Integrand size = 31, antiderivative size = 785 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=-\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g (1-c x) (1+c x)}+\frac {b \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {a \sqrt {c^2 f^2-g^2} \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {-1+c^2 x^2}}\right )}{g^2 (1-c x) (1+c x)}+\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}} \]
a*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/g/(-c*x+1)/(c*x+1)+b*arccosh(c*x)*(-c^ 2*d*x^2+d)^(1/2)/g-b*c*x*(-c^2*d*x^2+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+1)^(1/2 )-1/2*c*x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/g/(c*x-1)^(1/2)/(c*x +1)^(1/2)+1/2*(1-c^2*f^2/g^2)*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/ c/(g*x+f)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*(-c^2*x^2+1)*(a+b*arccosh(c*x))^ 2*(-c^2*d*x^2+d)^(1/2)/b/c/(g*x+f)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*arccosh(c *x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(c ^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*a rccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1 /2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(c*x-1)^(1/2)/(c*x+1)^( 1/2)+b*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^( 1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(c*x-1)^(1/2)/(c*x+1)^ (1/2)-b*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^ (1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(c*x-1)^(1/2)/(c*x+1) ^(1/2)-a*arctanh((c^2*f*x+g)/(c^2*f^2-g^2)^(1/2)/(c^2*x^2-1)^(1/2))*(c^2*f ^2-g^2)^(1/2)*(c^2*x^2-1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(-c*x+1)/(c*x+1)
Result contains complex when optimal does not.
Time = 2.96 (sec) , antiderivative size = 1121, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx =\text {Too large to display} \]
(2*a*g*Sqrt[d - c^2*d*x^2] - 2*a*c*Sqrt[d]*f*ArcTan[(c*x*Sqrt[d - c^2*d*x^ 2])/(Sqrt[d]*(-1 + c^2*x^2))] + 2*a*Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Log[f + g*x] - 2*a*Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Log[d*(g + c^2*f*x) + Sqrt[d]*S qrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]] + b*Sqrt[d - c^2*d*x^2]*((2*c*g *x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + 2*g*ArcCosh[c*x] + (c*f*Sqrt[(- 1 + c*x)/(1 + c*x)]*ArcCosh[c*x]^2)/(1 - c*x) + (2*(-(c*f) + g)*(c*f + g)* (2*ArcCosh[c*x]*ArcTan[((c*f + g)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - (2*I)*ArcCos[-((c*f)/g)]*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2] )/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] + 2*(ArcTan[((c*f + g)*Cot h[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Tanh[Arc Cosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2 ]*E^(ArcCosh[c*x]/2)*Sqrt[g]*Sqrt[c*(f + g*x)])] + (ArcCos[-((c*f)/g)] - 2 *(ArcTan[((c*f + g)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan [((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^(Arc Cosh[c*x]/2)*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*(f + g*x)])] - (ArcCos[-((c*f)/g)] + 2*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[ -(c^2*f^2) + g^2]])*Log[((c*f + g)*(c*f - g + I*Sqrt[-(c^2*f^2) + g^2])*(- 1 + Tanh[ArcCosh[c*x]/2]))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[Arc Cosh[c*x]/2]))] - (ArcCos[-((c*f)/g)] - 2*ArcTan[((-(c*f) + g)*Tanh[ArcCos h[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(-(c*f) + g + I*Sqrt...
Time = 3.75 (sec) , antiderivative size = 538, normalized size of antiderivative = 0.69, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {6387, 6391, 6385, 25, 6408, 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.
| \(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx\) |
| \(\Big \downarrow \) 6387 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{f+g x}dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6391 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {\int \frac {\left (g x^2 c^2+2 f x c^2+g\right ) (a+b \text {arccosh}(c x))^2}{(f+g x)^2}dx}{2 b c}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2}{2 b c (f+g x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6385 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {-2 b c \int -\frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (\frac {f^2}{f+g x}+g x\right )}{g^2}\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) (a+b \text {arccosh}(c x))^2}{f+g x}+\frac {c^2 x (a+b \text {arccosh}(c x))^2}{g}}{2 b c}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2}{2 b c (f+g x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {2 b c \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (\frac {f^2}{f+g x}+g x\right )}{g^2}\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) (a+b \text {arccosh}(c x))^2}{f+g x}+\frac {c^2 x (a+b \text {arccosh}(c x))^2}{g}}{2 b c}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2}{2 b c (f+g x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6408 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {2 b c \int \left (-\frac {b \text {arccosh}(c x) \left (f^2 c^2+g^2 x^2 c^2+f g x c^2-g^2\right )}{g^2 \sqrt {c x-1} \sqrt {c x+1} (f+g x)}-\frac {a \left (f^2 c^2+g^2 x^2 c^2+f g x c^2-g^2\right )}{g^2 \sqrt {c x-1} \sqrt {c x+1} (f+g x)}\right )dx-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) (a+b \text {arccosh}(c x))^2}{f+g x}+\frac {c^2 x (a+b \text {arccosh}(c x))^2}{g}}{2 b c}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2}{2 b c (f+g x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {2 b c \left (-\frac {a \sqrt {c^2 x^2-1} \sqrt {c^2 f^2-g^2} \text {arctanh}\left (\frac {c^2 f x+g}{\sqrt {c^2 x^2-1} \sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {a \left (1-c^2 x^2\right )}{g \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {b \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {b \text {arccosh}(c x) \sqrt {c^2 f^2-g^2} \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g^2}+\frac {b \text {arccosh}(c x) \sqrt {c^2 f^2-g^2} \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{g}+\frac {b c x}{g}\right )-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) (a+b \text {arccosh}(c x))^2}{f+g x}+\frac {c^2 x (a+b \text {arccosh}(c x))^2}{g}}{2 b c}-\frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2}{2 b c (f+g x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
