\(\int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\) [100]

Optimal result

Integrand size = 27, antiderivative size = 221 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {3 b x^2 \sqrt {d-c^2 d x^2}}{16 c^3 d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{16 c d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{4 c^2 d}-\frac {3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^5 d \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

3/16*b*x^2*(-c^2*d*x^2+d)^(1/2)/c^3/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*x 
^4*(-c^2*d*x^2+d)^(1/2)/c/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/8*x*(-c^2*d*x^2+ 
d)^(1/2)*(a+b*arccosh(c*x))/c^4/d-1/4*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos 
h(c*x))/c^2/d-3/16*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/c^5/d/(c*x- 
1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.77 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-\frac {16 a c x \left (3+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{d}-\frac {48 a \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{\sqrt {d}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (-16 \cosh (2 \text {arccosh}(c x))-\cosh (4 \text {arccosh}(c x))+4 \text {arccosh}(c x) (6 \text {arccosh}(c x)+8 \sinh (2 \text {arccosh}(c x))+\sinh (4 \text {arccosh}(c x))))}{\sqrt {d-c^2 d x^2}}}{128 c^5} \] Input:

Integrate[(x^4*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]
 

Output:

((-16*a*c*x*(3 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2])/d - (48*a*ArcTan[(c*x*Sqr 
t[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/Sqrt[d] + (b*Sqrt[(-1 + c*x)/ 
(1 + c*x)]*(1 + c*x)*(-16*Cosh[2*ArcCosh[c*x]] - Cosh[4*ArcCosh[c*x]] + 4* 
ArcCosh[c*x]*(6*ArcCosh[c*x] + 8*Sinh[2*ArcCosh[c*x]] + Sinh[4*ArcCosh[c*x 
]])))/Sqrt[d - c^2*d*x^2])/(128*c^5)
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6353, 15, 6353, 15, 6307}

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.

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6307

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

Input:

Int[(x^4*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]
 

Output:

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

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 6307

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d 
 + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x 
] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
 

rule 6353

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcCosh[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2 
*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] 
- Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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*Ar 
cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* 
d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(567\) vs. \(2(189)=378\).

Time = 0.34 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.57

method	result	size
default	\(-\frac {a \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 d \,c^{5} \left (c^{2} x^{2}-1\right )}\right )\)	\(568\)
parts	\(-\frac {a \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 d \,c^{5} \left (c^{2} x^{2}-1\right )}\right )\)	\(568\)

Input:

int(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4*a*x^3/c^2/d*(-c^2*d*x^2+d)^(1/2)-3/8*a/c^4*x/d*(-c^2*d*x^2+d)^(1/2)+3 
/8*a/c^4/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-3/ 
16*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/c^5/(c^2*x^2-1)*ar 
ccosh(c*x)^2-1/256*(-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))/d/c^5/(c^2*x^2-1)-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))*(-1+2*arccosh(c*x))/d/c^5/(c^2*x^2-1)-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)*(1+2*arccosh(c*x))/d/c^5/(c^2*x^2-1)-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*arccosh(c*x))/d/c^5/(c^2*x^2-1))
 

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:

integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas 
")
 

Output:

integral(-(b*x^4*arccosh(c*x) + a*x^4)*sqrt(-c^2*d*x^2 + d)/(c^2*d*x^2 - d 
), x)
 

Sympy [F]

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

integrate(x**4*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral(x**4*(a + b*acosh(c*x))/sqrt(-d*(c*x - 1)*(c*x + 1)), x)
 

Maxima [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:

integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima 
")
 

Output:

-1/8*a*(2*sqrt(-c^2*d*x^2 + d)*x^3/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*x/(c^4 
*d) - 3*arcsin(c*x)/(c^5*sqrt(d))) + b*integrate(x^4*log(c*x + sqrt(c*x + 
1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d), x)
 

Giac [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:

integrate(x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)*x^4/sqrt(-c^2*d*x^2 + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:

int((x^4*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(1/2),x)
 

Output:

int((x^4*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {3 \mathit {asin} \left (c x \right ) a -2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} x^{3}-3 \sqrt {-c^{2} x^{2}+1}\, a c x +8 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{4}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{5}}{8 \sqrt {d}\, c^{5}} \] Input:

int(x^4*(a+b*acosh(c*x))/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

(3*asin(c*x)*a - 2*sqrt( - c**2*x**2 + 1)*a*c**3*x**3 - 3*sqrt( - c**2*x** 
2 + 1)*a*c*x + 8*int((acosh(c*x)*x**4)/sqrt( - c**2*x**2 + 1),x)*b*c**5)/( 
8*sqrt(d)*c**5)