Integrand size = 18, antiderivative size = 707 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \arctan \left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {b c \arctan \left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \]
[Out]
Time = 0.76 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5793, 5828, 739, 210, 5827, 5680, 2221, 2317, 2438} \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=-\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b c \arctan \left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d-e}}-\frac {b c \arctan \left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d-e}} \]
[In]
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Rule 210
Rule 739
Rule 2221
Rule 2317
Rule 2438
Rule 5680
Rule 5793
Rule 5827
Rule 5828
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {e (a+b \text {arcsinh}(c x))}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e (a+b \text {arcsinh}(c x))}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {e (a+b \text {arcsinh}(c x))}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx \\ & = -\frac {e \int \frac {a+b \text {arcsinh}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{4 d}-\frac {e \int \frac {a+b \text {arcsinh}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{4 d}-\frac {e \int \frac {a+b \text {arcsinh}(c x)}{-d e-e^2 x^2} \, dx}{2 d} \\ & = -\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(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} (a+b \text {arcsinh}(c x))}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} (a+b \text {arcsinh}(c x))}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 d} \\ & = -\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{3/2}}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{3/2}}-\frac {(b c) \text {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) \text {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 \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \arctan \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 \arctan \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 {\text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{c \sqrt {-d}-\sqrt {e} \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{4 (-d)^{3/2}}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{c \sqrt {-d}+\sqrt {e} \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{4 (-d)^{3/2}} \\ & = -\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \arctan \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 \arctan \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 {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d+e}-\sqrt {e} e^x} \, dx,x,\text {arcsinh}(c x)\right )}{4 (-d)^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d+e}-\sqrt {e} e^x} \, dx,x,\text {arcsinh}(c x)\right )}{4 (-d)^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d+e}+\sqrt {e} e^x} \, dx,x,\text {arcsinh}(c x)\right )}{4 (-d)^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d+e}+\sqrt {e} e^x} \, dx,x,\text {arcsinh}(c x)\right )}{4 (-d)^{3/2}} \\ & = -\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \arctan \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 \arctan \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 {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 (-d)^{3/2} \sqrt {e}} \\ & = -\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \arctan \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 \arctan \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 {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{4 (-d)^{3/2} \sqrt {e}} \\ & = -\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \arctan \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 \arctan \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 {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.32 (sec) , antiderivative size = 622, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {a x}{d^2+d e x^2}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e}}+\frac {b \left (-2 \sqrt {d} \left (-\frac {\text {arcsinh}(c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {c \arctan \left (\frac {\sqrt {e}-i c^2 \sqrt {d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{\sqrt {c^2 d-e}}\right )+2 i \sqrt {d} \left (\frac {\text {arcsinh}(c x)}{\sqrt {d}+i \sqrt {e} x}+\frac {c \text {arctanh}\left (\frac {i \sqrt {e}-c^2 \sqrt {d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{\sqrt {c^2 d-e}}\right )+i \left (\text {arcsinh}(c x) \left (-\text {arcsinh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )-i \left (\text {arcsinh}(c x) \left (-\text {arcsinh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )\right )}{4 d^{3/2} \sqrt {e}}\right ) \]
[In]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 7.96 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.20
method | result | size |
parts | \(\frac {a x}{2 d \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+\frac {b \left (\frac {c^{3} \operatorname {arcsinh}\left (c x \right ) x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{4 d}+\frac {c^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{4 d}+\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) c^{2} \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right )}{2 d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right ) c^{2}}{2 d \,e^{3}}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) c^{2} \arctan \left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right )}{2 d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right ) c^{2}}{2 d \,e^{3}}\right )}{c}\) | \(848\) |
derivativedivides | \(\frac {\frac {a \,c^{3} x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\operatorname {arcsinh}\left (c x \right ) x}{2 c d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}}{4 c^{2} d}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}}{4 c^{2} d}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right )}{2 c^{2} d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right )}{2 c^{2} d \,e^{3}}+\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right )}{2 c^{2} d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right )}{2 c^{2} d \,e^{3}}\right )}{c}\) | \(863\) |
default | \(\frac {\frac {a \,c^{3} x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\operatorname {arcsinh}\left (c x \right ) x}{2 c d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}}{4 c^{2} d}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}}{4 c^{2} d}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right )}{2 c^{2} d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \arctan \left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right )}{2 c^{2} d \,e^{3}}+\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right )}{2 c^{2} d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \operatorname {arctanh}\left (\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right )}{2 c^{2} d \,e^{3}}\right )}{c}\) | \(863\) |
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\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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