Optimal. Leaf size=325 \[ \frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}}-\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}}+\frac {b \sqrt {c^2 x^2+1} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}}-\frac {b \sqrt {c^2 x^2+1} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}} \]
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Rubi [A] time = 0.55, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {5835, 5831, 3322, 2264, 2190, 2279, 2391} \[ \frac {b \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}}-\frac {b \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}}+\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}}-\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3322
Rule 5831
Rule 5835
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{(f+g x) \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) \sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}\\ &=\frac {\sqrt {1+c^2 x^2} \operatorname {Subst}\left (\int \frac {a+b x}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}}\\ &=\frac {\left (2 \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 )}{\sqrt {d+c^2 d x^2}}\\ &=\frac {\left (2 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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {\left (2 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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}\\ &=\frac {\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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {\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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {\left (b \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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}+\frac {\left (b \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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}\\ &=\frac {\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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {\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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {\left (b \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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}+\frac {\left (b \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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}\\ &=\frac {\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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {\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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}+\frac {b \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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {b \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 )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 256, normalized size = 0.79 \[ \frac {-\frac {a \log \left (\sqrt {d} \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}+d \left (g-c^2 f x\right )\right )}{\sqrt {d}}+\frac {a \log (f+g x)}{\sqrt {d}}+\frac {b \sqrt {c^2 x^2+1} \left (\text {Li}_2\left (\frac {e^{\sinh ^{-1}(c x)} g}{\sqrt {c^2 f^2+g^2}-c f}\right )-\text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )+\sinh ^{-1}(c x) \left (\log \left (\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )-\log \left (\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )\right )\right )}{\sqrt {c^2 d x^2+d}}}{\sqrt {c^2 f^2+g^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{c^{2} d g x^{3} + c^{2} d f x^{2} + d g x + d f}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {c^{2} d x^{2} + d} {\left (g x + f\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 678, normalized size = 2.09 \[ -\frac {a \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \arcsinh \left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \arcsinh \left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \dilog \left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \dilog \left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )} \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {c^{2} d x^{2} + d} {\left (g x + f\right )}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{\left (f+g\,x\right )\,\sqrt {d\,c^2\,x^2+d}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (f + g x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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