Integrand size = 31, antiderivative size = 1270 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}+\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a d \left (2+3 c x-2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{6 g}+\frac {b c d x \left (-12-9 c x+4 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{36 g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^3}-\frac {a d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{2 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (2+3 c x-2 c^2 x^2\right ) \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{6 g}-\frac {b d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2}{4 g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {2 a d (c f-g)^{3/2} (c f+g)^{3/2} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {1+c x}}{\sqrt {c f-g} \sqrt {-1+c x}}\right )}{g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d (c f-g) (c f+g) \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^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d (c f-g) (c f+g) \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^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d (c f-g) (c f+g) \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^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d (c f-g) (c f+g) \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^4 \sqrt {-1+c x} \sqrt {1+c x}} \]
-a*d*(c*f-g)*(c*f+g)*(-c^2*d*x^2+d)^(1/2)/g^3+1/6*a*d*(-2*c^2*x^2+3*c*x+2) *(-c^2*d*x^2+d)^(1/2)/g-b*d*(c*f-g)*(c*f+g)*arccosh(c*x)*(-c^2*d*x^2+d)^(1 /2)/g^3+1/6*b*d*(-2*c^2*x^2+3*c*x+2)*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/g+1 /2*c*d*(c*f-g)*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/g^2+b*c*d*(c*f-g) *(c*f+g)*x*(-c^2*d*x^2+d)^(1/2)/g^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4*b*c^2* d*(c*f-g)*x^2*(-c^2*d*x^2+d)^(1/2)/g^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/36*b* c*d*x*(4*c^2*x^2-9*c*x-12)*(-c^2*d*x^2+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+1)^(1 /2)-1/2*a*d*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+1)^(1/2 )-1/4*b*d*arccosh(c*x)^2*(-c^2*d*x^2+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+1)^(1/2 )-1/4*d*(c*f-g)*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/g^2/(c*x-1)^(1 /2)/(c*x+1)^(1/2)+1/2*c*d*(c*f-g)*(c*f+g)*x*(a+b*arccosh(c*x))^2*(-c^2*d*x ^2+d)^(1/2)/b/g^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*d*(c*f-g)^2*(c*f+g)^2*(a +b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/g^4/(g*x+f)/(c*x-1)^(1/2)/(c*x +1)^(1/2)+1/2*d*(c*f-g)*(c*f+g)*(-c^2*x^2+1)*(a+b*arccosh(c*x))^2*(-c^2*d* x^2+d)^(1/2)/b/c/g^2/(g*x+f)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2*a*d*(c*f-g)^(3/ 2)*(c*f+g)^(3/2)*arctanh((c*f+g)^(1/2)*(c*x+1)^(1/2)/(c*f-g)^(1/2)/(c*x-1) ^(1/2))*(-c^2*d*x^2+d)^(1/2)/g^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*d*(c*f-g)*( c*f+g)*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^4/(c*x-1)^(1/2)/( c*x+1)^(1/2)+b*d*(c*f-g)*(c*f+g)*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(...
Result contains complex when optimal does not.
Time = 11.36 (sec) , antiderivative size = 3068, normalized size of antiderivative = 2.42 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Result too large to show} \]
Sqrt[-(d*(-1 + c^2*x^2))]*((a*d*(-3*c^2*f^2 + 4*g^2))/(3*g^3) + (a*c^2*d*f *x)/(2*g^2) - (a*c^2*d*x^2)/(3*g)) + (a*c*d^(3/2)*f*(2*c^2*f^2 - 3*g^2)*Ar cTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(2*g^4) + (a*d^(3/2)*(-(c^2*f^2) + g^2)^(3/2)*Log[f + g*x])/g^4 - (a*d^(3/2)*(-(c^2* f^2) + g^2)^(3/2)*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqr t[-(d*(-1 + c^2*x^2))]])/g^4 + (b*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*((-2*c *g*x)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + 2*g*ArcCosh[c*x] - (c*f*Arc Cosh[c*x]^2)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(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)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Ta nh[ArcCosh[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 + c*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^(ArcCosh[c*x]/2)*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*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]...
