Integrand size = 18, antiderivative size = 1234 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^3} \, dx=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (c \sqrt {-d}+\sqrt {e}\right )^{3/2} \sqrt {e}}+\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}+\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (c \sqrt {-d}+\sqrt {e}\right )^{3/2} \sqrt {e}}-\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \]
[Out]
Time = 0.97 (sec) , antiderivative size = 1234, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5909, 5963, 98, 95, 214, 5962, 5681, 2221, 2317, 2438} \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^3} \, dx=-\frac {b \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right ) c^3}{8 d \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (\sqrt {-d} c+\sqrt {e}\right )^{3/2} \sqrt {e}}+\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right ) c^3}{8 d \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (\sqrt {-d} c+\sqrt {e}\right )^{3/2} \sqrt {e}}+\frac {3 b \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right ) c}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {e}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right ) c}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {e}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b \sqrt {c x-1} \sqrt {c x+1} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )^2}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \]
[In]
[Out]
Rule 95
Rule 98
Rule 214
Rule 2221
Rule 2317
Rule 2438
Rule 5681
Rule 5909
Rule 5962
Rule 5963
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {e^{3/2} (a+b \text {arccosh}(c x))}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-e x\right )^3}-\frac {3 e (a+b \text {arccosh}(c x))}{16 d^2 \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e^{3/2} (a+b \text {arccosh}(c x))}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+e x\right )^3}-\frac {3 e (a+b \text {arccosh}(c x))}{16 d^2 \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {3 e (a+b \text {arccosh}(c x))}{8 d^2 \left (-d e-e^2 x^2\right )}\right ) \, dx \\ & = -\frac {(3 e) \int \frac {a+b \text {arccosh}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{16 d^2}-\frac {(3 e) \int \frac {a+b \text {arccosh}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{16 d^2}-\frac {(3 e) \int \frac {a+b \text {arccosh}(c x)}{-d e-e^2 x^2} \, dx}{8 d^2}-\frac {e^{3/2} \int \frac {a+b \text {arccosh}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^3} \, dx}{8 (-d)^{3/2}}-\frac {e^{3/2} \int \frac {a+b \text {arccosh}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^3} \, dx}{8 (-d)^{3/2}} \\ & = -\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {(3 b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}-e x\right )} \, dx}{16 d^2}-\frac {(3 b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}+e x\right )} \, dx}{16 d^2}+\frac {\left (b c \sqrt {e}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{16 (-d)^{3/2}}-\frac {\left (b c \sqrt {e}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{16 (-d)^{3/2}}-\frac {(3 e) \int \left (-\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{8 d^2} \\ & = -\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {3 \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{16 (-d)^{5/2}}-\frac {3 \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{16 (-d)^{5/2}}+\frac {(3 b c) \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 )}{8 d^2}-\frac {(3 b c) \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 )}{8 d^2}+\frac {\left (b c^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}-e x\right )} \, dx}{16 d \left (c^2 d+e\right )}-\frac {\left (b c^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}+e x\right )} \, dx}{16 d \left (c^2 d+e\right )} \\ & = -\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}-\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}-\frac {3 \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{16 (-d)^{5/2}}-\frac {3 \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{16 (-d)^{5/2}}+\frac {\left (b c^3\right ) \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 )}{8 d \left (c^2 d+e\right )}-\frac {\left (b c^3\right ) \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 )}{8 d \left (c^2 d+e\right )} \\ & = -\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}+\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e} \left (c^2 d+e\right )}-\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e} \left (c^2 d+e\right )}-\frac {3 \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 {arccosh}(c x)\right )}{16 (-d)^{5/2}}-\frac {3 \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 {arccosh}(c x)\right )}{16 (-d)^{5/2}}-\frac {3 \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 {arccosh}(c x)\right )}{16 (-d)^{5/2}}-\frac {3 \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 {arccosh}(c x)\right )}{16 (-d)^{5/2}} \\ & = -\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}+\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e} \left (c^2 d+e\right )}-\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e} \left (c^2 d+e\right )}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 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 {arccosh}(c x)\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 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 {arccosh}(c x)\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 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 {arccosh}(c x)\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 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 {arccosh}(c x)\right )}{16 (-d)^{5/2} \sqrt {e}} \\ & = -\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}+\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e} \left (c^2 d+e\right )}-\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e} \left (c^2 d+e\right )}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 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 {arccosh}(c x)}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 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 {arccosh}(c x)}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 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 {arccosh}(c x)}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 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 {arccosh}(c x)}\right )}{16 (-d)^{5/2} \sqrt {e}} \\ & = -\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 (a+b \text {arccosh}(c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}+\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e} \left (c^2 d+e\right )}-\frac {3 b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e}}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {e} \left (c^2 d+e\right )}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.35 (sec) , antiderivative size = 1161, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^3} \, dx=\frac {\frac {8 a d^{3/2} x}{\left (d+e x^2\right )^2}+\frac {12 a \sqrt {d} x}{d+e x^2}+\frac {12 a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {6 b \sqrt {d} \left (\frac {\text {arccosh}(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 )}{\sqrt {e}}-\frac {6 b \sqrt {d} \left (-\frac {\text {arccosh}(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 )}{\sqrt {e}}+2 i b d \left (\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}-\frac {\text {arccosh}(c x)}{\sqrt {e} \left (-i \sqrt {d}+\sqrt {e} x\right )^2}+\frac {c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}-c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\sqrt {e} \left (c^2 d+e\right )^{3/2}}\right )-2 i b d \left (\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}-\frac {\text {arccosh}(c x)}{\sqrt {e} \left (i \sqrt {d}+\sqrt {e} x\right )^2}-\frac {c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}+c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\sqrt {e} \left (c^2 d+e\right )^{3/2}}\right )+\frac {3 i b \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{\sqrt {e}}+\frac {3 i b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )-2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )-2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{\sqrt {e}}}{32 d^{5/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 29.03 (sec) , antiderivative size = 1778, normalized size of antiderivative = 1.44
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1778\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1803\) |
default | \(\text {Expression too large to display}\) | \(1803\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
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