Integrand size = 27, antiderivative size = 245 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {3 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{64 b c^4}+\frac {3 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{64 b c^4}-\frac {\text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{64 b c^4}-\frac {\text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {7 a}{b}\right )}{64 b c^4}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^4}-\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}+\frac {\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}+\frac {\cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4} \]
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Time = 0.33 (sec) , antiderivative size = 245, 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} \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^4}+\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}-\frac {\sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}-\frac {\sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^4}-\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}+\frac {\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}+\frac {\cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4} \]
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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 ^4\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^4} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {\sinh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{64 x}+\frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{64 x}-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{64 x}-\frac {3 \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{64 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^4} \\ & = -\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^4}-\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^4}+\frac {3 \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^4}+\frac {3 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^4} \\ & = -\frac {\left (3 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^4}-\frac {\left (3 \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 {arcsinh}(c x)\right )}{64 b c^4}+\frac {\cosh \left (\frac {5 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^4}+\frac {\cosh \left (\frac {7 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^4}+\frac {\left (3 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^4}+\frac {\left (3 \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 {arcsinh}(c x)\right )}{64 b c^4}-\frac {\sinh \left (\frac {5 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^4}-\frac {\sinh \left (\frac {7 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^4} \\ & = \frac {3 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{64 b c^4}+\frac {3 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{64 b c^4}-\frac {\text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{64 b c^4}-\frac {\text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {7 a}{b}\right )}{64 b c^4}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^4}-\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}+\frac {\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4}+\frac {\cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^4} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {3 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+3 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-\text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {7 a}{b}\right )-3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{64 b c^4} \]
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Time = 0.24 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {{\mathrm e}^{\frac {7 a}{b}} \operatorname {Ei}_{1}\left (7 \,\operatorname {arcsinh}\left (c x \right )+\frac {7 a}{b}\right )+{\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right )-3 \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )-3 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )+3 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right )+3 \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )-{\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right )-{\mathrm e}^{-\frac {7 a}{b}} \operatorname {Ei}_{1}\left (-7 \,\operatorname {arcsinh}\left (c x \right )-\frac {7 a}{b}\right )}{128 c^{4} b}\) | \(196\) |
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\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^{3} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
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\[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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Exception generated. \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {x^3 \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^3\,{\left (c^2\,x^2+1\right )}^{3/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]
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