Optimal. Leaf size=846 \[ -\frac {3 \sqrt {e} \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {b c \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^2}-\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}}}+\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}}}+\frac {3 b \sqrt {e} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}} \]
[Out]
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Rubi [A] time = 2.02, antiderivative size = 846, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {5792, 5662, 92, 205, 5707, 5802, 93, 208, 5800, 5562, 2190, 2279, 2391} \[ -\frac {3 \sqrt {e} \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {b c \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^2}-\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}}}+\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}}}+\frac {3 b \sqrt {e} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 92
Rule 93
Rule 205
Rule 208
Rule 2190
Rule 2279
Rule 2391
Rule 5562
Rule 5662
Rule 5707
Rule 5792
Rule 5800
Rule 5802
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx &=\int \left (\frac {a+b \cosh ^{-1}(c x)}{d^2 x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d \left (d+e x^2\right )^2}-\frac {e \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^2} \, dx}{d^2}-\frac {e \int \frac {a+b \cosh ^{-1}(c x)}{d+e x^2} \, dx}{d^2}-\frac {e \int \frac {a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{d}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {(b c) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d^2}-\frac {e \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 {e \int \left (-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx}{d}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\left (b c^2\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 \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 (-d)^{5/2}}+\frac {e \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)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{4 d^2}+\frac {e^2 \int \frac {a+b \cosh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{4 d^2}+\frac {e^2 \int \frac {a+b \cosh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 d^2}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e \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 \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 e) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}-e x\right )} \, dx}{4 d^2}+\frac {(b c e) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}+e x\right )} \, dx}{4 d^2}+\frac {e^2 \int \left (-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 d^2}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{5/2}}+\frac {e \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{5/2}}+\frac {e \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 \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 \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 \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 e) \operatorname {Subst}\left (\int \frac {1}{c \sqrt {-d} \sqrt {e}+e-\left (c \sqrt {-d} \sqrt {e}-e\right ) x^2} \, dx,x,\frac {\sqrt {1+c x}}{\sqrt {-1+c x}}\right )}{2 d^2}+\frac {(b c e) \operatorname {Subst}\left (\int \frac {1}{c \sqrt {-d} \sqrt {e}-e-\left (c \sqrt {-d} \sqrt {e}+e\right ) x^2} \, dx,x,\frac {\sqrt {1+c x}}{\sqrt {-1+c x}}\right )}{2 d^2}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}-\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}+\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}-\frac {\sqrt {e} \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 {\sqrt {e} \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 {\sqrt {e} \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 {\sqrt {e} \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 \sqrt {e}\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 \sqrt {e}\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 \sqrt {e}\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 \sqrt {e}\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 {e \operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac {e \operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}-\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}+\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}-\frac {\sqrt {e} \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 {\sqrt {e} \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 {\sqrt {e} \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 {\sqrt {e} \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 \sqrt {e}\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 \sqrt {e}\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 \sqrt {e}\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 \sqrt {e}\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 {e \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 )}{4 (-d)^{5/2}}+\frac {e \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 )}{4 (-d)^{5/2}}+\frac {e \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 )}{4 (-d)^{5/2}}+\frac {e \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 )}{4 (-d)^{5/2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}-\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}+\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {b \sqrt {e} \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 \sqrt {e} \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 \sqrt {e} \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 \sqrt {e} \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 {\left (b \sqrt {e}\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 )}{4 (-d)^{5/2}}-\frac {\left (b \sqrt {e}\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 )}{4 (-d)^{5/2}}+\frac {\left (b \sqrt {e}\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 )}{4 (-d)^{5/2}}-\frac {\left (b \sqrt {e}\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 )}{4 (-d)^{5/2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}-\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}+\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {b \sqrt {e} \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 \sqrt {e} \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 \sqrt {e} \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 \sqrt {e} \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 {\left (b \sqrt {e}\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 )}{4 (-d)^{5/2}}-\frac {\left (b \sqrt {e}\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 )}{4 (-d)^{5/2}}+\frac {\left (b \sqrt {e}\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 )}{4 (-d)^{5/2}}-\frac {\left (b \sqrt {e}\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 )}{4 (-d)^{5/2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}-\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}+\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \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 )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 2.85, size = 821, normalized size = 0.97 \[ \frac {-12 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) a-\frac {8 \sqrt {d} a}{x}-\frac {4 \sqrt {d} e x a}{e x^2+d}+b \left (8 \sqrt {d} \left (\frac {c \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\cosh ^{-1}(c x)}{x}\right )-2 \sqrt {d} \sqrt {e} \left (\frac {\cosh ^{-1}(c x)}{\sqrt {e} x-i \sqrt {d}}+\frac {c \log \left (\frac {2 e \left (\sqrt {d} x c^2+i \sqrt {e}-i \sqrt {-d c^2-e} \sqrt {c x-1} \sqrt {c x+1}\right )}{c \sqrt {-d c^2-e} \left (i \sqrt {e} x+\sqrt {d}\right )}\right )}{\sqrt {-d c^2-e}}\right )+2 \sqrt {d} \sqrt {e} \left (-\frac {\cosh ^{-1}(c x)}{\sqrt {e} x+i \sqrt {d}}-\frac {c \log \left (\frac {2 e \left (-i \sqrt {d} x c^2-\sqrt {e}+\sqrt {-d c^2-e} \sqrt {c x-1} \sqrt {c x+1}\right )}{c \sqrt {-d c^2-e} \left (\sqrt {e} x+i \sqrt {d}\right )}\right )}{\sqrt {-d c^2-e}}\right )-3 i \sqrt {e} \left (\cosh ^{-1}(c x) \left (2 \left (\log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{i c \sqrt {d}-\sqrt {-d c^2-e}}+1\right )+\log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{i \sqrt {d} c+\sqrt {-d c^2-e}}+1\right )\right )-\cosh ^{-1}(c x)\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d c^2-e}-i c \sqrt {d}}\right )+2 \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i \sqrt {d} c+\sqrt {-d c^2-e}}\right )\right )+3 i \sqrt {e} \left (\cosh ^{-1}(c x) \left (2 \left (\log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{\sqrt {-d c^2-e}-i c \sqrt {d}}+1\right )+\log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i \sqrt {d} c+\sqrt {-d c^2-e}}\right )\right )-\cosh ^{-1}(c x)\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}-\sqrt {-d c^2-e}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i \sqrt {d} c+\sqrt {-d c^2-e}}\right )\right )\right )}{8 d^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcosh}\left (c x\right ) + a}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.42, size = 1821, normalized size = 2.15 \[ \text {result too large to display} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {3 \, e x^{2} + 2 \, d}{d^{2} e x^{3} + d^{3} x} + \frac {3 \, e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} d^{2}}\right )} + b \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,{\left (e\,x^2+d\right )}^2} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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