Integrand size = 31, antiderivative size = 1744 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx =\text {Too large to display} \]
-1/2*d^2*(c^2*f^2-g^2)^2*(-c^2*x^2+1)*(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*c^2*d^2*f*(c^2*f^2-2 *g^2)*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/g^4-1/5*c^2*d^2*x^2*(-c*x+ 1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/g+b*d^2*(c^2*f^2-g^2)^2 *arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/g^5+2/15*b*c*d^2*x*(-c^2*d*x^2+d)^(1/2) /g/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/45*b*c^3*d^2*x^3*(-c^2*d*x^2+d)^(1/2)/g/( c*x-1)^(1/2)/(c*x+1)^(1/2)-1/25*b*c^5*d^2*x^5*(-c^2*d*x^2+d)^(1/2)/g/(c*x- 1)^(1/2)/(c*x+1)^(1/2)-1/9*b*c^3*d^2*(c^2*f^2-2*g^2)*x^3*(-c^2*d*x^2+d)^(1 /2)/g^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*c^5*d^2*f*x^4*(-c^2*d*x^2+d)^(1 /2)/g^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*c*d^2*f*(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)+1/8*c^2*d^2*f*x*(a+b*ar ccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/g^2-1/4*c^4*d^2*f*x^3*(a+b*arccosh(c*x))* (-c^2*d*x^2+d)^(1/2)/g^2-1/3*d^2*(c^2*f^2-2*g^2)*(-c*x+1)*(c*x+1)*(a+b*arc cosh(c*x))*(-c^2*d*x^2+d)^(1/2)/g^3-1/2*d^2*(c^2*f^2-g^2)^3*(a+b*arccosh(c *x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/g^6/(g*x+f)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/ 4*b*c^3*d^2*f*(c^2*f^2-2*g^2)*x^2*(-c^2*d*x^2+d)^(1/2)/g^4/(c*x-1)^(1/2)/( c*x+1)^(1/2)+1/4*c*d^2*f*(c^2*f^2-2*g^2)*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+ d)^(1/2)/b/g^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*c*d^2*(c^2*f^2-g^2)^2*x*(a+ b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/g^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1 /3*b*c*d^2*(c^2*f^2-2*g^2)*x*(-c^2*d*x^2+d)^(1/2)/g^3/(c*x-1)^(1/2)/(c*...
Result contains complex when optimal does not.
Time = 16.31 (sec) , antiderivative size = 6244, normalized size of antiderivative = 3.58 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Result too large to show} \]
Time = 4.92 (sec) , antiderivative size = 1070, normalized size of antiderivative = 0.61, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6387, 6392, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx\) |
| \(\Big \downarrow \) 6387 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{f+g x}dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6392 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^4}{g}-\frac {f x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^4}{g^2}-\frac {f \left (c^2 f^2-2 g^2\right ) \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^2}{g^4}+\frac {\left (c^2 f^2-2 g^2\right ) x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^2}{g^3}+\frac {\left (g^2-c^2 f^2\right )^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{g^4 (f+g x)}\right )dx}{\sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b x^5 c^5}{25 g}+\frac {b f x^4 c^5}{16 g^2}-\frac {f x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^4}{4 g^2}-\frac {b \left (c^2 f^2-2 g^2\right ) x^3 c^3}{9 g^3}+\frac {b x^3 c^3}{45 g}+\frac {b f \left (c^2 f^2-2 g^2\right ) x^2 c^3}{4 g^4}-\frac {b f x^2 c^3}{16 g^2}+\frac {x^2 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x)) c^2}{5 g}-\frac {f \left (c^2 f^2-2 g^2\right ) x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^2}{2 g^4}+\frac {f x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^2}{8 g^2}+\frac {f \left (c^2 f^2-2 g^2\right ) (a+b \text {arccosh}(c x))^2 c}{4 b g^4}-\frac {\left (c^2 f^2-g^2\right )^2 x (a+b \text {arccosh}(c x))^2 c}{2 b g^5}+\frac {f (a+b \text {arccosh}(c x))^2 c}{16 b g^2}-\frac {b \left (c^2 f^2-g^2\right )^2 x c}{g^5}+\frac {b \left (c^2 f^2-2 g^2\right ) x c}{3 g^3}+\frac {2 b x c}{15 g}-\frac {a \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right )}{g^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (c^2 f^2-g^2\right )^2 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{g^5}+\frac {\left (c^2 f^2-2 