\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx\) [49]

Optimal result

Integrand size = 31, antiderivative size = 1270 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx =\text {Too large to display} \] Output:

-a*d*(c*f-g)*(c*f+g)*(-c^2*d*x^2+d)^(1/2)/g^3+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)+1/6*a*d*(-2*c^2*x^2+3* 
c*x+2)*(-c^2*d*x^2+d)^(1/2)/g+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)-b*d*(c*f-g)*(c*f+g)*(-c^2*d*x^2+d 
)^(1/2)*arccosh(c*x)/g^3-1/2*a*d*(-c^2*d*x^2+d)^(1/2)*arccosh(c*x)/g/(c*x- 
1)^(1/2)/(c*x+1)^(1/2)+1/6*b*d*(-2*c^2*x^2+3*c*x+2)*(-c^2*d*x^2+d)^(1/2)*a 
rccosh(c*x)/g-1/4*b*d*(-c^2*d*x^2+d)^(1/2)*arccosh(c*x)^2/g/(c*x-1)^(1/2)/ 
(c*x+1)^(1/2)+1/2*c*d*(c*f-g)*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/g^ 
2-1/4*d*(c*f-g)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^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*(-c^2*d*x^2+d)^(1/2)*(a+b*arcc 
osh(c*x))^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*(- 
c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/c/g^4/(c*x-1)^(1/2)/(c*x+1)^(1/2 
)/(g*x+f)+1/2*d*(c*f-g)*(c*f+g)*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arc 
cosh(c*x))^2/b/c/g^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(g*x+f)-2*a*d*(c*f-g)^(3/ 
2)*(c*f+g)^(3/2)*(-c^2*d*x^2+d)^(1/2)*arctanh((c*f+g)^(1/2)*(c*x+1)^(1/2)/ 
(c*f-g)^(1/2)/(c*x-1)^(1/2))/g^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*d*(c*f-g)*( 
c*f+g)*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)*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)))/g^4/(c*x-1)^(1/2)/( 
c*x+1)^(1/2)+b*d*(c*f-g)*(c*f+g)*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 11.18 (sec) , antiderivative size = 3068, normalized size of antiderivative = 2.42 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Result too large to show} \] Input:

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(f + g*x),x]
 

Output:

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]...
 

Rubi [F]

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

\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \left (\frac {c \int (c x-1)^{3/2} \sqrt {c x+1} (a+b \text {arccosh}(c x))dx}{g}-\frac {\left (1-c^2 x^2\right ) (c f-g) (c f+g) (a+b \text {arccosh}(c x))^2}{2 b c g^2 (f+g x)}-\frac {(c f-g)^2 (c f+g)^2 (a+b \text {arccosh}(c x))^2}{2 b c g^4 (f+g x)}-\frac {c x (c f-g) (c f+g) (a+b \text {arccosh}(c x))^2}{2 b g^3}-\frac {c x \sqrt {c x-1} \sqrt {c x+1} (c f-g) (a+b \text {arccosh}(c x))}{2 g^2}+\frac {(c f-g) (a+b \text {arccosh}(c x))^2}{4 b g^2}+\frac {a \sqrt {c^2 x^2-1} (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \text {arctanh}\left (\frac {c^2 f x+g}{\sqrt {c^2 x^2-1} \sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {a \left (1-c^2 x^2\right ) (c f-g) (c f+g)}{g^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {b (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4}+\frac {b \text {arccosh}(c x) (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g^4}-\frac {b \text {arccosh}(c x) (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g^4}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x) (c f-g) (c f+g)}{g^3}+\frac {b c^2 x^2 (c f-g)}{4 g^2}-\frac {b c x (c f-g) (c f+g)}{g^3}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(f + g*x),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 1659, normalized size of antiderivative = 1.31

method	result	size
default	\(\text {Expression too large to display}\)	\(1659\)
parts	\(\text {Expression too large to display}\)	\(1659\)

Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/(g*x+f),x,method=_RETURNVERBOS 
E)
 

Output:

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*(x+f/g)*c^2*d+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))))-1/4*b*(-d*(c^2*x^2-1))^(1/2)*f*c^3*d/(c*x-1)^(1/2)/(c*x+1 
)^(1/2)/g^2*x^2+b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^3 
*x*c^3*f^2+1/2*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccos 
h(c*x)^2*f^3*c^3*d/g^4-3/4*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^ 
(1/2)*arccosh(c*x)^2*f*c*d/g^2-1/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x-1)/(c*x 
+1)/g*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)/ 
g*arccosh(c*x)*x^2*c^2+b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x-1)/(c*x+1)/g^3*arcc 
osh(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-f*c+(c^2*f^2-g^2)^(1/2))/(-f*c+(c^2*f^2-g^2)^(1/2)))+b*(c^2*f^2-g^2)^...
 

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{g x + f} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="fr 
icas")
 

Output:

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)
 

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{f + g x}\, dx \] Input:

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))/(g*x+f),x)
 

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))/(f + g*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="ma 
xima")
 

Output:

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)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="gi 
ac")
 

Output:

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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{f+g\,x} \,d x \] Input:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/(f + g*x),x)
                                                                                    
                                                                                    
 

Output:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/(f + g*x), x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\frac {\sqrt {d}\, d \left (-6 \mathit {asin} \left (c x \right ) a \,c^{3} f^{3}+9 \mathit {asin} \left (c x \right ) a c f \,g^{2}+12 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f^{2}-12 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,g^{2}-6 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g +3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f \,g^{2} x -2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g^{3} x^{2}+8 \sqrt {-c^{2} x^{2}+1}\, a \,g^{3}-6 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}}{g x +f}d x \right ) b \,c^{2} g^{4}+6 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{g x +f}d x \right ) b \,g^{4}-2 a \,c^{2} f^{2} g \right )}{6 g^{4}} \] Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*acosh(c*x))/(g*x+f),x)
 

Output:

(sqrt(d)*d*( - 6*asin(c*x)*a*c**3*f**3 + 9*asin(c*x)*a*c*f*g**2 + 12*sqrt( 
c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - g**2))* 
a*c**2*f**2 - 12*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sq 
rt(c**2*f**2 - g**2))*a*g**2 - 6*sqrt( - c**2*x**2 + 1)*a*c**2*f**2*g + 3* 
sqrt( - c**2*x**2 + 1)*a*c**2*f*g**2*x - 2*sqrt( - c**2*x**2 + 1)*a*c**2*g 
**3*x**2 + 8*sqrt( - c**2*x**2 + 1)*a*g**3 - 6*int((sqrt( - c**2*x**2 + 1) 
*acosh(c*x)*x**2)/(f + g*x),x)*b*c**2*g**4 + 6*int((sqrt( - c**2*x**2 + 1) 
*acosh(c*x))/(f + g*x),x)*b*g**4 - 2*a*c**2*f**2*g))/(6*g**4)