\(\int \frac {(f+g x)^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\) [55]

Optimal result

Integrand size = 31, antiderivative size = 288 \[ \int \frac {(f+g x)^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {2 b f g x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c \sqrt {d-c^2 d x^2}}-\frac {2 f g (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}} \] Output:

-2*b*f*g*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-1/4*b*g^2*x^ 
2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-2*f*g*(-c*x+1)*(c*x+1 
)*(a+b*arccosh(c*x))/c^2/(-c^2*d*x^2+d)^(1/2)-1/2*g^2*x*(-c*x+1)*(c*x+1)*( 
a+b*arccosh(c*x))/c^2/(-c^2*d*x^2+d)^(1/2)+1/2*f^2*(c*x-1)^(1/2)*(c*x+1)^( 
1/2)*(a+b*arccosh(c*x))^2/b/c/(-c^2*d*x^2+d)^(1/2)+1/4*g^2*(c*x-1)^(1/2)*( 
c*x+1)^(1/2)*(a+b*arccosh(c*x))^2/b/c^3/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.99 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.99 \[ \int \frac {(f+g x)^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {4 c \sqrt {d} g \left (-1+c^2 x^2\right ) \left (-4 b f+a \sqrt {\frac {-1+c x}{1+c x}} (4 f+g x)\right )+2 b \sqrt {d} \left (2 c^2 f^2+g^2\right ) (-1+c x) \text {arccosh}(c x)^2-4 a \left (2 c^2 f^2+g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-b \sqrt {d} g^2 (-1+c x) \cosh (2 \text {arccosh}(c x))+2 b \sqrt {d} g (-1+c x) \text {arccosh}(c x) \left (8 c f \sqrt {\frac {-1+c x}{1+c x}} (1+c x)+g \sinh (2 \text {arccosh}(c x))\right )}{8 c^3 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(4*c*Sqrt[d]*g*(-1 + c^2*x^2)*(-4*b*f + a*Sqrt[(-1 + c*x)/(1 + c*x)]*(4*f 
+ g*x)) + 2*b*Sqrt[d]*(2*c^2*f^2 + g^2)*(-1 + c*x)*ArcCosh[c*x]^2 - 4*a*(2 
*c^2*f^2 + g^2)*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x 
*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - b*Sqrt[d]*g^2*(-1 + c*x) 
*Cosh[2*ArcCosh[c*x]] + 2*b*Sqrt[d]*g*(-1 + c*x)*ArcCosh[c*x]*(8*c*f*Sqrt[ 
(-1 + c*x)/(1 + c*x)]*(1 + c*x) + g*Sinh[2*ArcCosh[c*x]]))/(8*c^3*Sqrt[d]* 
Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.60, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6387, 6390, 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.

\(\displaystyle \int \frac {(f+g x)^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\)

\(\Big \downarrow \) 6387

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {(f+g x)^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 6390

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \left (\frac {(a+b \text {arccosh}(c x)) f^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 g x (a+b \text {arccosh}(c x)) f}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {g^2 x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}\right )dx}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {g^2 (a+b \text {arccosh}(c x))^2}{4 b c^3}+\frac {2 f g \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}+\frac {g^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}+\frac {f^2 (a+b \text {arccosh}(c x))^2}{2 b c}-\frac {2 b f g x}{c}-\frac {b g^2 x^2}{4 c}\right )}{\sqrt {d-c^2 d x^2}}\)

Input:

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

Output:

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((-2*b*f*g*x)/c - (b*g^2*x^2)/(4*c) + (2*f*g 
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c^2 + (g^2*x*Sqrt[-1 + 
 c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c^2) + (f^2*(a + b*ArcCosh[c* 
x])^2)/(2*b*c) + (g^2*(a + b*ArcCosh[c*x])^2)/(4*b*c^3)))/Sqrt[d - c^2*d*x 
^2]
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6387

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]
 

rule 6390

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( 
d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand 
Integrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, 
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && 
 EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[ 
d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1 
] || (EqQ[m, 2] && LtQ[p, -2]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(515\) vs. \(2(252)=504\).

Time = 0.52 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.79

method	result	size
default	\(a \left (\frac {f^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{2} \left (-\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )-\frac {2 f g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2} \left (2 c^{2} f^{2}+g^{2}\right )}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) g^{2} \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{c^{2} \left (c^{2} x^{2}-1\right ) d}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (1+\operatorname {arccosh}\left (c x \right )\right )}{c^{2} \left (c^{2} x^{2}-1\right ) d}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) g^{2} \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(516\)
parts	\(a \left (\frac {f^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{2} \left (-\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )-\frac {2 f g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2} \left (2 c^{2} f^{2}+g^{2}\right )}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) g^{2} \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{c^{2} \left (c^{2} x^{2}-1\right ) d}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) f g \left (1+\operatorname {arccosh}\left (c x \right )\right )}{c^{2} \left (c^{2} x^{2}-1\right ) d}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) g^{2} \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(516\)

Input:

int((g*x+f)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERB 
OSE)
 

Output:

a*(f^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+g^2*(-1/ 
2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2/c^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)* 
x/(-c^2*d*x^2+d)^(1/2)))-2*f*g/c^2/d*(-c^2*d*x^2+d)^(1/2))+b*(-1/4*(-d*(c^ 
2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/c^3/(c^2*x^2-1)*arccosh(c*x) 
^2*(2*c^2*f^2+g^2)-1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x-1)^ 
(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*g^2*(-1+2*arccosh 
(c*x))/d/c^3/(c^2*x^2-1)-(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/ 
2)*c*x+c^2*x^2-1)*f*g*(-1+arccosh(c*x))/c^2/(c^2*x^2-1)/d-(-d*(c^2*x^2-1)) 
^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*f*g*(1+arccosh(c*x))/c 
^2/(c^2*x^2-1)/d-1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/ 
2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*g^2*(1+2*arccosh(c 
*x))/d/c^3/(c^2*x^2-1))
 

Fricas [F]

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

integrate((g*x+f)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm=" 
fricas")
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(a*g^2*x^2 + 2*a*f*g*x + a*f^2 + (b*g^2*x^2 
 + 2*b*f*g*x + b*f^2)*arccosh(c*x))/(c^2*d*x^2 - d), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((g*x+f)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm=" 
maxima")
 

Output:

-1/2*a*g^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt(d))) + 
a*f^2*arcsin(c*x)/(c*sqrt(d)) - 2*sqrt(-c^2*d*x^2 + d)*a*f*g/(c^2*d) + int 
egrate(b*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + 
d) + 2*b*f*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d) 
 + b*f^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d), x)
 

Giac [F]

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

integrate((g*x+f)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm=" 
giac")
 

Output:

integrate((g*x + f)^2*(b*arccosh(c*x) + a)/sqrt(-c^2*d*x^2 + d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*asin(c*x)*a*c**2*f**2 + asin(c*x)*a*g**2 - 4*sqrt( - c**2*x**2 + 1)*a*c 
*f*g - sqrt( - c**2*x**2 + 1)*a*c*g**2*x + 2*int(acosh(c*x)/sqrt( - c**2*x 
**2 + 1),x)*b*c**3*f**2 + 2*int((acosh(c*x)*x**2)/sqrt( - c**2*x**2 + 1),x 
)*b*c**3*g**2 + 4*int((acosh(c*x)*x)/sqrt( - c**2*x**2 + 1),x)*b*c**3*f*g 
+ 4*a*c*f*g)/(2*sqrt(d)*c**3)