Optimal. Leaf size=444 \[ -\frac {g \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right ) (f+g x)}+\frac {c^2 f \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac {c^2 f \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}+\frac {b c^2 f \sqrt {c^2 x^2+1} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac {b c^2 f \sqrt {c^2 x^2+1} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}+\frac {b c \sqrt {c^2 x^2+1} \log (f+g x)}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )} \]
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Rubi [A] time = 0.66, antiderivative size = 444, normalized size of antiderivative = 1.00, 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 31
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3322
Rule 3324
Rule 5831
Rule 5835
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*}
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Mathematica [A] time = 2.17, size = 448, normalized size = 1.01 \[ \frac {-a g \left (c^2 d x^2+d\right ) \sqrt {c^2 f^2+g^2}-a c^2 \sqrt {d} f \sqrt {c^2 d x^2+d} (f+g x) \log \left (\sqrt {d} \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}+d \left (g-c^2 f x\right )\right )+a c^2 \sqrt {d} f \sqrt {c^2 d x^2+d} (f+g x) \log (f+g x)-b d \sqrt {c^2 x^2+1} \left (-c^2 f (f+g x) \text {Li}_2\left (\frac {e^{\sinh ^{-1}(c x)} g}{\sqrt {c^2 f^2+g^2}-c f}\right )+c^2 f (f+g x) \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )+g \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2} \sinh ^{-1}(c x)-c \sqrt {c^2 f^2+g^2} (f+g x) \log (c (f+g x))+c^2 (-f) \sinh ^{-1}(c x) (f+g x) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )+c^2 f \sinh ^{-1}(c x) (f+g x) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )\right )}{d \sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2} (f+g x)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{c^{2} d g^{2} x^{4} + 2 \, c^{2} d f g x^{3} + 2 \, d f g x + d f^{2} + {\left (c^{2} d f^{2} + d g^{2}\right )} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.69, size = 1770, normalized size = 3.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (f+g\,x\right )}^2\,\sqrt {d\,c^2\,x^2+d}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (f + g x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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