Optimal. Leaf size=562 \[ \frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e^2}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^2}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^2}+\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^2}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^2}+\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^2}-\frac {b c \sqrt {d} \sqrt {c^2 x^2-1} \tanh ^{-1}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 e^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d+e}} \]
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Rubi [A] time = 0.99, antiderivative size = 562, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5792, 5788, 519, 377, 208, 5800, 5562, 2190, 2279, 2391} \[ \frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^2}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^2}+\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^2}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e^2}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^2}-\frac {b c \sqrt {d} \sqrt {c^2 x^2-1} \tanh ^{-1}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 e^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d+e}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 377
Rule 519
Rule 2190
Rule 2279
Rule 2391
Rule 5562
Rule 5788
Rule 5792
Rule 5800
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac {d x \left (a+b \cosh ^{-1}(c x)\right )}{e \left (d+e x^2\right )^2}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d+e x^2} \, dx}{e}-\frac {d \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {(b c d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )} \, dx}{2 e^2}+\frac {\int \left (-\frac {a+b \cosh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e}\\ &=\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {\int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^{3/2}}+\frac {\int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^{3/2}}-\frac {\left (b c d \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{2 e^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{3/2}}-\frac {\left (b c d \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{d-\left (c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^2}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{3/2}}\\ &=\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^2}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^2}\\ &=\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^2}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^2}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^2}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^2}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^2}\\ &=\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^2}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}\\ \end {align*}
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Mathematica [C] time = 2.15, size = 693, normalized size = 1.23 \[ \frac {\frac {2 a d}{d+e x^2}+2 a \log \left (d+e x^2\right )+b \left (2 \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d c^2-e}-i c \sqrt {d}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d c^2-e}-i c \sqrt {d}}\right )+2 \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i \sqrt {d} c+\sqrt {-d c^2-e}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i \sqrt {d} c+\sqrt {-d c^2-e}}\right )+2 \cosh ^{-1}(c x) \left (\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}-i c \sqrt {d}}\right )+\log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+i c \sqrt {d}}\right )\right )+2 \cosh ^{-1}(c x) \left (\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{-\sqrt {c^2 (-d)-e}+i c \sqrt {d}}\right )+\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+i c \sqrt {d}}\right )\right )-i \sqrt {d} \left (\frac {c \log \left (\frac {2 e \left (-i \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 (-d)-e}+c^2 \sqrt {d} x+i \sqrt {e}\right )}{c \sqrt {c^2 (-d)-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {c^2 (-d)-e}}+\frac {\cosh ^{-1}(c x)}{\sqrt {e} x-i \sqrt {d}}\right )-i \sqrt {d} \left (-\frac {c \log \left (\frac {2 e \left (\sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 (-d)-e}-i c^2 \sqrt {d} x-\sqrt {e}\right )}{c \sqrt {c^2 (-d)-e} \left (\sqrt {e} x+i \sqrt {d}\right )}\right )}{\sqrt {c^2 (-d)-e}}-\frac {\cosh ^{-1}(c x)}{\sqrt {e} x+i \sqrt {d}}\right )-2 \cosh ^{-1}(c x)^2\right )}{4 e^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{3} \operatorname {arcosh}\left (c x\right ) + a x^{3}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.93, size = 2964, normalized size = 5.27 \[ \text {output too large to display} \]
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 \[ \frac {1}{2} \, a {\left (\frac {d}{e^{3} x^{2} + d e^{2}} + \frac {\log \left (e x^{2} + d\right )}{e^{2}}\right )} + b \int \frac {x^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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 \[ \int \frac {x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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