Integrand size = 25, antiderivative size = 339 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {15 \sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{16 b c \sqrt {-1+c x}}-\frac {15 \sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c \sqrt {-1+c x}} \]
15/32*Chi(2*(a+b*arccosh(c*x))/b)*cosh(2*a/b)*(-c*x+1)^(1/2)/b/c/(c*x-1)^( 1/2)-3/16*Chi(4*(a+b*arccosh(c*x))/b)*cosh(4*a/b)*(-c*x+1)^(1/2)/b/c/(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/( c*x-1)^(1/2)-5/16*ln(a+b*arccosh(c*x))*(-c*x+1)^(1/2)/b/c/(c*x-1)^(1/2)-15 /32*Shi(2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)*(-c*x+1)^(1/2)/b/c/(c*x-1)^(1/ 2)+3/16*Shi(4*(a+b*arccosh(c*x))/b)*sinh(4*a/b)*(-c*x+1)^(1/2)/b/c/(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/(c* x-1)^(1/2)
Time = 0.76 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.56 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c^2 x^2} \left (15 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-6 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-10 \log (a+b \text {arccosh}(c x))-15 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+6 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{32 b c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
(Sqrt[1 - c^2*x^2]*(15*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])] - 6*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])] + Cosh[(6*a)/b]*Cos hIntegral[6*(a/b + ArcCosh[c*x])] - 10*Log[a + b*ArcCosh[c*x]] - 15*Sinh[( 2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 6*Sinh[(4*a)/b]*SinhIntegra l[4*(a/b + ArcCosh[c*x])] - Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c* x])]))/(32*b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
Time = 0.57 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.56, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6321, 3042, 25, 3793, 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.
| \(\displaystyle \int \frac {\left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx\) |
| \(\Big \downarrow \) 6321 |
| \(\displaystyle \frac {\sqrt {1-c x} \int \frac {\sinh ^6\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {1-c x} \int -\frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^6}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {1-c x} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^6}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1}}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {\sqrt {1-c x} \int \left (-\frac {\cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}+\frac {3 \cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}-\frac {15 \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}+\frac {5}{16 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c x} \left (\frac {15}{32} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{16} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{32} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )-\frac {15}{32} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {3}{16} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{16} \log (a+b \text {arccosh}(c x))\right )}{b c \sqrt {c x-1}}\) |
(Sqrt[1 - c*x]*((15*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b] )/32 - (3*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/16 + (Co sh[(6*a)/b]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b])/32 - (5*Log[a + b*Ar cCosh[c*x]])/16 - (15*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/ b])/32 + (3*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/16 - ( Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/32))/(b*c*Sqrt[-1 + c*x])
3.3.87.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
Time = 0.95 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (20 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+20 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {Ei}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}+\operatorname {Ei}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}-6 \,\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+15 \,\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}+15 \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-6 \,\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}\right )}{64 \left (c x -1\right ) \left (c x +1\right ) c b}\) | \(297\) |
1/64*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(20*( c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(a+b*arccosh(c*x))+20*ln(a+b*arccosh(c*x))*c* x+Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)+Ei(1,-6*arccosh(c *x)-6*a/b)*exp(-(-b*arccosh(c*x)+6*a)/b)-6*Ei(1,4*arccosh(c*x)+4*a/b)*exp( (b*arccosh(c*x)+4*a)/b)+15*Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+ 2*a)/b)+15*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-(-b*arccosh(c*x)+2*a)/b)-6*Ei( 1,-4*arccosh(c*x)-4*a/b)*exp(-(-b*arccosh(c*x)+4*a)/b))/(c*x-1)/(c*x+1)/c/ b
\[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\text {Timed out} \]
\[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]