Integrand size = 28, antiderivative size = 350 \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {\sqrt {1-c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 c^4 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}}+\frac {5 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}} \]
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Time = 0.49 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5942, 5887, 5556, 3384, 3379, 3382} \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^4 \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^4 \sqrt {c x-1}}-\frac {5 \sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^4 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^4 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^4 \sqrt {c x-1}}+\frac {5 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^4 \sqrt {c x-1}}-\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5887
Rule 5942
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}-\frac {\left (3 \sqrt {1-c x}\right ) \int \frac {x^2}{a+b \text {arccosh}(c x)} \, dx}{b c \sqrt {-1+c x}}+\frac {\left (5 c \sqrt {1-c x}\right ) \int \frac {x^4}{a+b \text {arccosh}(c x)} \, dx}{b \sqrt {-1+c x}} \\ & = -\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (3 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^4 \sqrt {-1+c x}}-\frac {\left (5 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^4 \sqrt {-1+c x}} \\ & = -\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (3 \sqrt {1-c x}\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^4 \sqrt {-1+c x}}-\frac {\left (5 \sqrt {1-c x}\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}+\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^4 \sqrt {-1+c x}} \\ & = -\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}-\frac {\left (5 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^4 \sqrt {-1+c x}}-\frac {\left (5 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^4 \sqrt {-1+c x}}+\frac {\left (3 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^4 \sqrt {-1+c x}}+\frac {\left (3 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^4 \sqrt {-1+c x}}-\frac {\left (15 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^4 \sqrt {-1+c x}} \\ & = -\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (5 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^4 \sqrt {-1+c x}}-\frac {\left (3 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^4 \sqrt {-1+c x}}-\frac {\left (3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^4 \sqrt {-1+c x}}+\frac {\left (15 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^4 \sqrt {-1+c x}}+\frac {\left (5 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^4 \sqrt {-1+c x}}-\frac {\left (5 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^4 \sqrt {-1+c x}}+\frac {\left (3 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^4 \sqrt {-1+c x}}+\frac {\left (3 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^4 \sqrt {-1+c x}}-\frac {\left (15 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^4 \sqrt {-1+c x}}-\frac {\left (5 \sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^4 \sqrt {-1+c x}} \\ & = -\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {\sqrt {1-c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 c^4 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}}+\frac {5 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2} \left (16 b c^3 x^3-16 b c^5 x^5+2 (a+b \text {arccosh}(c x)) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-3 (a+b \text {arccosh}(c x)) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-5 a \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-5 b \text {arccosh}(c x) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-2 a \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-2 b \text {arccosh}(c x) \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+3 a \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 b \text {arccosh}(c x) \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 a \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 b \text {arccosh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{16 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))} \]
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Time = 1.31 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.59
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^{5} x^{5}+32 b \,c^{6} x^{6}-32 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{3} x^{3}-32 b \,c^{4} x^{4}+5 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}+3 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}-2 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-5 \,\operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-3 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+2 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}} b \,\operatorname {arccosh}\left (c x \right )+5 a \,\operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}+3 a \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}-2 a \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-5 \,\operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}} a -3 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}} a +2 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}} a \right )}{32 \left (c x +1\right ) c^{4} \left (c x -1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\) | \(555\) |
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\[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{3}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^{3} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{3}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^3\,\sqrt {1-c^2\,x^2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
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