Integrand size = 19, antiderivative size = 77 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=-\frac {c^2 \left (1+a^2 x^2\right )^{5/2}}{a \text {arcsinh}(a x)}+\frac {5 c^2 \text {Shi}(\text {arcsinh}(a x))}{8 a}+\frac {15 c^2 \text {Shi}(3 \text {arcsinh}(a x))}{16 a}+\frac {5 c^2 \text {Shi}(5 \text {arcsinh}(a x))}{16 a} \]
-c^2*(a^2*x^2+1)^(5/2)/a/arcsinh(a*x)+5/8*c^2*Shi(arcsinh(a*x))/a+15/16*c^ 2*Shi(3*arcsinh(a*x))/a+5/16*c^2*Shi(5*arcsinh(a*x))/a
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=\frac {c^2 \left (-16 \left (1+a^2 x^2\right )^{5/2}+10 \text {arcsinh}(a x) \text {Shi}(\text {arcsinh}(a x))+15 \text {arcsinh}(a x) \text {Shi}(3 \text {arcsinh}(a x))+5 \text {arcsinh}(a x) \text {Shi}(5 \text {arcsinh}(a x))\right )}{16 a \text {arcsinh}(a x)} \]
(c^2*(-16*(1 + a^2*x^2)^(5/2) + 10*ArcSinh[a*x]*SinhIntegral[ArcSinh[a*x]] + 15*ArcSinh[a*x]*SinhIntegral[3*ArcSinh[a*x]] + 5*ArcSinh[a*x]*SinhInteg ral[5*ArcSinh[a*x]]))/(16*a*ArcSinh[a*x])
Time = 0.47 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6205, 6234, 5971, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a^2 c x^2+c\right )^2}{\text {arcsinh}(a x)^2} \, dx\) |
| \(\Big \downarrow \) 6205 |
| \(\displaystyle 5 a c^2 \int \frac {x \left (a^2 x^2+1\right )^{3/2}}{\text {arcsinh}(a x)}dx-\frac {c^2 \left (a^2 x^2+1\right )^{5/2}}{a \text {arcsinh}(a x)}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {5 c^2 \int \frac {a x \left (a^2 x^2+1\right )^2}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a}-\frac {c^2 \left (a^2 x^2+1\right )^{5/2}}{a \text {arcsinh}(a x)}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {5 c^2 \int \left (\frac {a x}{8 \text {arcsinh}(a x)}+\frac {3 \sinh (3 \text {arcsinh}(a x))}{16 \text {arcsinh}(a x)}+\frac {\sinh (5 \text {arcsinh}(a x))}{16 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a}-\frac {c^2 \left (a^2 x^2+1\right )^{5/2}}{a \text {arcsinh}(a x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 c^2 \left (\frac {1}{8} \text {Shi}(\text {arcsinh}(a x))+\frac {3}{16} \text {Shi}(3 \text {arcsinh}(a x))+\frac {1}{16} \text {Shi}(5 \text {arcsinh}(a x))\right )}{a}-\frac {c^2 \left (a^2 x^2+1\right )^{5/2}}{a \text {arcsinh}(a x)}\) |
-((c^2*(1 + a^2*x^2)^(5/2))/(a*ArcSinh[a*x])) + (5*c^2*(SinhIntegral[ArcSi nh[a*x]]/8 + (3*SinhIntegral[3*ArcSinh[a*x]])/16 + SinhIntegral[5*ArcSinh[ a*x]]/16))/a
3.5.7.3.1 Defintions of rubi rules used
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[Simp[Sqrt[1 + c^2*x^2]*(d + e*x^2)^p]*((a + b*ArcSinh[c*x] )^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Simp[(d + e*x ^2)^p/(1 + c^2*x^2)^p] Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x]) ^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && LtQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
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 | \(\frac {c^{2} \left (10 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+15 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+5 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )-5 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )-\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )-10 \sqrt {a^{2} x^{2}+1}\right )}{16 a \,\operatorname {arcsinh}\left (a x \right )}\) | \(84\) |
1/16/a*c^2*(10*Shi(arcsinh(a*x))*arcsinh(a*x)+15*Shi(3*arcsinh(a*x))*arcsi nh(a*x)+5*Shi(5*arcsinh(a*x))*arcsinh(a*x)-5*cosh(3*arcsinh(a*x))-cosh(5*a rcsinh(a*x))-10*(a^2*x^2+1)^(1/2))/arcsinh(a*x)
\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=c^{2} \left (\int \frac {2 a^{2} x^{2}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{4}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx\right ) \]
c**2*(Integral(2*a**2*x**2/asinh(a*x)**2, x) + Integral(a**4*x**4/asinh(a* x)**2, x) + Integral(asinh(a*x)**(-2), x))
\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
-(a^7*c^2*x^7 + 3*a^5*c^2*x^5 + 3*a^3*c^2*x^3 + a*c^2*x + (a^6*c^2*x^6 + 3 *a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*sqrt(a^2*x^2 + 1))/((a^3*x^2 + sqrt(a^ 2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt(a^2*x^2 + 1))) + integrate((5*a^8*c^2 *x^8 + 16*a^6*c^2*x^6 + 18*a^4*c^2*x^4 + 8*a^2*c^2*x^2 + (5*a^6*c^2*x^6 + 9*a^4*c^2*x^4 + 3*a^2*c^2*x^2 - c^2)*(a^2*x^2 + 1) + c^2 + 5*(2*a^7*c^2*x^ 7 + 5*a^5*c^2*x^5 + 4*a^3*c^2*x^3 + a*c^2*x)*sqrt(a^2*x^2 + 1))/((a^4*x^4 + (a^2*x^2 + 1)*a^2*x^2 + 2*a^2*x^2 + 2*(a^3*x^3 + a*x)*sqrt(a^2*x^2 + 1) + 1)*log(a*x + sqrt(a^2*x^2 + 1))), x)
\[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)^2} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^2}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]