Integrand size = 18, antiderivative size = 245 \[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{4 b c^3}+\frac {e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {arcsinh}(c x)\right )}{4 b c^3}-\frac {d e \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b c^2}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}+\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{4 b c^3}+\frac {d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{b c^2}-\frac {e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {arcsinh}(c x)\right )}{4 b c^3} \]
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Time = 0.54 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5830, 6874, 3384, 3379, 3382, 5556} \[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{4 b c^3}+\frac {e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {arcsinh}(c x)\right )}{4 b c^3}+\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{4 b c^3}-\frac {e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {arcsinh}(c x)\right )}{4 b c^3}-\frac {d e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{b c^2}+\frac {d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{b c^2}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5830
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (x) (c d+e \sinh (x))^2}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {c^2 d^2 \cosh (x)}{a+b x}+\frac {e^2 \cosh (x) \sinh ^2(x)}{a+b x}+\frac {c d e \sinh (2 x)}{a+b x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^3} \\ & = \frac {d^2 \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c}+\frac {(d e) \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c^2}+\frac {e^2 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c^3} \\ & = \frac {e^2 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 (a+b x)}+\frac {\cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^3}+\frac {\left (d^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c}+\frac {\left (d e \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c^2}-\frac {\left (d^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c}-\frac {\left (d e \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{c^2} \\ & = \frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {d e \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b c^2}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}+\frac {d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{b c^2}-\frac {e^2 \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3}+\frac {e^2 \text {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3} \\ & = \frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {d e \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b c^2}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}+\frac {d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{b c^2}-\frac {\left (e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3}+\frac {\left (e^2 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3}+\frac {\left (e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3}-\frac {\left (e^2 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3} \\ & = \frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{4 b c^3}+\frac {e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {arcsinh}(c x)\right )}{4 b c^3}-\frac {d e \text {Chi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b c^2}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b c}+\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{4 b c^3}+\frac {d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {arcsinh}(c x)\right )}{b c^2}-\frac {e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {arcsinh}(c x)\right )}{4 b c^3} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\frac {\left (4 c^2 d^2-e^2\right ) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-4 c d e \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-4 c^2 d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+4 c d e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{4 b c^3} \]
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Time = 1.00 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {-\frac {e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{8 c^{2} b}+\frac {d e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c b}-\frac {d e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c b}}{c}\) | \(254\) |
default | \(\frac {-\frac {e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{8 c^{2} b}+\frac {d e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c b}-\frac {d e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c b}}{c}\) | \(254\) |
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\[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\left (d + e x\right )^{2}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
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\[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^2}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]
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