Integrand size = 27, antiderivative size = 268 \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{128 b c^3}-\frac {5 \log (a+b \text {arcsinh}(c x))}{128 b c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{128 b c^3} \]
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Time = 0.34 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5819, 5556, 3384, 3379, 3382} \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{128 b c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{128 b c^3}-\frac {5 \log (a+b \text {arcsinh}(c x))}{128 b c^3} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh ^6\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^3} \\ & = \frac {\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 {arcsinh}(c x)\right )}{b c^3} \\ & = -\frac {5 \log (a+b \text {arcsinh}(c x))}{128 b c^3}+\frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {8 a}{b}-\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{128 b c^3}+\frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3}+\frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3}-\frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3} \\ & = -\frac {5 \log (a+b \text {arcsinh}(c x))}{128 b c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{128 b c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{32 b c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{128 b c^3} \\ & = -\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{128 b c^3}-\frac {5 \log (a+b \text {arcsinh}(c x))}{128 b c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arcsinh}(c x))}{b}\right )}{128 b c^3} \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.74 \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {-4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+4 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+4 \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (8 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-5 \log (a+b \text {arcsinh}(c x))+4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-4 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-4 \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{128 b c^3} \]
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Time = 0.51 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {{\mathrm e}^{\frac {8 a}{b}} \operatorname {Ei}_{1}\left (8 \,\operatorname {arcsinh}\left (c x \right )+\frac {8 a}{b}\right )+4 \,{\mathrm e}^{\frac {6 a}{b}} \operatorname {Ei}_{1}\left (6 \,\operatorname {arcsinh}\left (c x \right )+\frac {6 a}{b}\right )+4 \,{\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (c x \right )+\frac {4 a}{b}\right )-4 \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )-4 \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )+4 \,{\mathrm e}^{-\frac {4 a}{b}} \operatorname {Ei}_{1}\left (-4 \,\operatorname {arcsinh}\left (c x \right )-\frac {4 a}{b}\right )+4 \,{\mathrm e}^{-\frac {6 a}{b}} \operatorname {Ei}_{1}\left (-6 \,\operatorname {arcsinh}\left (c x \right )-\frac {6 a}{b}\right )+{\mathrm e}^{-\frac {8 a}{b}} \operatorname {Ei}_{1}\left (-8 \,\operatorname {arcsinh}\left (c x \right )-\frac {8 a}{b}\right )+10 \ln \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{3} b}\) | \(211\) |
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\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^{2} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
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\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^2\,{\left (c^2\,x^2+1\right )}^{5/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]
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