Integrand size = 28, antiderivative size = 439 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{128 b c^3 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{128 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{128 b c^3 \sqrt {-1+c x}} \]
1/32*Chi(2*(a+b*arccosh(c*x))/b)*cosh(2*a/b)*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^ (1/2)+1/32*Chi(4*(a+b*arccosh(c*x))/b)*cosh(4*a/b)*(-c*x+1)^(1/2)/b/c^3/(c *x-1)^(1/2)-1/32*Chi(6*(a+b*arccosh(c*x))/b)*cosh(6*a/b)*(-c*x+1)^(1/2)/b/ c^3/(c*x-1)^(1/2)+1/128*Chi(8*(a+b*arccosh(c*x))/b)*cosh(8*a/b)*(-c*x+1)^( 1/2)/b/c^3/(c*x-1)^(1/2)-5/128*ln(a+b*arccosh(c*x))*(-c*x+1)^(1/2)/b/c^3/( c*x-1)^(1/2)-1/32*Shi(2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)*(-c*x+1)^(1/2)/b /c^3/(c*x-1)^(1/2)-1/32*Shi(4*(a+b*arccosh(c*x))/b)*sinh(4*a/b)*(-c*x+1)^( 1/2)/b/c^3/(c*x-1)^(1/2)+1/32*Shi(6*(a+b*arccosh(c*x))/b)*sinh(6*a/b)*(-c* x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)-1/128*Shi(8*(a+b*arccosh(c*x))/b)*sinh(8*a/ b)*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)
Time = 1.04 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.53 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c^2 x^2} \left (4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-4 \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (8 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-5 \log (a+b \text {arccosh}(c x))-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-4 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{128 c^3 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]
(Sqrt[1 - c^2*x^2]*(4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])] + 4*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])] - 4*Cosh[(6*a)/b]*Co shIntegral[6*(a/b + ArcCosh[c*x])] + Cosh[(8*a)/b]*CoshIntegral[8*(a/b + A rcCosh[c*x])] - 5*Log[a + b*ArcCosh[c*x]] - 4*Sinh[(2*a)/b]*SinhIntegral[2 *(a/b + ArcCosh[c*x])] - 4*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x ])] + 4*Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])] - Sinh[(8*a)/b] *SinhIntegral[8*(a/b + ArcCosh[c*x])]))/(128*c^3*Sqrt[(-1 + c*x)/(1 + c*x) ]*(b + b*c*x))
Time = 0.75 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {6367, 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.
| \(\displaystyle \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx\) |
| \(\Big \downarrow \) 6367 |
| \(\displaystyle \frac {\sqrt {1-c x} \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^6\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c^3 \sqrt {c x-1}}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\sqrt {1-c x} \int \left (\frac {\cosh \left (\frac {8 a}{b}-\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{128 (a+b \text {arccosh}(c x))}-\frac {\cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}-\frac {5}{128 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b c^3 \sqrt {c x-1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c x} \left (\frac {1}{32} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{32} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{128} \cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{32} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{128} \sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{128} \log (a+b \text {arccosh}(c x))\right )}{b c^3 \sqrt {c x-1}}\) |
(Sqrt[1 - c*x]*((Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/3 2 + (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/32 - (Cosh[(6 *a)/b]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b])/32 + (Cosh[(8*a)/b]*CoshI ntegral[(8*(a + b*ArcCosh[c*x]))/b])/128 - (5*Log[a + b*ArcCosh[c*x]])/128 - (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/32 - (Sinh[(4* a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/32 + (Sinh[(6*a)/b]*SinhIn tegral[(6*(a + b*ArcCosh[c*x]))/b])/32 - (Sinh[(8*a)/b]*SinhIntegral[(8*(a + b*ArcCosh[c*x]))/b])/128))/(b*c^3*Sqrt[-1 + c*x])
3.3.85.3.1 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_.) + ArcCosh[(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*x )^p*(-1 + c*x)^p)] Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && Eq Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.80 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-10 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-10 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +4 \,\operatorname {Ei}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}-\operatorname {Ei}_{1}\left (8 \,\operatorname {arccosh}\left (c x \right )+\frac {8 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+8 a}{b}}-\operatorname {Ei}_{1}\left (-8 \,\operatorname {arccosh}\left (c x \right )-\frac {8 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+8 a}{b}}+4 \,\operatorname {Ei}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}-4 \,\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-4 \,\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-4 \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-4 \,\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right ) b}\) | \(365\) |
-1/256*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(-1 0*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(a+b*arccosh(c*x))-10*ln(a+b*arccosh(c*x)) *c*x+4*Ei(1,-6*arccosh(c*x)-6*a/b)*exp(-(-b*arccosh(c*x)+6*a)/b)-Ei(1,8*ar ccosh(c*x)+8*a/b)*exp((b*arccosh(c*x)+8*a)/b)-Ei(1,-8*arccosh(c*x)-8*a/b)* exp(-(-b*arccosh(c*x)+8*a)/b)+4*Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh( c*x)+6*a)/b)-4*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)-4*Ei (1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)-4*Ei(1,-2*arccosh(c*x )-2*a/b)*exp(-(-b*arccosh(c*x)+2*a)/b)-4*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(- (-b*arccosh(c*x)+4*a)/b))/(c*x+1)/c^3/(c*x-1)/b
\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^2\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]