Integrand size = 32, antiderivative size = 332 \[ \int \frac {(a+b \text {arcsinh}(c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\frac {m (a+b \text {arcsinh}(c x))^3}{6 b^2 c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{2 b c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{2 b c}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c} \]
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Time = 0.38 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5783, 5846, 5827, 5680, 2221, 2611, 2320, 6724} \[ \int \frac {(a+b \text {arcsinh}(c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\frac {m (a+b \text {arcsinh}(c x))^3}{6 b^2 c}-\frac {m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{2 b c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{2 b c}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 5680
Rule 5783
Rule 5827
Rule 5846
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {(g m) \int \frac {(a+b \text {arcsinh}(c x))^2}{f+g x} \, dx}{2 b c} \\ & = \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {(g m) \text {Subst}\left (\int \frac {(a+b x)^2 \cosh (x)}{c f+g \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{2 b c} \\ & = \frac {m (a+b \text {arcsinh}(c x))^3}{6 b^2 c}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {(g m) \text {Subst}\left (\int \frac {e^x (a+b x)^2}{c f+e^x g-\sqrt {c^2 f^2+g^2}} \, dx,x,\text {arcsinh}(c x)\right )}{2 b c}-\frac {(g m) \text {Subst}\left (\int \frac {e^x (a+b x)^2}{c f+e^x g+\sqrt {c^2 f^2+g^2}} \, dx,x,\text {arcsinh}(c x)\right )}{2 b c} \\ & = \frac {m (a+b \text {arcsinh}(c x))^3}{6 b^2 c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{2 b c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{2 b c}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}+\frac {m \text {Subst}\left (\int (a+b x) \log \left (1+\frac {e^x g}{c f-\sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c}+\frac {m \text {Subst}\left (\int (a+b x) \log \left (1+\frac {e^x g}{c f+\sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c} \\ & = \frac {m (a+b \text {arcsinh}(c x))^3}{6 b^2 c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{2 b c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{2 b c}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {(b m) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {e^x g}{c f-\sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c}+\frac {(b m) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {e^x g}{c f+\sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c} \\ & = \frac {m (a+b \text {arcsinh}(c x))^3}{6 b^2 c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{2 b c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{2 b c}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {(b m) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {g x}{-c f+\sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c}+\frac {(b m) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {g x}{c f+\sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c} \\ & = \frac {m (a+b \text {arcsinh}(c x))^3}{6 b^2 c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{2 b c}-\frac {m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{2 b c}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {b m \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \text {arcsinh}(c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\frac {\frac {m (a+b \text {arcsinh}(c x))^3}{3 b}-m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )-m (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )+(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )+2 b m \left (-\left ((a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )\right )+b \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )\right )+2 b m \left (-\left ((a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )+b \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{2 b c} \]
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\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right ) \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {c^{2} x^{2}+1}}d x\]
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\[ \int \frac {(a+b \text {arcsinh}(c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {c^{2} x^{2} + 1}}\, dx \]
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\[ \int \frac {(a+b \text {arcsinh}(c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {c^2\,x^2+1}} \,d x \]
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