Integrand size = 19, antiderivative size = 77 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=-\frac {c^2 \left (1+a^2 x^2\right )^{5/2}}{a \text {arcsinh}(a x)}+\frac {5 c^2 \text {Shi}(\text {arcsinh}(a x))}{8 a}+\frac {15 c^2 \text {Shi}(3 \text {arcsinh}(a x))}{16 a}+\frac {5 c^2 \text {Shi}(5 \text {arcsinh}(a x))}{16 a} \]
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Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5790, 5819, 5556, 3379} \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=-\frac {c^2 \left (a^2 x^2+1\right )^{5/2}}{a \text {arcsinh}(a x)}+\frac {5 c^2 \text {Shi}(\text {arcsinh}(a x))}{8 a}+\frac {15 c^2 \text {Shi}(3 \text {arcsinh}(a x))}{16 a}+\frac {5 c^2 \text {Shi}(5 \text {arcsinh}(a x))}{16 a} \]
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Rule 3379
Rule 5556
Rule 5790
Rule 5819
Rubi steps \begin{align*} \text {integral}& = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2}}{a \text {arcsinh}(a x)}+\left (5 a c^2\right ) \int \frac {x \left (1+a^2 x^2\right )^{3/2}}{\text {arcsinh}(a x)} \, dx \\ & = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2}}{a \text {arcsinh}(a x)}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a} \\ & = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2}}{a \text {arcsinh}(a x)}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \left (\frac {\sinh (x)}{8 x}+\frac {3 \sinh (3 x)}{16 x}+\frac {\sinh (5 x)}{16 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a} \\ & = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2}}{a \text {arcsinh}(a x)}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{8 a}+\frac {\left (15 c^2\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a} \\ & = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2}}{a \text {arcsinh}(a x)}+\frac {5 c^2 \text {Shi}(\text {arcsinh}(a x))}{8 a}+\frac {15 c^2 \text {Shi}(3 \text {arcsinh}(a x))}{16 a}+\frac {5 c^2 \text {Shi}(5 \text {arcsinh}(a x))}{16 a} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=\frac {c^2 \left (-16 \left (1+a^2 x^2\right )^{5/2}+10 \text {arcsinh}(a x) \text {Shi}(\text {arcsinh}(a x))+15 \text {arcsinh}(a x) \text {Shi}(3 \text {arcsinh}(a x))+5 \text {arcsinh}(a x) \text {Shi}(5 \text {arcsinh}(a x))\right )}{16 a \text {arcsinh}(a x)} \]
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Time = 0.34 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {c^{2} \left (10 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+15 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+5 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )-5 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )-\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )-10 \sqrt {a^{2} x^{2}+1}\right )}{16 a \,\operatorname {arcsinh}\left (a x \right )}\) | \(84\) |
default | \(\frac {c^{2} \left (10 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+15 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+5 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )-5 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )-\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )-10 \sqrt {a^{2} x^{2}+1}\right )}{16 a \,\operatorname {arcsinh}\left (a x \right )}\) | \(84\) |
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\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=c^{2} \left (\int \frac {2 a^{2} x^{2}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{4}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx\right ) \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^2}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]
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