Integrand size = 28, antiderivative size = 354 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c^3 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}} \]
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Time = 0.58 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5942, 5912, 5952, 5556, 3384, 3379, 3382} \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3 \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5912
Rule 5942
Rule 5952
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (2 \sqrt {1-c x}\right ) \int \frac {x (-1+c x) (1+c x)}{a+b \text {arccosh}(c x)} \, dx}{b c \sqrt {-1+c x}}-\frac {\left (6 c \sqrt {1-c x}\right ) \int \frac {x^3 (-1+c x) (1+c x)}{a+b \text {arccosh}(c x)} \, dx}{b \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (2 \sqrt {1-c x}\right ) \int \frac {x \left (-1+c^2 x^2\right )}{a+b \text {arccosh}(c x)} \, dx}{b c \sqrt {-1+c x}}-\frac {\left (6 c \sqrt {1-c x}\right ) \int \frac {x^3 \left (-1+c^2 x^2\right )}{a+b \text {arccosh}(c x)} \, dx}{b \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\left (2 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3 \sqrt {-1+c x}}+\frac {\left (6 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\cosh ^3\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\left (2 \sqrt {1-c x}\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3 \sqrt {-1+c x}}+\frac {\left (6 \sqrt {1-c x}\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{32 x}-\frac {3 \sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{32 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (3 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (9 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\left (\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (9 \sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (9 \sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (3 \sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c^3 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}} \\ \end{align*}
Time = 0.85 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.95 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (16 b c^2 x^2-32 b c^4 x^4+16 b c^6 x^6-(a+b \text {arccosh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-4 (a+b \text {arccosh}(c x)) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+3 a \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+3 b \text {arccosh}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+b \text {arccosh}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 b \text {arccosh}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-3 b \text {arccosh}(c x) \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{16 b^2 c^3 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \]
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Time = 0.79 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.69
method | result | size |
default | \(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (32 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{6} x^{6}+32 b \,c^{7} x^{7}-64 x^{4} c^{4} b \sqrt {c x -1}\, \sqrt {c x +1}-64 b \,c^{5} x^{5}+32 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{2} x^{2}+32 b \,c^{3} x^{3}+3 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}-\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-4 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-3 \,\operatorname {Ei}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+4 \,\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+3 a \,\operatorname {Ei}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}-a \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-4 a \,\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-3 \,\operatorname {Ei}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}} a +4 \,\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} a +\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} a \right )}{32 \left (c x +1\right ) c^{3} \left (c x -1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\) | \(599\) |
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\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^{2} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^2\,{\left (1-c^2\,x^2\right )}^{3/2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
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