Integrand size = 25, antiderivative size = 246 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c \sqrt {-1+c x}} \]
cosh(2*a/b)*Shi(2*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c/(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/(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/(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/(c *x-1)^(1/2)-(-c^2*x^2+1)^(3/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccos h(c*x))
Time = 0.49 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.94 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (-2 b+4 b c^2 x^2-2 b c^4 x^4+2 (a+b \text {arccosh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-(a+b \text {arccosh}(c x)) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )-2 a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-2 b \text {arccosh}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+b \text {arccosh}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{2 b^2 c \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \]
(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-2*b + 4*b*c^2*x^2 - 2*b*c^4*x^4 + 2*(a + b *ArcCosh[c*x])*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] - (a + b *ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] - 2*a*Co sh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 2*b*ArcCosh[c*x]*Cosh[( 2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + a*Cosh[(4*a)/b]*SinhIntegra l[4*(a/b + ArcCosh[c*x])] + b*ArcCosh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a /b + ArcCosh[c*x])]))/(2*b^2*c*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))
Time = 0.76 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.73, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6319, 25, 6327, 6367, 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.
| \(\displaystyle \int \frac {\left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx\) |
| \(\Big \downarrow \) 6319 |
| \(\displaystyle -\frac {4 c \sqrt {1-c x} \int -\frac {x (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 c \sqrt {1-c x} \int \frac {x (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {4 c \sqrt {1-c x} \int \frac {x \left (1-c^2 x^2\right )}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6367 |
| \(\displaystyle -\frac {4 \sqrt {1-c x} \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \sqrt {1-c x} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {4 \sqrt {1-c x} \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 (a+b \text {arccosh}(c x))}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{4 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt {1-c x} \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*(a + b*ArcCosh[c *x]))) - (4*Sqrt[1 - c*x]*((CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[ (2*a)/b])/4 - (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]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/8))/(b^2*c*Sqrt[-1 + c*x])
3.4.30.3.1 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_)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p]*((a + b*A rcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Si mp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(1 + c*x)^(p - 1/2)*(- 1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IntegerQ[2*p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(444\) vs. \(2(220)=440\).
Time = 0.92 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.81
| 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 (4 x^{4} c^{4} b \sqrt {c x -1}\, \sqrt {c x +1}+4 b \,c^{5} x^{5}-8 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{2} x^{2}-8 b \,c^{3} x^{3}+\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 )+4 b \sqrt {c x -1}\, \sqrt {c x +1}+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 +4 b c x \right )}{4 \left (c x -1\right ) \left (c x +1\right ) c \,b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\) | \(445\) |
-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^5-8*(c*x-1)^(1/2)*(c*x+1)^(1 /2)*b*c^2*x^2-8*b*c^3*x^3+arccosh(c*x)*b*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(- (-b*arccosh(c*x)+4*a)/b)-2*arccosh(c*x)*b*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)*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)+4*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)+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*Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)*a+4*b*c* x)/(c*x-1)/(c*x+1)/c/b^2/(a+b*arccosh(c*x))
\[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
((c^4*x^4 - 2*c^2*x^2 + 1)*(c*x + 1)*sqrt(c*x - 1) + (c^5*x^5 - 2*c^3*x^3 + c*x)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c ^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) - integrate(((4*c^4* x^4 - 3*c^2*x^2 - 1)*(c*x + 1)^(3/2)*(c*x - 1) + 4*(2*c^5*x^5 - 3*c^3*x^3 + c*x)*(c*x + 1)*sqrt(c*x - 1) + (4*c^6*x^6 - 9*c^4*x^4 + 6*c^2*x^2 - 1)*s qrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^4*x^4 + (c*x + 1)*(c*x - 1)*a*b*c^2*x^ 2 - 2*a*b*c^2*x^2 + 2*(a*b*c^3*x^3 - a*b*c*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + a*b + (b^2*c^4*x^4 + (c*x + 1)*(c*x - 1)*b^2*c^2*x^2 - 2*b^2*c^2*x^2 + 2 *(b^2*c^3*x^3 - b^2*c*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + b^2)*log(c*x + sqrt (c*x + 1)*sqrt(c*x - 1))), x)
\[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]