2} x^{2}}{a x \sqrt{a + b \operatorname{asinh}{\left (c x \right )}} + b x \sqrt{a + b \operatorname{asinh}{\left (c x \right )}} \operatorname{asinh}{\left (c x \right )}}\, dx + \int \frac{1}{a x \sqrt{a + b \operatorname{asinh}{\left (c x \right )}} + b x \sqrt{a + b \operatorname{asinh}{\left (c x \right )}} \operatorname{asinh}{\left (c x \right )}}\, dx\right ) \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{c^{2} d x^{2} + d}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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