Optimal. Leaf size=624 \[ \frac {e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 (-d)^{5/2}}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {b c^3 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d}-\frac {b e^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}-\frac {b c e \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d x^2} \]
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Rubi [A] time = 0.98, antiderivative size = 624, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5792, 5662, 103, 12, 92, 205, 5707, 5800, 5562, 2190, 2279, 2391} \[ -\frac {b e^{3/2} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 (-d)^{5/2}}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {b c^3 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d}-\frac {b c e \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d x^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 92
Rule 103
Rule 205
Rule 2190
Rule 2279
Rule 2391
Rule 5562
Rule 5662
Rule 5707
Rule 5792
Rule 5800
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^4 \left (d+e x^2\right )} \, dx &=\int \left (\frac {a+b \cosh ^{-1}(c x)}{d x^4}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x^2}+\frac {e^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \cosh ^{-1}(c x)}{x^4} \, dx}{d}-\frac {e \int \frac {a+b \cosh ^{-1}(c x)}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \cosh ^{-1}(c x)}{d+e x^2} \, dx}{d^2}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}+\frac {(b c) \int \frac {1}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 d}-\frac {(b c e) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d^2}+\frac {e^2 \int \left (\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^2}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}+\frac {(b c) \int \frac {c^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d}-\frac {\left (b c^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}-\frac {e^2 \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 (-d)^{5/2}}-\frac {e^2 \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 (-d)^{5/2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}-\frac {b c e \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {\left (b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d}-\frac {e^2 \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 (-d)^{5/2}}-\frac {e^2 \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 (-d)^{5/2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}-\frac {b c e \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {\left (b c^4\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {e^2 \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 (-d)^{5/2}}-\frac {e^2 \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 (-d)^{5/2}}-\frac {e^2 \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 (-d)^{5/2}}-\frac {e^2 \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 (-d)^{5/2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}+\frac {b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} \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 (-d)^{5/2}}-\frac {e^{3/2} \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 (-d)^{5/2}}+\frac {e^{3/2} \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 (-d)^{5/2}}-\frac {e^{3/2} \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 (-d)^{5/2}}-\frac {\left (b e^{3/2}\right ) \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 (-d)^{5/2}}+\frac {\left (b e^{3/2}\right ) \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 (-d)^{5/2}}-\frac {\left (b e^{3/2}\right ) \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 (-d)^{5/2}}+\frac {\left (b e^{3/2}\right ) \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 (-d)^{5/2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}+\frac {b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} \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 (-d)^{5/2}}-\frac {e^{3/2} \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 (-d)^{5/2}}+\frac {e^{3/2} \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 (-d)^{5/2}}-\frac {e^{3/2} \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 (-d)^{5/2}}-\frac {\left (b e^{3/2}\right ) \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 (-d)^{5/2}}+\frac {\left (b e^{3/2}\right ) \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 (-d)^{5/2}}-\frac {\left (b e^{3/2}\right ) \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 (-d)^{5/2}}+\frac {\left (b e^{3/2}\right ) \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 (-d)^{5/2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}+\frac {b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} \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 (-d)^{5/2}}-\frac {e^{3/2} \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 (-d)^{5/2}}+\frac {e^{3/2} \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 (-d)^{5/2}}-\frac {e^{3/2} \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 (-d)^{5/2}}-\frac {b e^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 1.61, size = 641, normalized size = 1.03 \[ \frac {1}{6} \left (-\frac {3 e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{(-d)^{5/2}}+\frac {3 e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}-c \sqrt {-d}}+1\right )}{(-d)^{5/2}}+\frac {3 e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{(-d)^{5/2}}-\frac {3 e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{(-d)^{5/2}}+\frac {6 e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{d x^3}-\frac {6 b c e \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b e^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{(-d)^{5/2}}-\frac {3 b e^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d c^2-e}-c \sqrt {-d}}\right )}{(-d)^{5/2}}-\frac {3 b e^{3/2} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{(-d)^{5/2}}+\frac {3 b e^{3/2} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{(-d)^{5/2}}+\frac {b c \left (c^2 x^2+c^2 x^2 \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )-1\right )}{d x^2 \sqrt {c x-1} \sqrt {c x+1}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcosh}\left (c x\right ) + a}{e x^{6} + d x^{4}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.79, size = 410, normalized size = 0.66 \[ -\frac {a}{3 d \,x^{3}}+\frac {a e}{d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}+\frac {b c \sqrt {c x -1}\, \sqrt {c x +1}}{6 d \,x^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e}{d^{2} x}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{3 d \,x^{3}}-\frac {2 c b e \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{8 c \,d^{3}}+\frac {c^{3} b \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d}+\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{8 c \,d^{3}} \]
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 \[ \frac {1}{3} \, a {\left (\frac {3 \, e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} d^{2}} + \frac {3 \, e x^{2} - d}{d^{2} x^{3}}\right )} + b \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e x^{6} + d x^{4}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,\left (e\,x^2+d\right )} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{4} \left (d + e x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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