Integrand size = 28, antiderivative size = 136 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^2 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}-\frac {\sqrt {-1+c x} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c^3 \sqrt {1-c x}}+\frac {\sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c^3 \sqrt {1-c x}} \]
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Time = 0.19 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5950, 5887, 5556, 12, 3384, 3379, 3382} \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {c x-1} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c^3 \sqrt {1-c x}}+\frac {\sqrt {c x-1} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c^3 \sqrt {1-c x}}-\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))} \]
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5887
Rule 5950
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}+\frac {\left (2 \sqrt {-1+c x}\right ) \int \frac {x}{a+b \text {arccosh}(c x)} \, dx}{b c \sqrt {1-c x}} \\ & = -\frac {x^2 \sqrt {-1+c x}}{b c \sqrt {1-c x} (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 \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}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}-\frac {\left (2 \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 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}}{b c \sqrt {1-c x} (a+b \text {arccosh}(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 )}{b^2 c^3 \sqrt {1-c x}} \\ & = -\frac {x^2 \sqrt {-1+c x}}{b c \sqrt {1-c x} (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 )}{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 )}{b^2 c^3 \sqrt {1-c x}} \\ & = -\frac {x^2 \sqrt {-1+c x}}{b c \sqrt {1-c x} (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 )}{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 )}{b^2 c^3 \sqrt {1-c x}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2} \left (b c^2 x^2+(a+b \text {arccosh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-(a+b \text {arccosh}(c x)) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))} \]
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Time = 0.77 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.79
method | result | size |
default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{2} x^{2}+2 b \,c^{3} x^{3}+\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+a \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} a \right )}{2 c^{3} \left (c^{2} x^{2}-1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\) | \(243\) |
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\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{2}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^{2}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{2}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{2}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]
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