Integrand size = 19, antiderivative size = 94 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=-\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \text {arcsinh}(a x)}+\frac {35 c^3 \text {Shi}(\text {arcsinh}(a x))}{64 a}+\frac {63 c^3 \text {Shi}(3 \text {arcsinh}(a x))}{64 a}+\frac {35 c^3 \text {Shi}(5 \text {arcsinh}(a x))}{64 a}+\frac {7 c^3 \text {Shi}(7 \text {arcsinh}(a x))}{64 a} \]
-c^3*(a^2*x^2+1)^(7/2)/a/arcsinh(a*x)+35/64*c^3*Shi(arcsinh(a*x))/a+63/64* c^3*Shi(3*arcsinh(a*x))/a+35/64*c^3*Shi(5*arcsinh(a*x))/a+7/64*c^3*Shi(7*a rcsinh(a*x))/a
Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=\frac {c^3 \left (-64 \left (1+a^2 x^2\right )^{7/2}+35 \text {arcsinh}(a x) \text {Shi}(\text {arcsinh}(a x))+63 \text {arcsinh}(a x) \text {Shi}(3 \text {arcsinh}(a x))+35 \text {arcsinh}(a x) \text {Shi}(5 \text {arcsinh}(a x))+7 \text {arcsinh}(a x) \text {Shi}(7 \text {arcsinh}(a x))\right )}{64 a \text {arcsinh}(a x)} \]
(c^3*(-64*(1 + a^2*x^2)^(7/2) + 35*ArcSinh[a*x]*SinhIntegral[ArcSinh[a*x]] + 63*ArcSinh[a*x]*SinhIntegral[3*ArcSinh[a*x]] + 35*ArcSinh[a*x]*SinhInte gral[5*ArcSinh[a*x]] + 7*ArcSinh[a*x]*SinhIntegral[7*ArcSinh[a*x]]))/(64*a *ArcSinh[a*x])
Time = 0.49 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.84, 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.
| \(\displaystyle \int \frac {\left (a^2 c x^2+c\right )^3}{\text {arcsinh}(a x)^2} \, dx\) |
| \(\Big \downarrow \) 6205 |
| \(\displaystyle 7 a c^3 \int \frac {x \left (a^2 x^2+1\right )^{5/2}}{\text {arcsinh}(a x)}dx-\frac {c^3 \left (a^2 x^2+1\right )^{7/2}}{a \text {arcsinh}(a x)}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {7 c^3 \int \frac {a x \left (a^2 x^2+1\right )^3}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a}-\frac {c^3 \left (a^2 x^2+1\right )^{7/2}}{a \text {arcsinh}(a x)}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {7 c^3 \int \left (\frac {5 a x}{64 \text {arcsinh}(a x)}+\frac {9 \sinh (3 \text {arcsinh}(a x))}{64 \text {arcsinh}(a x)}+\frac {5 \sinh (5 \text {arcsinh}(a x))}{64 \text {arcsinh}(a x)}+\frac {\sinh (7 \text {arcsinh}(a x))}{64 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a}-\frac {c^3 \left (a^2 x^2+1\right )^{7/2}}{a \text {arcsinh}(a x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7 c^3 \left (\frac {5}{64} \text {Shi}(\text {arcsinh}(a x))+\frac {9}{64} \text {Shi}(3 \text {arcsinh}(a x))+\frac {5}{64} \text {Shi}(5 \text {arcsinh}(a x))+\frac {1}{64} \text {Shi}(7 \text {arcsinh}(a x))\right )}{a}-\frac {c^3 \left (a^2 x^2+1\right )^{7/2}}{a \text {arcsinh}(a x)}\) |
-((c^3*(1 + a^2*x^2)^(7/2))/(a*ArcSinh[a*x])) + (7*c^3*((5*SinhIntegral[Ar cSinh[a*x]])/64 + (9*SinhIntegral[3*ArcSinh[a*x]])/64 + (5*SinhIntegral[5* ArcSinh[a*x]])/64 + SinhIntegral[7*ArcSinh[a*x]]/64))/a
3.5.6.3.1 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 = 0.24 (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 | \(\frac {c^{3} \left (35 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+63 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+35 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )+7 \,\operatorname {Shi}\left (7 \,\operatorname {arcsinh}\left (a x \right )\right ) \operatorname {arcsinh}\left (a x \right )-35 \sqrt {a^{2} x^{2}+1}-21 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )-7 \cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )-\cosh \left (7 \,\operatorname {arcsinh}\left (a x \right )\right )\right )}{64 a \,\operatorname {arcsinh}\left (a x \right )}\) | \(106\) |
1/64/a*c^3*(35*Shi(arcsinh(a*x))*arcsinh(a*x)+63*Shi(3*arcsinh(a*x))*arcsi nh(a*x)+35*Shi(5*arcsinh(a*x))*arcsinh(a*x)+7*Shi(7*arcsinh(a*x))*arcsinh( a*x)-35*(a^2*x^2+1)^(1/2)-21*cosh(3*arcsinh(a*x))-7*cosh(5*arcsinh(a*x))-c osh(7*arcsinh(a*x)))/arcsinh(a*x)
\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx\right ) \]
c**3*(Integral(3*a**2*x**2/asinh(a*x)**2, x) + Integral(3*a**4*x**4/asinh( a*x)**2, x) + Integral(a**6*x**6/asinh(a*x)**2, x) + Integral(asinh(a*x)** (-2), x))
\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
-(a^9*c^3*x^9 + 4*a^7*c^3*x^7 + 6*a^5*c^3*x^5 + 4*a^3*c^3*x^3 + a*c^3*x + (a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 + 4*a^2*c^3*x^2 + c^3)*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((7*a^10*c^3*x^10 + 29*a^8*c^3*x^8 + 46*a^6*c^3*x^6 + 34*a^4*c^3*x^4 + 11*a^2*c^3*x^2 + c^3 + (7*a^8*c^3*x^8 + 20*a^6*c^3*x^6 + 18*a^4*c^3*x^4 + 4*a^2*c^3*x^2 - c^3)*(a^2*x^2 + 1) + 7*(2*a^9*c^3*x^9 + 7*a^7*c^3*x^7 + 9*a^5*c^3*x^5 + 5*a^3*c^3*x^3 + a*c^3*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)
\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)^2} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^3}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]