Integrand size = 19, antiderivative size = 50 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)} \, dx=\frac {5 c^2 \text {Chi}(\text {arcsinh}(a x))}{8 a}+\frac {5 c^2 \text {Chi}(3 \text {arcsinh}(a x))}{16 a}+\frac {c^2 \text {Chi}(5 \text {arcsinh}(a x))}{16 a} \]
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Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5791, 3393, 3382} \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)} \, dx=\frac {5 c^2 \text {Chi}(\text {arcsinh}(a x))}{8 a}+\frac {5 c^2 \text {Chi}(3 \text {arcsinh}(a x))}{16 a}+\frac {c^2 \text {Chi}(5 \text {arcsinh}(a x))}{16 a} \]
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Rule 3382
Rule 3393
Rule 5791
Rubi steps \begin{align*} \text {integral}& = \frac {c^2 \text {Subst}\left (\int \frac {\cosh ^5(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a} \\ & = \frac {c^2 \text {Subst}\left (\int \left (\frac {5 \cosh (x)}{8 x}+\frac {5 \cosh (3 x)}{16 x}+\frac {\cosh (5 x)}{16 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a} \\ & = \frac {c^2 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{8 a} \\ & = \frac {5 c^2 \text {Chi}(\text {arcsinh}(a x))}{8 a}+\frac {5 c^2 \text {Chi}(3 \text {arcsinh}(a x))}{16 a}+\frac {c^2 \text {Chi}(5 \text {arcsinh}(a x))}{16 a} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)} \, dx=\frac {c^2 (10 \text {Chi}(\text {arcsinh}(a x))+5 \text {Chi}(3 \text {arcsinh}(a x))+\text {Chi}(5 \text {arcsinh}(a x)))}{16 a} \]
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Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {c^{2} \left (10 \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )+5 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )+\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )\right )}{16 a}\) | \(33\) |
default | \(\frac {c^{2} \left (10 \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )+5 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )+\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )\right )}{16 a}\) | \(33\) |
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\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)} \, dx=c^{2} \left (\int \frac {2 a^{2} x^{2}}{\operatorname {asinh}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{4}}{\operatorname {asinh}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asinh}{\left (a x \right )}}\, dx\right ) \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^2}{\mathrm {asinh}\left (a\,x\right )} \,d x \]
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