Optimal. Leaf size=839 \[ \frac {a x}{e^2}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}+\frac {b x \cosh ^{-1}(c x)}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}-\frac {b c d \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}-\frac {3 b \sqrt {-d} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}-\frac {3 b \sqrt {-d} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.59, antiderivative size = 839, normalized size of antiderivative = 1.00, number of steps
used = 49, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5959, 5879,
75, 5909, 5963, 95, 214, 5962, 5681, 2221, 2317, 2438} \begin {gather*} \frac {x \cosh ^{-1}(c x) b}{e^2}+\frac {c d \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 ) b}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}-\frac {c d \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 ) b}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}-\frac {3 \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac {3 \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac {3 \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac {3 \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac {\sqrt {c x-1} \sqrt {c x+1} b}{c e^2}+\frac {a x}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right )}{4 e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 75
Rule 95
Rule 214
Rule 2221
Rule 2317
Rule 2438
Rule 5681
Rule 5879
Rule 5909
Rule 5959
Rule 5962
Rule 5963
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (\frac {a+b \cosh ^{-1}(c x)}{e^2}+\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )^2}-\frac {2 d \left (a+b \cosh ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{e^2}-\frac {(2 d) \int \frac {a+b \cosh ^{-1}(c x)}{d+e x^2} \, dx}{e^2}+\frac {d^2 \int \frac {a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{e^2}\\ &=\frac {a x}{e^2}+\frac {b \int \cosh ^{-1}(c x) \, dx}{e^2}-\frac {(2 d) \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}{e^2}+\frac {d^2 \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}{e^2}\\ &=\frac {a x}{e^2}+\frac {b x \cosh ^{-1}(c x)}{e^2}-\frac {(b c) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e^2}-\frac {\sqrt {-d} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{e^2}-\frac {\sqrt {-d} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{e^2}-\frac {d \int \frac {a+b \cosh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{4 e}-\frac {d \int \frac {a+b \cosh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{4 e}-\frac {d \int \frac {a+b \cosh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 e}\\ &=\frac {a x}{e^2}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}+\frac {b x \cosh ^{-1}(c x)}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{e^2}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{e^2}+\frac {(b c d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}-e x\right )} \, dx}{4 e^2}-\frac {(b c d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}+e x\right )} \, dx}{4 e^2}-\frac {d \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 e}\\ &=\frac {a x}{e^2}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}+\frac {b x \cosh ^{-1}(c x)}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\sqrt {-d} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 e^2}+\frac {\sqrt {-d} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 e^2}-\frac {\sqrt {-d} \text {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 )}{e^2}-\frac {\sqrt {-d} \text {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 )}{e^2}-\frac {\sqrt {-d} \text {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 )}{e^2}-\frac {\sqrt {-d} \text {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 )}{e^2}+\frac {(b c d) \text {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 e^2}-\frac {(b c d) \text {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 e^2}\\ &=\frac {a x}{e^2}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}+\frac {b x \cosh ^{-1}(c x)}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}-\frac {b c d \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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}+\frac {\sqrt {-d} \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 )}{e^{5/2}}-\frac {\sqrt {-d} \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 )}{e^{5/2}}+\frac {\sqrt {-d} \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 )}{e^{5/2}}-\frac {\sqrt {-d} \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 )}{e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \text {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 )}{e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \text {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 )}{e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \text {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 )}{e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \text {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 )}{e^{5/2}}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{4 e^2}+\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{4 e^2}\\ &=\frac {a x}{e^2}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}+\frac {b x \cosh ^{-1}(c x)}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}-\frac {b c d \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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}+\frac {\sqrt {-d} \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 )}{e^{5/2}}-\frac {\sqrt {-d} \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 )}{e^{5/2}}+\frac {\sqrt {-d} \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 )}{e^{5/2}}-\frac {\sqrt {-d} \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 )}{e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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^{\cosh ^{-1}(c x)}\right )}{e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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^{\cosh ^{-1}(c x)}\right )}{e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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^{\cosh ^{-1}(c x)}\right )}{e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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^{\cosh ^{-1}(c x)}\right )}{e^{5/2}}+\frac {\sqrt {-d} \text {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 e^2}+\frac {\sqrt {-d} \text {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 e^2}+\frac {\sqrt {-d} \text {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 e^2}+\frac {\sqrt {-d} \text {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 e^2}\\ &=\frac {a x}{e^2}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}+\frac {b x \cosh ^{-1}(c x)}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}-\frac {b c d \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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}+\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {3 \sqrt {-d} \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 e^{5/2}}+\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{e^{5/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{e^{5/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{e^{5/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \text {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 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \text {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 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \text {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 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \text {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 e^{5/2}}\\ &=\frac {a x}{e^2}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}+\frac {b x \cosh ^{-1}(c x)}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}-\frac {b c d \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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}+\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {3 \sqrt {-d} \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 e^{5/2}}+\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{e^{5/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{e^{5/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{e^{5/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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^{\cosh ^{-1}(c x)}\right )}{4 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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^{\cosh ^{-1}(c x)}\right )}{4 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \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^{\cosh ^{-1}(c x)}\right )}{4 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \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^{\cosh ^{-1}(c x)}\right )}{4 e^{5/2}}\\ &=\frac {a x}{e^2}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}+\frac {b x \cosh ^{-1}(c x)}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}-\frac {b c d \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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}+\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {3 \sqrt {-d} \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 e^{5/2}}+\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {3 b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}-\frac {3 b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 e^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.35, size = 777, normalized size = 0.93 \begin {gather*} \frac {8 a \sqrt {e} x+\frac {4 a d \sqrt {e} x}{d+e x^2}-12 a \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+b \left (\frac {8 \sqrt {e} \left (-\sqrt {\frac {-1+c x}{1+c x}} (1+c x)+c x \cosh ^{-1}(c x)\right )}{c}+2 d \left (\frac {\cosh ^{-1}(c x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )+2 d \left (\frac {\cosh ^{-1}(c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )-3 i \sqrt {d} \left (\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )+3 i \sqrt {d} \left (\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )}{8 e^{5/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 75.66, size = 1749, normalized size = 2.08 \[\text {Expression too large to display}\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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