Integrand size = 28, antiderivative size = 439 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{128 b c^3 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{128 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{128 b c^3 \sqrt {-1+c x}} \]
[Out]
Time = 0.37 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5952, 5556, 3384, 3379, 3382} \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{128 b c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{128 b c^3 \sqrt {c x-1}}-\frac {5 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{128 b c^3 \sqrt {c x-1}} \]
[In]
[Out]
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5952
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^6\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^3 \sqrt {-1+c x}} \\ & = \frac {\sqrt {1-c x} \text {Subst}\left (\int \left (-\frac {5}{128 x}+\frac {\cosh \left (\frac {8 a}{b}-\frac {8 x}{b}\right )}{128 x}-\frac {\cosh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{32 x}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{32 x}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{32 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^3 \sqrt {-1+c x}} \\ & = -\frac {5 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{128 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {8 a}{b}-\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{128 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b c^3 \sqrt {-1+c x}} \\ & = -\frac {5 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{128 b c^3 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \cosh \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 )}{32 b 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 {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \cosh \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 )}{32 b c^3 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{128 b 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 {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b 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 {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \sinh \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 )}{32 b c^3 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \sinh \left (\frac {8 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{128 b c^3 \sqrt {-1+c x}} \\ & = \frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{128 b c^3 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{128 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{128 b c^3 \sqrt {-1+c x}} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.53 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c^2 x^2} \left (4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-4 \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (8 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-5 \log (a+b \text {arccosh}(c x))-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-4 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{128 c^3 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]
[In]
[Out]
Time = 0.80 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-10 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-10 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +4 \,\operatorname {Ei}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}-\operatorname {Ei}_{1}\left (8 \,\operatorname {arccosh}\left (c x \right )+\frac {8 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+8 a}{b}}-\operatorname {Ei}_{1}\left (-8 \,\operatorname {arccosh}\left (c x \right )-\frac {8 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+8 a}{b}}+4 \,\operatorname {Ei}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}-4 \,\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-4 \,\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-4 \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-4 \,\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right ) b}\) | \(365\) |
[In]
[Out]
\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^2\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]
[In]
[Out]