Integrand size = 28, antiderivative size = 236 \[ \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^4 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}-\frac {\sqrt {-1+c x} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c^5 \sqrt {1-c x}}-\frac {\sqrt {-1+c x} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c^5 \sqrt {1-c x}}+\frac {\sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c^5 \sqrt {1-c x}}+\frac {\sqrt {-1+c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^5 \sqrt {1-c x}} \]
-x^4*(c*x-1)^(1/2)/b/c/(a+b*arccosh(c*x))/(-c*x+1)^(1/2)+cosh(2*a/b)*Shi(2 *(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)/b^2/c^5/(-c*x+1)^(1/2)+1/2*cosh(4*a/b )*Shi(4*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)/b^2/c^5/(-c*x+1)^(1/2)-Chi(2*( a+b*arccosh(c*x))/b)*sinh(2*a/b)*(c*x-1)^(1/2)/b^2/c^5/(-c*x+1)^(1/2)-1/2* Chi(4*(a+b*arccosh(c*x))/b)*sinh(4*a/b)*(c*x-1)^(1/2)/b^2/c^5/(-c*x+1)^(1/ 2)
Time = 0.37 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.63 \[ \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2} \left (\frac {2 b c^4 x^4}{a+b \text {arccosh}(c x)}+2 \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+\text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )-2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{2 b^2 c^5 \sqrt {-1+c x} \sqrt {1+c x}} \]
(Sqrt[1 - c^2*x^2]*((2*b*c^4*x^4)/(a + b*ArcCosh[c*x]) + 2*CoshIntegral[2* (a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] + CoshIntegral[4*(a/b + ArcCosh[c*x])] *Sinh[(4*a)/b] - 2*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - Co sh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]))/(2*b^2*c^5*Sqrt[-1 + c* x]*Sqrt[1 + c*x])
Time = 0.69 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6365, 6302, 25, 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.
| \(\displaystyle \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx\) |
| \(\Big \downarrow \) 6365 |
| \(\displaystyle \frac {4 \sqrt {c x-1} \int \frac {x^3}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {1-c x}}-\frac {x^4 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6302 |
| \(\displaystyle \frac {4 \sqrt {c x-1} \int -\frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^5 \sqrt {1-c x}}-\frac {x^4 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 \sqrt {c x-1} \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^5 \sqrt {1-c x}}-\frac {x^4 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle -\frac {4 \sqrt {c x-1} \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 (a+b \text {arccosh}(c x))}+\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{4 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c^5 \sqrt {1-c x}}-\frac {x^4 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \sqrt {c x-1} \left (-\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^5 \sqrt {1-c x}}-\frac {x^4 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}\) |
-((x^4*Sqrt[-1 + c*x])/(b*c*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) + (4*Sqrt [-1 + c*x]*(-1/4*(CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b]) - (CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/8 + (Cosh[(2*a) /b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/4 + (Cosh[(4*a)/b]*SinhInteg ral[(4*(a + b*ArcCosh[c*x]))/b])/8))/(b^2*c^5*Sqrt[1 - c*x])
3.4.43.3.1 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_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/( b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], x] - Si mp[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])] Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c , d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
Time = 0.87 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.64
| method | result | size |
| default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-4 x^{4} c^{4} b \sqrt {c x -1}\, \sqrt {c x +1}+4 b \,c^{5} x^{5}+\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+2 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-2 \,\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+a \,\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+2 a \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} a -2 \,\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} a \right )}{4 c^{5} \left (c^{2} x^{2}-1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\) | \(387\) |
1/4*(-c^2*x^2+1)^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(-4*x^4 *c^4*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*b*c^5*x^5+arccosh(c*x)*b*Ei(1,-4*arcc osh(c*x)-4*a/b)*exp(-(b*arccosh(c*x)+4*a)/b)+2*arccosh(c*x)*b*Ei(1,-2*arcc osh(c*x)-2*a/b)*exp(-(b*arccosh(c*x)+2*a)/b)-Ei(1,4*arccosh(c*x)+4*a/b)*ex p((-b*arccosh(c*x)+4*a)/b)*b*arccosh(c*x)-2*Ei(1,2*arccosh(c*x)+2*a/b)*exp ((-b*arccosh(c*x)+2*a)/b)*b*arccosh(c*x)+a*Ei(1,-4*arccosh(c*x)-4*a/b)*exp (-(b*arccosh(c*x)+4*a)/b)+2*a*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-(b*arccosh( c*x)+2*a)/b)-Ei(1,4*arccosh(c*x)+4*a/b)*exp((-b*arccosh(c*x)+4*a)/b)*a-2*E i(1,2*arccosh(c*x)+2*a/b)*exp((-b*arccosh(c*x)+2*a)/b)*a)/c^5/(c^2*x^2-1)/ b^2/(a+b*arccosh(c*x))
\[ \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{4}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
integral(-sqrt(-c^2*x^2 + 1)*x^4/(a^2*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arccos h(c*x)^2 - a^2 + 2*(a*b*c^2*x^2 - a*b)*arccosh(c*x)), x)
\[ \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^{4}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{4}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
-(c^3*x^7 - c*x^5 + (c^2*x^6 - x^4)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x + (b^2*c^3*x^2 - b^2*c)*sqrt(c*x + 1))*sqrt(-c* x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)*sqrt(c*x - 1)*a *b*c^2*x + (a*b*c^3*x^2 - a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)) + integrat e((4*c^5*x^8 - 9*c^3*x^6 + 5*c*x^4 + (4*c^3*x^6 - 3*c*x^4)*(c*x + 1)*(c*x - 1) + 4*(2*c^4*x^7 - 3*c^2*x^5 + x^3)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 1)^(3/2)*(c*x - 1)*b^2*c^3*x^2 + 2*(b^2*c^4*x^3 - b^2*c^2*x)*(c*x + 1)* sqrt(c*x - 1) + (b^2*c^5*x^4 - 2*b^2*c^3*x^2 + b^2*c)*sqrt(c*x + 1))*sqrt( -c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)^(3/2)*(c*x - 1)*a*b*c^3*x^2 + 2*(a*b*c^4*x^3 - a*b*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (a *b*c^5*x^4 - 2*a*b*c^3*x^2 + a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)), x)
\[ \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {x^{4}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^4}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]