Optimal. Leaf size=600 \[ \frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (3,-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (3,-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.63, antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {5892, 5973,
5893, 5983, 5962, 5681, 2221, 2611, 2320, 6724} \begin {gather*} \frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {c x-1} \sqrt {c x+1} \text {Li}_3\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {c x-1} \sqrt {c x+1} \text {Li}_3\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 5681
Rule 5892
Rule 5893
Rule 5962
Rule 5973
Rule 5983
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (h (f+g x)^m\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {\left (g m \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{f+g x} \, dx}{2 b c \sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {\left (g m \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {(a+b x)^2 \sinh (x)}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt {1-c^2 x^2}}\\ &=\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {\left (g m \sqrt {-1+c x} \sqrt {1+c x}\right ) \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,\cosh ^{-1}(c x)\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {\left (g m \sqrt {-1+c x} \sqrt {1+c x}\right ) \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,\cosh ^{-1}(c x)\right )}{2 b c \sqrt {1-c^2 x^2}}\\ &=\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\left (m \sqrt {-1+c x} \sqrt {1+c x}\right ) \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,\cosh ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}+\frac {\left (m \sqrt {-1+c x} \sqrt {1+c x}\right ) \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,\cosh ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {\left (b m \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \text {Li}_2\left (-\frac {e^x g}{c f-\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}+\frac {\left (b m \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \text {Li}_2\left (-\frac {e^x g}{c f+\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {\left (b m \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {g x}{-c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c \sqrt {1-c^2 x^2}}+\frac {\left (b m \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_3\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_3\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [F]
time = 1.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {1-c^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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