g^2\right ) (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 g^3}+\frac {2 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{15 g}+\frac {a \left (c^2 f^2-g^2\right )^{5/2} \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {c^2 x^2-1}}\right )}{g^6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (c^2 f^2-g^2\right )^{5/2} \text {arccosh}(c x) \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g^6}-\frac {b \left (c^2 f^2-g^2\right )^{5/2} \text {arccosh}(c x) \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )}{g^6}+\frac {b \left (c^2 f^2-g^2\right )^{5/2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6}-\frac {b \left (c^2 f^2-g^2\right )^{5/2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6}-\frac {\left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2}{2 b g^4 (f+g x) c}-\frac {\left (c^2 f^2-g^2\right )^3 (a+b \text {arccosh}(c x))^2}{2 b g^6 (f+g x) c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\) |
(d^2*Sqrt[d - c^2*d*x^2]*((2*b*c*x)/(15*g) + (b*c*(c^2*f^2 - 2*g^2)*x)/(3* g^3) - (b*c*(c^2*f^2 - g^2)^2*x)/g^5 - (b*c^3*f*x^2)/(16*g^2) + (b*c^3*f*( c^2*f^2 - 2*g^2)*x^2)/(4*g^4) + (b*c^3*x^3)/(45*g) - (b*c^3*(c^2*f^2 - 2*g ^2)*x^3)/(9*g^3) + (b*c^5*f*x^4)/(16*g^2) - (b*c^5*x^5)/(25*g) - (a*(c^2*f ^2 - g^2)^2*(1 - c^2*x^2))/(g^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(c^2*f^ 2 - g^2)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/g^5 + (c^2*f*x*Sqrt[ -1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(8*g^2) - (c^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*g^4) - (c^4 *f*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(4*g^2) + (2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(15*g) + ((c^2*f^2 - 2 *g^2)*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(3*g^3) + (c^ 2*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(5*g) + (c*f* (a + b*ArcCosh[c*x])^2)/(16*b*g^2) + (c*f*(c^2*f^2 - 2*g^2)*(a + b*ArcCosh [c*x])^2)/(4*b*g^4) - (c*(c^2*f^2 - g^2)^2*x*(a + b*ArcCosh[c*x])^2)/(2*b* g^5) - ((c^2*f^2 - g^2)^3*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^6*(f + g*x)) - ((c^2*f^2 - g^2)^2*(1 - c^2*x^2)*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^4*(f + g *x)) + (a*(c^2*f^2 - g^2)^(5/2)*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^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x ]) + (b*(c^2*f^2 - g^2)^(5/2)*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^6 - (b*(c^2*f^2 - g^2)^(5/2)*ArcCosh[c*x]*L...
3.1.65.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(3768\) vs. \(2(1608)=3216\).
Time = 2.14 (sec) , antiderivative size = 3769, normalized size of antiderivative = 2.16
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3769\) |
| parts | \(\text {Expression too large to display}\) | \(3769\) |
1/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)/g*arccosh(c*x)*x^6*c^6+34 /15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)/g*arccosh(c*x)*x^2*c^2+7/ 3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)/g^3*arccosh(c*x)*c^2*f^2+b* d^2*(-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*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)))*arccosh(c*x)-b*d^2*(-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*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)))*arccosh(c*x )-1/2*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*f^5*arccosh(c*x )^2*d^2*c^5/g^6+a/g*(1/5*(-(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)^(5/2)+c^2*d*f/g*(-1/8*(-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)^(3/2)-3/16*(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*(-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*(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)...
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{g x + f} \,d x } \]
integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c ^2*d^2*x^2 + b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/(g*x + f), x)
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{f+g\,x} \,d x \]