Integrand size = 27, antiderivative size = 315 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^5} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 x^2}+\frac {c^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b c^4 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b c^4 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/4*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^4+1/8*c^2*(a+b*arccosh(c*x) )*(-c^2*d*x^2+d)^(1/2)/x^2-1/12*b*c*(-c^2*d*x^2+d)^(1/2)/x^3/(c*x-1)^(1/2) /(c*x+1)^(1/2)+1/8*b*c^3*(-c^2*d*x^2+d)^(1/2)/x/(c*x-1)^(1/2)/(c*x+1)^(1/2 )+1/4*c^4*(a+b*arccosh(c*x))*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-c^2 *d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/8*I*b*c^4*polylog(2,-I*(c*x+ (c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^( 1/2)+1/8*I*b*c^4*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^ 2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.85 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\frac {1}{24} \left (\frac {3 a \left (-2+c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{x^4}-3 a c^4 \sqrt {d} \log (x)+3 a c^4 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b \sqrt {d-c^2 d x^2} \left (-2 c x+3 c^3 x^3-6 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)+3 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-3 i c^4 x^4 \text {arccosh}(c x) \left (\log \left (1-i e^{-\text {arccosh}(c x)}\right )-\log \left (1+i e^{-\text {arccosh}(c x)}\right )\right )-3 i c^4 x^4 \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )\right )}{x^4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right ) \]
((3*a*(-2 + c^2*x^2)*Sqrt[d - c^2*d*x^2])/x^4 - 3*a*c^4*Sqrt[d]*Log[x] + 3 *a*c^4*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*Sqrt[d - c^2*d*x^ 2]*(-2*c*x + 3*c^3*x^3 - 6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c* x] + 3*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - (3*I)*c ^4*x^4*ArcCosh[c*x]*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]] ) - (3*I)*c^4*x^4*(PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^ArcCos h[c*x]])))/(x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/24
Time = 1.18 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.68, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6339, 15, 6348, 15, 6362, 3042, 4668, 2715, 2838}
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.
| \(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^5} \, dx\) |
| \(\Big \downarrow \) 6339 |
| \(\displaystyle \frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \sqrt {d-c^2 d x^2} \int \frac {1}{x^4}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 x^4}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 x^4}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6348 |
| \(\displaystyle \frac {c^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{2} c^2 \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{2} b c \int \frac {1}{x^2}dx+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 x^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 x^4}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {c^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{2} c^2 \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c}{2 x}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 x^4}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6362 |
| \(\displaystyle \frac {c^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{2} c^2 \int \frac {a+b \text {arccosh}(c x)}{c x}d\text {arccosh}(c x)+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c}{2 x}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 x^4}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {c^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{2} c^2 \int (a+b \text {arccosh}(c x)) \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c}{2 x}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 x^4}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {c^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{2} c^2 \left (-i b \int \log \left (1-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i b \int \log \left (1+i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c}{2 x}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 x^4}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {c^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{2} c^2 \left (-i b \int e^{-\text {arccosh}(c x)} \log \left (1-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+i b \int e^{-\text {arccosh}(c x)} \log \left (1+i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c}{2 x}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 x^4}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {c^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{2} c^2 \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )\right )+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {b c}{2 x}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 x^4}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/12*(b*c*Sqrt[d - c^2*d*x^2])/(x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt [d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(4*x^4) + (c^2*Sqrt[d - c^2*d*x^2]*( (b*c)/(2*x) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*x^2) + (c^2*(2*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]] - I*b*PolyLog[2, (-I )*E^ArcCosh[c*x]] + I*b*PolyLog[2, I*E^ArcCosh[c*x]]))/2))/(4*Sqrt[-1 + c* x]*Sqrt[1 + c*x])
3.1.70.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Cosh[c*x])^n/(f*(m + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*Simp[Sqrt[d + e* x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x ])^(n - 1), x], x] - Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^(m + 2)*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(f*x)^(m + 1) *(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d1*d2*f*( m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)* (d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp[b*c*(n/(f *(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCos h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && Eq Q[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[m, -1]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 _.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Subst [Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte gerQ[m]
Time = 1.11 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.72
| method | result | size |
| default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 d \,x^{4}}-\frac {a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8 d \,x^{2}}+\frac {a \,c^{4} \sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{8}-\frac {a \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) c^{4}}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3}}{8 \sqrt {c x +1}\, x \sqrt {c x -1}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) c^{2}}{8 \left (c x +1\right ) x^{2} \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c}{12 \sqrt {c x +1}\, x^{3} \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{4 \left (c x +1\right ) x^{4} \left (c x -1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}\) | \(541\) |
| parts | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 d \,x^{4}}-\frac {a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8 d \,x^{2}}+\frac {a \,c^{4} \sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{8}-\frac {a \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) c^{4}}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3}}{8 \sqrt {c x +1}\, x \sqrt {c x -1}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) c^{2}}{8 \left (c x +1\right ) x^{2} \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c}{12 \sqrt {c x +1}\, x^{3} \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{4 \left (c x +1\right ) x^{4} \left (c x -1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}\) | \(541\) |
-1/4*a/d/x^4*(-c^2*d*x^2+d)^(3/2)-1/8*a*c^2/d/x^2*(-c^2*d*x^2+d)^(3/2)+1/8 *a*c^4*d^(1/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)-1/8*a*c^4*(-c^2* d*x^2+d)^(1/2)+1/8*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)/(c*x-1)*arccosh(c*x)*c ^4+1/8*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/x/(c*x-1)^(1/2)*c^3-3/8*b*(- d*(c^2*x^2-1))^(1/2)/(c*x+1)/x^2/(c*x-1)*arccosh(c*x)*c^2-1/12*b*(-d*(c^2* x^2-1))^(1/2)/(c*x+1)^(1/2)/x^3/(c*x-1)^(1/2)*c+1/4*b*(-d*(c^2*x^2-1))^(1/ 2)/(c*x+1)/x^4/(c*x-1)*arccosh(c*x)-1/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1) ^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)) )*c^4+1/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c *x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^4-1/8*I*b*(-d*(c^2*x^2-1)) ^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1 /2)))*c^4+1/8*I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*dilog (1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^4
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{5}} \,d x } \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{5}}\, dx \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{5}} \,d x } \]
1/8*(c^4*sqrt(d)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - sqrt(-c^2*d*x^2 + d)*c^4 - (-c^2*d*x^2 + d)^(3/2)*c^2/(d*x^2) - 2*(-c^2*d *x^2 + d)^(3/2)/(d*x^4))*a + b*integrate(sqrt(-c^2*d*x^2 + d)*log(c*x + sq rt(c*x + 1)*sqrt(c*x - 1))/x^5, x)
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^5} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^5} \,d x \]