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx\) |
| \(\Big \downarrow \) 6387 |
| \(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \int \frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{f+g x}dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6392 |
| \(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \int \left (\frac {c \sqrt {c x+1} (a+b \text {arccosh}(c x)) (c x-1)^{3/2}}{g}-\frac {c (c f-g) \sqrt {c x+1} (a+b \text {arccosh}(c x)) \sqrt {c x-1}}{g^2}+\frac {(c f-g) (c f+g) \sqrt {c x+1} (a+b \text {arccosh}(c x)) \sqrt {c x-1}}{g^2 (f+g x)}\right )dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \left (\frac {c \int (c x-1)^{3/2} \sqrt {c x+1} (a+b \text {arccosh}(c x))dx}{g}-\frac {\left (1-c^2 x^2\right ) (c f-g) (c f+g) (a+b \text {arccosh}(c x))^2}{2 b c g^2 (f+g x)}-\frac {(c f-g)^2 (c f+g)^2 (a+b \text {arccosh}(c x))^2}{2 b c g^4 (f+g x)}-\frac {c x (c f-g) (c f+g) (a+b \text {arccosh}(c x))^2}{2 b g^3}-\frac {c x \sqrt {c x-1} \sqrt {c x+1} (c f-g) (a+b \text {arccosh}(c x))}{2 g^2}+\frac {(c f-g) (a+b \text {arccosh}(c x))^2}{4 b g^2}+\frac {a \sqrt {c^2 x^2-1} (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \text {arctanh}\left (\frac {c^2 f x+g}{\sqrt {c^2 x^2-1} \sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {a \left (1-c^2 x^2\right ) (c f-g) (c f+g)}{g^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {b (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4}+\frac {b \text {arccosh}(c x) (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g^4}-\frac {b \text {arccosh}(c x) (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g^4}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x) (c f-g) (c f+g)}{g^3}+\frac {b c^2 x^2 (c f-g)}{4 g^2}-\frac {b c x (c f-g) (c f+g)}{g^3}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
3.1.61.3.1 Defintions of rubi rules used
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_.)*((d1_) + (e1_.)*(x_))^(p_)*(( d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand Integrand[Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n, (f + g*x) ^m*(d1 + e1*x)^(p - 1/2)*(d2 + e2*x)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d 1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && Integ erQ[m] && IGtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
Time = 1.92 (sec) , antiderivative size = 1659, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1659\) |
| parts | \(\text {Expression too large to display}\) | \(1659\) |
a/g*(1/3*(-(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)^(3/2)+ c^2*d*f/g*(-1/4*(-2*c^2*d*(x+f/g)+2*c^2*d*f/g)/c^2/d*(-(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)-1/8*(4*c^2*d^2*(c^2*f^2-g^2)/g ^2-4*c^4*d^2*f^2/g^2)/c^2/d/(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*((-(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))))-4/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)/g*arccosh( c*x)+1/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g*x^3*c^3- 4/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g*x*c+1/2*b*(-d *(c^2*x^2-1))^(1/2)*f*c^4*d/(c*x+1)/(c*x-1)/g^2*arccosh(c*x)*x^3-1/2*b*(-d *(c^2*x^2-1))^(1/2)*f*c^2*d/(c*x+1)/(c*x-1)/g^2*arccosh(c*x)*x+1/8*b*(-d*( c^2*x^2-1))^(1/2)*f*c*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)/g^2-b*(-d*(c^2*x^2-1)) ^(1/2)*d/(c*x+1)/(c*x-1)/g^3*arccosh(c*x)*x^2*c^4*f^2+b*(-d*(c^2*x^2-1))^( 1/2)*d/(c*x+1)/(c*x-1)/g^3*arccosh(c*x)*c^2*f^2-b*(c^2*f^2-g^2)^(3/2)*d*(- d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*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^...
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{g x + f} \,d x } \]
integral(-(a*c^2*d*x^2 - a*d + (b*c^2*d*x^2 - b*d)*arccosh(c*x))*sqrt(-c^2 *d*x^2 + d)/(g*x + f), x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{f + g x}\, dx \]
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{f+g\,x} \,d x \]