Optimal. Leaf size=707 \[ -\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d-e}}-\frac {b c \tan ^{-1}\left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d-e}} \]
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Rubi [A] time = 1.07, antiderivative size = 707, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5706, 5801, 725, 204, 5799, 5561, 2190, 2279, 2391} \[ \frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d-e}}-\frac {b c \tan ^{-1}\left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d-e}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 725
Rule 2190
Rule 2279
Rule 2391
Rule 5561
Rule 5706
Rule 5799
Rule 5801
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac {e \left (a+b \sinh ^{-1}(c x)\right )}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e \left (a+b \sinh ^{-1}(c x)\right )}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {e \left (a+b \sinh ^{-1}(c x)\right )}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac {e \int \frac {a+b \sinh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{4 d}-\frac {e \int \frac {a+b \sinh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{4 d}-\frac {e \int \frac {a+b \sinh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 d}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {(b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}-e x\right ) \sqrt {1+c^2 x^2}} \, dx}{4 d}-\frac {(b c) \int \frac {1}{\left (\sqrt {-d} \sqrt {e}+e x\right ) \sqrt {1+c^2 x^2}} \, dx}{4 d}-\frac {e \int \left (-\frac {\sqrt {-d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 d}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{3/2}}+\frac {\int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{3/2}}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{-c^2 d e+e^2-x^2} \, dx,x,\frac {-e-c^2 \sqrt {-d} \sqrt {e} x}{\sqrt {1+c^2 x^2}}\right )}{4 d}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{-c^2 d e+e^2-x^2} \, dx,x,\frac {e-c^2 \sqrt {-d} \sqrt {e} x}{\sqrt {1+c^2 x^2}}\right )}{4 d}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}+\frac {\operatorname {Subst}\left (\int \frac {(a+b x) \cosh (x)}{c \sqrt {-d}-\sqrt {e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {(a+b x) \cosh (x)}{c \sqrt {-d}+\sqrt {e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}+\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,\sinh ^{-1}(c x)\right )}{4 (-d)^{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,\sinh ^{-1}(c x)\right )}{4 (-d)^{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,\sinh ^{-1}(c x)\right )}{4 (-d)^{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,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\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,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}-\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,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}+\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,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}-\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,\sinh ^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\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^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}-\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^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}+\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^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}-\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^{\sinh ^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \sinh ^{-1}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}\\ \end {align*}
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Mathematica [C] time = 1.72, size = 622, normalized size = 0.88 \[ \frac {1}{2} \left (\frac {a \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e}}+\frac {a x}{d^2+d e x^2}+\frac {b \left (i \left (2 \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}-i c \sqrt {d}}\right )+2 \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{i \sqrt {d} c+\sqrt {e-c^2 d}}\right )+\sinh ^{-1}(c x) \left (-\sinh ^{-1}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{-\sqrt {e-c^2 d}+i c \sqrt {d}}\right )+\log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+i c \sqrt {d}}\right )\right )\right )\right )-i \left (2 \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}-i c \sqrt {d}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{i \sqrt {d} c+\sqrt {e-c^2 d}}\right )+\sinh ^{-1}(c x) \left (-\sinh ^{-1}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}-i c \sqrt {d}}\right )+\log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+i c \sqrt {d}}\right )\right )\right )\right )-2 \sqrt {d} \left (\frac {c \tan ^{-1}\left (\frac {\sqrt {e}-i c^2 \sqrt {d} x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{\sqrt {c^2 d-e}}-\frac {\sinh ^{-1}(c x)}{\sqrt {e} x+i \sqrt {d}}\right )+2 i \sqrt {d} \left (\frac {c \tanh ^{-1}\left (\frac {-c^2 \sqrt {d} x+i \sqrt {e}}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{\sqrt {c^2 d-e}}+\frac {\sinh ^{-1}(c x)}{\sqrt {d}+i \sqrt {e} x}\right )\right )}{4 d^{3/2} \sqrt {e}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsinh}\left (c x\right ) + a}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.16, size = 1745, normalized size = 2.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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