Integrand size = 28, antiderivative size = 397 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {3 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{128 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^4 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{128 b c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 (a+b \text {arccosh}(c x))}{b}\right )}{256 b c^4 \sqrt {-1+c x}} \]
-3/128*Chi((a+b*arccosh(c*x))/b)*cosh(a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1)^(1 /2)+1/32*Chi(3*(a+b*arccosh(c*x))/b)*cosh(3*a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x -1)^(1/2)-3/256*Chi(7*(a+b*arccosh(c*x))/b)*cosh(7*a/b)*(-c*x+1)^(1/2)/b/c ^4/(c*x-1)^(1/2)+1/256*Chi(9*(a+b*arccosh(c*x))/b)*cosh(9*a/b)*(-c*x+1)^(1 /2)/b/c^4/(c*x-1)^(1/2)+3/128*Shi((a+b*arccosh(c*x))/b)*sinh(a/b)*(-c*x+1) ^(1/2)/b/c^4/(c*x-1)^(1/2)-1/32*Shi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b)*(- c*x+1)^(1/2)/b/c^4/(c*x-1)^(1/2)+3/256*Shi(7*(a+b*arccosh(c*x))/b)*sinh(7* a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1)^(1/2)-1/256*Shi(9*(a+b*arccosh(c*x))/b)* sinh(9*a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1)^(1/2)
Time = 1.05 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.54 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-6 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+8 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-3 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (9 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+6 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-8 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (9 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{256 c^4 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]
(Sqrt[1 - c^2*x^2]*(-6*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + 8*Cosh [(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] - 3*Cosh[(7*a)/b]*CoshInteg ral[7*(a/b + ArcCosh[c*x])] + Cosh[(9*a)/b]*CoshIntegral[9*(a/b + ArcCosh[ c*x])] + 6*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 8*Sinh[(3*a)/b]*Si nhIntegral[3*(a/b + ArcCosh[c*x])] + 3*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])] - Sinh[(9*a)/b]*SinhIntegral[9*(a/b + ArcCosh[c*x])]))/(25 6*c^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))
Time = 0.73 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.56, 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.
| \(\displaystyle \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx\) |
| \(\Big \downarrow \) 6367 |
| \(\displaystyle \frac {\sqrt {1-c x} \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^6\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c^4 \sqrt {c x-1}}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\sqrt {1-c x} \int \left (\frac {\cosh \left (\frac {9 a}{b}-\frac {9 (a+b \text {arccosh}(c x))}{b}\right )}{256 (a+b \text {arccosh}(c x))}-\frac {3 \cosh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{256 (a+b \text {arccosh}(c x))}+\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}-\frac {3 \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{128 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b c^4 \sqrt {c x-1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c x} \left (-\frac {3}{128} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{32} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{256} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{256} \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 (a+b \text {arccosh}(c x))}{b}\right )+\frac {3}{128} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{32} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {3}{256} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{256} \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^4 \sqrt {c x-1}}\) |
(Sqrt[1 - c*x]*((-3*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/128 + (Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/32 - (3*Cosh[(7*a )/b]*CoshIntegral[(7*(a + b*ArcCosh[c*x]))/b])/256 + (Cosh[(9*a)/b]*CoshIn tegral[(9*(a + b*ArcCosh[c*x]))/b])/256 + (3*Sinh[a/b]*SinhIntegral[(a + b *ArcCosh[c*x])/b])/128 - (Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x ]))/b])/32 + (3*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcCosh[c*x]))/b])/25 6 - (Sinh[(9*a)/b]*SinhIntegral[(9*(a + b*ArcCosh[c*x]))/b])/256))/(b*c^4* Sqrt[-1 + c*x])
3.3.84.3.1 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 = 0.60 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.81
| method | result | size |
| default | \(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (3 \,\operatorname {Ei}_{1}\left (7 \,\operatorname {arccosh}\left (c x \right )+\frac {7 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+7 a}{b}}-\operatorname {Ei}_{1}\left (9 \,\operatorname {arccosh}\left (c x \right )+\frac {9 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+9 a}{b}}-\operatorname {Ei}_{1}\left (-9 \,\operatorname {arccosh}\left (c x \right )-\frac {9 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+9 a}{b}}-8 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+6 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}+6 \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-8 \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+3 \,\operatorname {Ei}_{1}\left (-7 \,\operatorname {arccosh}\left (c x \right )-\frac {7 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+7 a}{b}}\right )}{512 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}\) | \(320\) |
-1/512*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(3* Ei(1,7*arccosh(c*x)+7*a/b)*exp((b*arccosh(c*x)+7*a)/b)-Ei(1,9*arccosh(c*x) +9*a/b)*exp((b*arccosh(c*x)+9*a)/b)-Ei(1,-9*arccosh(c*x)-9*a/b)*exp(-(-b*a rccosh(c*x)+9*a)/b)-8*Ei(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a)/ b)+6*Ei(1,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)+6*Ei(1,-arccosh(c*x) -a/b)*exp(-(-b*arccosh(c*x)+a)/b)-8*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-(-b*a rccosh(c*x)+3*a)/b)+3*Ei(1,-7*arccosh(c*x)-7*a/b)*exp(-(-b*arccosh(c*x)+7* a)/b))/(c*x+1)/c^4/(c*x-1)/b
\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\text {Timed out} \]
\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^3\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]