Integrand size = 27, antiderivative size = 292 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {4 b c d x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+d*(a+b*arccosh(c*x))*(-c^2*d*x ^2+d)^(1/2)-4/3*b*c*d*x*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1 /9*b*c^3*d*x^3*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2*d*(a+b*a rccosh(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)+I*b*d*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)-I*b*d*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 = 0.96 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.15 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {1}{3} a d \left (-4+c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {b d \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )}{36 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+a d^{3/2} \log (x)-a d^{3/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b d \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+c x \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+i \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
-1/3*(a*d*(-4 + c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (b*d*Sqrt[d - c^2*d*x^2]*( 9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3* ArcCosh[c*x]]))/(36*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + a*d^(3/2)*Log[ x] - a*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d*Sqrt[d - c^2*d* x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + c*x*Sqrt[(-1 + c* x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I* ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
Time = 1.17 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.79, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6345, 25, 39, 2009, 6341, 24, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx+\frac {b c d \sqrt {d-c^2 d x^2} \int -((1-c x) (c x+1))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx-\frac {b c d \sqrt {d-c^2 d x^2} \int (1-c x) (c x+1)dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int 1dx}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6362 |
| \(\displaystyle d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{c x}d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d \left (-\frac {\sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x)) \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-i b \int \log \left (1-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i b \int \log \left (1+i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle d \left (-\frac {\sqrt {d-c^2 d x^2} \left (-i b \int e^{-\text {arccosh}(c x)} \log \left (1-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+i b \int e^{-\text {arccosh}(c x)} \log \left (1+i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle d \left (-\frac {\sqrt {d-c^2 d x^2} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/3*(b*c*d*Sqrt[d - c^2*d*x^2]*(x - (c^2*x^3)/3))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/3 + d*(-((b*c*x*Sqr t[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (Sqrt[d - c^2*d*x^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]]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]))
3.1.84.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
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 + 2))), x] + (-Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/(Sq rt[1 + c*x]*Sqrt[-1 + c*x])] Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] - Simp[b*c*(n/(f*(m + 2)))*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]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Cosh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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*ArcCosh[c*x])^(n - 1) , x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[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 = 2.03 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.71
| method | result | size |
| default | \(\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a}{3}-a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right ) d^{\frac {3}{2}}+a d \sqrt {-c^{2} d \,x^{2}+d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3} x^{3}}{9 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c x}{3 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right )}{3 \left (c x +1\right ) \left (c x -1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right ) x^{4} c^{4}}{3 \left (c x +1\right ) \left (c x -1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right ) x^{2} c^{2}}{3 \left (c x +1\right ) \left (c x -1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}\) | \(499\) |
| parts | \(\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a}{3}-a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right ) d^{\frac {3}{2}}+a d \sqrt {-c^{2} d \,x^{2}+d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3} x^{3}}{9 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c x}{3 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right )}{3 \left (c x +1\right ) \left (c x -1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right ) x^{4} c^{4}}{3 \left (c x +1\right ) \left (c x -1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right ) x^{2} c^{2}}{3 \left (c x +1\right ) \left (c x -1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}\) | \(499\) |
1/3*(-c^2*d*x^2+d)^(3/2)*a-a*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)*d^ (3/2)+a*d*(-c^2*d*x^2+d)^(1/2)+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)))*d-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)))*d+1/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1 /2)/(c*x-1)^(1/2)*c^3*x^3-4/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/(c* x-1)^(1/2)*c*x-4/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*arccosh(c*x) -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)))*d-1/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1 )*arccosh(c*x)*x^4*c^4+5/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*arcc osh(c*x)*x^2*c^2+I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*di log(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x}\, dx \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
-1/3*(3*d^(3/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - (-c^2*d*x^2 + d)^(3/2) - 3*sqrt(-c^2*d*x^2 + d)*d)*a + b*integrate((-c^2*d *x^2 + d)^(3/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x} \,d x \]