Optimal. Leaf size=444 \[ \frac{b c^2 f \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}\right )}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac{b c^2 f \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}\right )}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac{g \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right ) (f+g x)}+\frac{c^2 f \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right )}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac{c^2 f \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right )}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}+\frac{b c \sqrt{c^2 x^2+1} \log (f+g x)}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )} \]
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Rubi [A] time = 0.660769, antiderivative size = 444, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5835, 5831, 3324, 3322, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{b c^2 f \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}\right )}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac{b c^2 f \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}\right )}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac{g \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right ) (f+g x)}+\frac{c^2 f \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right )}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac{c^2 f \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right )}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}+\frac{b c \sqrt{c^2 x^2+1} \log (f+g x)}{\sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )} \]
Antiderivative was successfully verified.
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Rule 5835
Rule 5831
Rule 3324
Rule 3322
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{(f+g x)^2 \sqrt{d+c^2 d x^2}} \, dx &=\frac{\sqrt{1+c^2 x^2} \int \frac{a+b \sinh ^{-1}(c x)}{(f+g x)^2 \sqrt{1+c^2 x^2}} \, dx}{\sqrt{d+c^2 d x^2}}\\ &=\frac{\left (c \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{a+b x}{(c f+g \sinh (x))^2} \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{d+c^2 d x^2}}\\ &=-\frac{g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt{d+c^2 d x^2}}+\frac{\left (c^2 f \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{a+b x}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}+\frac{\left (b c g \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}\\ &=-\frac{g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt{d+c^2 d x^2}}+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c f+x} \, dx,x,c g x\right )}{\left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}+\frac{\left (2 c^2 f \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 c e^x f-g+e^{2 x} g} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}\\ &=-\frac{g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt{d+c^2 d x^2}}+\frac{b c \sqrt{1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}+\frac{\left (2 c^2 f g \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 c f+2 e^x g-2 \sqrt{c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}-\frac{\left (2 c^2 f g \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 c f+2 e^x g+2 \sqrt{c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}\\ &=-\frac{g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt{d+c^2 d x^2}}+\frac{c^2 f \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}-\frac{c^2 f \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}+\frac{b c \sqrt{1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}-\frac{\left (b c^2 f \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f-2 \sqrt{c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}+\frac{\left (b c^2 f \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f+2 \sqrt{c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}\\ &=-\frac{g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt{d+c^2 d x^2}}+\frac{c^2 f \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}-\frac{c^2 f \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}+\frac{b c \sqrt{1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}-\frac{\left (b c^2 f \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f-2 \sqrt{c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}+\frac{\left (b c^2 f \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f+2 \sqrt{c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}\\ &=-\frac{g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt{d+c^2 d x^2}}+\frac{c^2 f \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}-\frac{c^2 f \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}+\frac{b c \sqrt{1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}}+\frac{b c^2 f \sqrt{1+c^2 x^2} \text{Li}_2\left (-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}-\frac{b c^2 f \sqrt{1+c^2 x^2} \text{Li}_2\left (-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 2.2051, size = 448, normalized size = 1.01 \[ \frac{-b d \sqrt{c^2 x^2+1} \left (-c^2 f (f+g x) \text{PolyLog}\left (2,\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}-c f}\right )+c^2 f (f+g x) \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}\right )+g \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2} \sinh ^{-1}(c x)-c \sqrt{c^2 f^2+g^2} (f+g x) \log (c (f+g x))+c^2 (-f) \sinh ^{-1}(c x) (f+g x) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right )+c^2 f \sinh ^{-1}(c x) (f+g x) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right )\right )-a g \left (c^2 d x^2+d\right ) \sqrt{c^2 f^2+g^2}-a c^2 \sqrt{d} f \sqrt{c^2 d x^2+d} (f+g x) \log \left (\sqrt{d} \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2}+d \left (g-c^2 f x\right )\right )+a c^2 \sqrt{d} f \sqrt{c^2 d x^2+d} (f+g x) \log (f+g x)}{d \sqrt{c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2} (f+g x)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.31, size = 1770, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (c x\right ) + a}{\sqrt{c^{2} d x^{2} + d}{\left (g x + f\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{c^{2} d g^{2} x^{4} + 2 \, c^{2} d f g x^{3} + 2 \, d f g x + d f^{2} +{\left (c^{2} d f^{2} + d g^{2}\right )} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asinh}{\left (c x \right )}}{\sqrt{d \left (c^{2} x^{2} + 1\right )} \left (f + g x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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