(Sqrt[d - c^2*d*x^2]*(-1/2*((1 - c^2*x^2)*(a + b*ArcCosh[c*x])^2)/(b*c*(f + g*x)) - ((c^2*x*(a + b*ArcCosh[c*x])^2)/g - ((1 - (c^2*f^2)/g^2)*(a + b* ArcCosh[c*x])^2)/(f + g*x) + 2*b*c*((b*c*x)/g + (a*(1 - c^2*x^2))/(g*Sqrt[ -1 + c*x]*Sqrt[1 + c*x]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/g - (a*Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2]*ArcTanh[(g + c^2*f*x)/(Sqrt[c ^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2])])/(g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ( b*Sqrt[c^2*f^2 - g^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[ c^2*f^2 - g^2])])/g^2 + (b*Sqrt[c^2*f^2 - g^2]*ArcCosh[c*x]*Log[1 + (E^Arc Cosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^2 - (b*Sqrt[c^2*f^2 - g^2]*Po lyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/g^2 + (b*Sqrt [c^2*f^2 - g^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2] ))])/g^2))/(2*b*c)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.1.56.3.1 Defintions of rubi rules used
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*( x_)^2)^(p_.))/((d_) + (e_.)*(x_))^2, x_Symbol] :> With[{u = IntHide[(f + g* x + h*x^2)^p/(d + e*x)^2, x]}, Simp[(a + b*ArcCosh[c*x])^n u, x] - Simp[b *c*n Int[SimplifyIntegrand[u*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[-1 + c*x ]*Sqrt[1 + c*x])), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGt Q[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra cPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p])) Int[(f + g*x)^m* (-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqr t[(d2_) + (e2_.)*(x_)]*((f_) + (g_.)*(x_))^(m_), x_Symbol] :> Simp[(f + g*x )^m*(d1*d2 + e1*e2*x^2)*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[(-d1)*d2]*( n + 1))), x] - Simp[1/(b*c*Sqrt[(-d1)*d2]*(n + 1)) Int[(d1*d2*g*m + 2*e1* e2*f*x + e1*e2*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1 ), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && ILtQ[m, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
Int[(ArcCosh[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d1_) + (e1_.)*(x_))^( p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(d1 + e1*x )^p*(d2 + e2*x)^p, RFx*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d1 , e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c* d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]
Time = 1.47 (sec) , antiderivative size = 925, normalized size of antiderivative = 1.18
| method | result | size |
| default | \(\frac {a \left (\sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}+\frac {c^{2} d f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g \sqrt {c^{2} d}}+\frac {d \left (c^{2} f^{2}-g^{2}\right ) \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{2} \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \operatorname {arccosh}\left (c x \right )^{2} c}{2 \sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} c^{2}}{\left (c x +1\right ) \left (c x -1\right ) g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x c}{\sqrt {c x +1}\, \sqrt {c x -1}\, g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{\left (c x +1\right ) \left (c x -1\right ) g}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {dilog}\left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {dilog}\left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}\) | \(925\) |
| parts | \(\frac {a \left (\sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}+\frac {c^{2} d f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g \sqrt {c^{2} d}}+\frac {d \left (c^{2} f^{2}-g^{2}\right ) \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{2} \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \operatorname {arccosh}\left (c x \right )^{2} c}{2 \sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} c^{2}}{\left (c x +1\right ) \left (c x -1\right ) g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x c}{\sqrt {c x +1}\, \sqrt {c x -1}\, g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{\left (c x +1\right ) \left (c x -1\right ) g}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {dilog}\left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {dilog}\left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}\) | \(925\) |
a/g*((-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)+c^2* d*f/g/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*( x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))+d*(c^2*f^2-g^2)/g^2/(-d*(c^2*f^2-g^2)/g ^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^ 2)/g^2)^(1/2)*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^( 1/2))/(x+f/g)))-1/2*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*f *arccosh(c*x)^2*c/g^2+b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)/(c*x-1)/g*arccosh(c *x)*x^2*c^2-b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g*x*c-b*( -d*(c^2*x^2-1))^(1/2)/(c*x+1)/(c*x-1)/g*arccosh(c*x)+b*(-d*(c^2*x^2-1))^(1 /2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*arccosh(c*x)*ln((- (c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^ 2-g^2)^(1/2)))-b*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/ (c*x+1)^(1/2)/g^2*arccosh(c*x)*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f +(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))+b*(-d*(c^2*x^2-1))^(1/2)* (c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*dilog((-(c*x+(c*x-1)^( 1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2))) -b*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/ g^2*dilog(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c *f+(c^2*f^2-g^2)^(1/2)))
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{g x + f} \,d x } \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{f + g x}\, dx \]
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for mor e details)
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{f+g\,x} \,d x \]