Integrand size = 19, antiderivative size = 94 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=-\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \text {arcsinh}(a x)}+\frac {35 c^3 \text {Shi}(\text {arcsinh}(a x))}{64 a}+\frac {63 c^3 \text {Shi}(3 \text {arcsinh}(a x))}{64 a}+\frac {35 c^3 \text {Shi}(5 \text {arcsinh}(a x))}{64 a}+\frac {7 c^3 \text {Shi}(7 \text {arcsinh}(a x))}{64 a} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5790, 5819, 5556, 3379} \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=-\frac {c^3 \left (a^2 x^2+1\right )^{7/2}}{a \text {arcsinh}(a x)}+\frac {35 c^3 \text {Shi}(\text {arcsinh}(a x))}{64 a}+\frac {63 c^3 \text {Shi}(3 \text {arcsinh}(a x))}{64 a}+\frac {35 c^3 \text {Shi}(5 \text {arcsinh}(a x))}{64 a}+\frac {7 c^3 \text {Shi}(7 \text {arcsinh}(a x))}{64 a} \]
[In]
[Out]
Rule 3379
Rule 5556
Rule 5790
Rule 5819
Rubi steps \begin{align*} \text {integral}& = -\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \text {arcsinh}(a x)}+\left (7 a c^3\right ) \int \frac {x \left (1+a^2 x^2\right )^{5/2}}{\text {arcsinh}(a x)} \, dx \\ & = -\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \text {arcsinh}(a x)}+\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\cosh ^6(x) \sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a} \\ & = -\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \text {arcsinh}(a x)}+\frac {\left (7 c^3\right ) \text {Subst}\left (\int \left (\frac {5 \sinh (x)}{64 x}+\frac {9 \sinh (3 x)}{64 x}+\frac {5 \sinh (5 x)}{64 x}+\frac {\sinh (7 x)}{64 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a} \\ & = -\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \text {arcsinh}(a x)}+\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\sinh (7 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a}+\frac {\left (63 c^3\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a} \\ & = -\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \text {arcsinh}(a x)}+\frac {35 c^3 \text {Shi}(\text {arcsinh}(a x))}{64 a}+\frac {63 c^3 \text {Shi}(3 \text {arcsinh}(a x))}{64 a}+\frac {35 c^3 \text {Shi}(5 \text {arcsinh}(a x))}{64 a}+\frac {7 c^3 \text {Shi}(7 \text {arcsinh}(a x))}{64 a} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=\frac {c^3 \left (-64 \left (1+a^2 x^2\right )^{7/2}+35 \text {arcsinh}(a x) \text {Shi}(\text {arcsinh}(a x))+63 \text {arcsinh}(a x) \text {Shi}(3 \text {arcsinh}(a x))+35 \text {arcsinh}(a x) \text {Shi}(5 \text {arcsinh}(a x))+7 \text {arcsinh}(a x) \text {Shi}(7 \text {arcsinh}(a x))\right )}{64 a \text {arcsinh}(a x)} \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {c^{3} \left (35 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+63 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+35 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+7 \,\operatorname {Shi}\left (7 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )-35 \sqrt {a^{2} x^{2}+1}-21 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )-7 \cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )-\cosh \left (7 \,\operatorname {arcsinh}\left (a x \right )\right )\right )}{64 a \,\operatorname {arcsinh}\left (a x \right )}\) | \(106\) |
default | \(\frac {c^{3} \left (35 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+63 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+35 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+7 \,\operatorname {Shi}\left (7 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )-35 \sqrt {a^{2} x^{2}+1}-21 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )-7 \cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )-\cosh \left (7 \,\operatorname {arcsinh}\left (a x \right )\right )\right )}{64 a \,\operatorname {arcsinh}\left (a x \right )}\) | \(106\) |
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx\right ) \]
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^3}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]
[In]
[Out]