\(\int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx\) [56]

Optimal result

Integrand size = 27, antiderivative size = 201 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {b x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

1/16*b*x^2*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/16*b*c*x^4 
*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/8*x*(-c^2*d*x^2+d)^(1/ 
2)*(a+b*arccosh(c*x))/c^2+1/4*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))- 
1/16*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/c^3/(c*x-1)^(1/2)/(c*x+1) 
^(1/2)
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.75 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {-16 a c x \left (-1+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}+16 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {b \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}}{128 c^3} \] Input:

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

Output:

-1/128*(-16*a*c*x*(-1 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2] + 16*a*Sqrt[d]*ArcT 
an[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (b*Sqrt[d - c^2*d 
*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*Arc 
Cosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/c^3
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6341, 15, 6354, 15, 6308}

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.

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

\(\Big \downarrow \) 6341

\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^3dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6354

\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6308

\(\displaystyle \frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} \left (\frac {(a+b \text {arccosh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-1/16*(b*c*x^4*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x^3* 
Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/4 - (Sqrt[d - c^2*d*x^2]*(-1/4*( 
b*x^2)/c + (x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c^2) + 
 (a + b*ArcCosh[c*x])^2/(4*b*c^3)))/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6341

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Cosh[c*x])^n/(f*(m + 2))), x] + (-Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/(Sq 
rt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 
+ c*x]*Sqrt[-1 + c*x])), x], x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e* 
x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x 
])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] 
 && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 
1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( 
m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1)))   Int[(f*x)^(m 
 - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f 
*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( 
-1 + c*x)^p]   Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( 
a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, 
p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(366\) vs. \(2(169)=338\).

Time = 0.25 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\)	\(367\)
parts	\(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\)	\(367\)

Input:

int(x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
 

Output:

-1/4*a*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/8*a/c^2*x*(-c^2*d*x^2+d)^(1/2)+1/8*a 
/c^2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-1/16 
*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^2+1/2 
56*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*c^4*x^4*(c*x-1)^(1/2)*(c 
*x+1)^(1/2)+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x 
+1)^(1/2))*(-1+4*arccosh(c*x))/(c*x+1)/c^3/(c*x-1)+1/256*(-d*(c^2*x^2-1))^ 
(1/2)*(-8*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+8*c^5*x^5+8*(c*x-1)^(1/2)*(c 
*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*(1+4*arc 
cosh(c*x))/(c*x+1)/c^3/(c*x-1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/8*a*(sqrt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d*x^2 + d)^(3/2)*x/(c^2*d) + s 
qrt(d)*arcsin(c*x)/c^3) + b*integrate(sqrt(-c^2*d*x^2 + d)*x^2*log(c*x + s 
qrt(c*x + 1)*sqrt(c*x - 1)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, \left (\mathit {asin} \left (c x \right ) a +2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} x^{3}-\sqrt {-c^{2} x^{2}+1}\, a c x +8 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{3}\right )}{8 c^{3}} \] Input:

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

Output:

(sqrt(d)*(asin(c*x)*a + 2*sqrt( - c**2*x**2 + 1)*a*c**3*x**3 - sqrt( - c** 
2*x**2 + 1)*a*c*x + 8*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**2,x)*b*c**3 
))/(8*c**3)