\(\int (a+b \text {arccosh}(c+d x))^4 \, dx\) [43]

Optimal result

Integrand size = 12, antiderivative size = 129 \[ \int (a+b \text {arccosh}(c+d x))^4 \, dx=24 b^4 x-\frac {24 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{d}+\frac {12 b^2 (c+d x) (a+b \text {arccosh}(c+d x))^2}{d}-\frac {4 b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{d}+\frac {(c+d x) (a+b \text {arccosh}(c+d x))^4}{d} \] Output:

24*b^4*x-24*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))/d+12* 
b^2*(d*x+c)*(a+b*arccosh(d*x+c))^2/d-4*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*( 
a+b*arccosh(d*x+c))^3/d+(d*x+c)*(a+b*arccosh(d*x+c))^4/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(261\) vs. \(2(129)=258\).

Time = 0.24 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.02 \[ \int (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {\left (a^4+12 a^2 b^2+24 b^4\right ) (c+d x)-4 a b \left (a^2+6 b^2\right ) \sqrt {-1+c+d x} \sqrt {1+c+d x}-4 b \left (-a^3 (c+d x)-6 a b^2 (c+d x)+3 a^2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}+6 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+6 b^2 \left (a^2 (c+d x)+2 b^2 (c+d x)-2 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2-4 b^3 \left (-a (c+d x)+b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^3+b^4 (c+d x) \text {arccosh}(c+d x)^4}{d} \] Input:

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

Output:

((a^4 + 12*a^2*b^2 + 24*b^4)*(c + d*x) - 4*a*b*(a^2 + 6*b^2)*Sqrt[-1 + c + 
 d*x]*Sqrt[1 + c + d*x] - 4*b*(-(a^3*(c + d*x)) - 6*a*b^2*(c + d*x) + 3*a^ 
2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + 6*b^3*Sqrt[-1 + c + d*x]*Sqrt[1 
 + c + d*x])*ArcCosh[c + d*x] + 6*b^2*(a^2*(c + d*x) + 2*b^2*(c + d*x) - 2 
*a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^2 - 4*b^3*(-(a 
*(c + d*x)) + b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^3 + 
 b^4*(c + d*x)*ArcCosh[c + d*x]^4)/d
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6410, 6294, 6330, 6294, 6330, 24}

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.

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

\(\Big \downarrow \) 6410

\(\displaystyle \frac {\int (a+b \text {arccosh}(c+d x))^4d(c+d x)}{d}\)

\(\Big \downarrow \) 6294

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

\(\Big \downarrow \) 6330

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

\(\Big \downarrow \) 6294

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

\(\Big \downarrow \) 6330

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

\(\Big \downarrow \) 24

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 6294

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcCosh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt 
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 6330

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p 
_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + 
e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 
*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ 
p]   Int[(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, p}, x] && EqQ[e1, c*d1] && E 
qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
 

rule 6410

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d 
   Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d 
, n}, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(274\) vs. \(2(121)=242\).

Time = 0.27 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.13

method	result	size
derivativedivides	\(\frac {\left (d x +c \right ) a^{4}+b^{4} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{4}-4 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+12 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}-24 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+24 d x +24 c \right )+4 a \,b^{3} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{3}-3 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+6 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )+6 a^{2} b^{2} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}-2 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )+4 a^{3} b \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}\)	\(275\)
default	\(\frac {\left (d x +c \right ) a^{4}+b^{4} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{4}-4 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+12 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}-24 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+24 d x +24 c \right )+4 a \,b^{3} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{3}-3 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+6 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )+6 a^{2} b^{2} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}-2 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )+4 a^{3} b \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}\)	\(275\)
parts	\(x \,a^{4}+\frac {b^{4} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{4}-4 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+12 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}-24 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+24 d x +24 c \right )}{d}+\frac {4 a \,b^{3} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{3}-3 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+6 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}+\frac {6 a^{2} b^{2} \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}-2 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )}{d}+\frac {4 a^{3} b \left (\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}\)	\(279\)
orering	\(\text {Expression too large to display}\)	\(815\)

Input:

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

Output:

1/d*((d*x+c)*a^4+b^4*((d*x+c)*arccosh(d*x+c)^4-4*arccosh(d*x+c)^3*(d*x+c-1 
)^(1/2)*(d*x+c+1)^(1/2)+12*(d*x+c)*arccosh(d*x+c)^2-24*arccosh(d*x+c)*(d*x 
+c-1)^(1/2)*(d*x+c+1)^(1/2)+24*d*x+24*c)+4*a*b^3*((d*x+c)*arccosh(d*x+c)^3 
-3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+6*(d*x+c)*arccosh(d*x+ 
c)-6*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))+6*a^2*b^2*((d*x+c)*arccosh(d*x+c)^2- 
2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+2*d*x+2*c)+4*a^3*b*((d*x+ 
c)*arccosh(d*x+c)-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (121) = 242\).

Time = 0.11 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.67 \[ \int (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {{\left (b^{4} d x + b^{4} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{4} + 4 \, {\left (a b^{3} d x + a b^{3} c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + {\left (a^{4} + 12 \, a^{2} b^{2} + 24 \, b^{4}\right )} d x - 6 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} a b^{3} - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} d x - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 4 \, {\left ({\left (a^{3} b + 6 \, a b^{3}\right )} d x + {\left (a^{3} b + 6 \, a b^{3}\right )} c - 3 \, {\left (a^{2} b^{2} + 2 \, b^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 4 \, {\left (a^{3} b + 6 \, a b^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d} \] Input:

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

Output:

((b^4*d*x + b^4*c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^4 + 4* 
(a*b^3*d*x + a*b^3*c - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*b^4)*log(d*x + c 
+ sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^3 + (a^4 + 12*a^2*b^2 + 24*b^4)*d*x - 
 6*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*a*b^3 - (a^2*b^2 + 2*b^4)*d*x - (a 
^2*b^2 + 2*b^4)*c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^2 + 4* 
((a^3*b + 6*a*b^3)*d*x + (a^3*b + 6*a*b^3)*c - 3*(a^2*b^2 + 2*b^4)*sqrt(d^ 
2*x^2 + 2*c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1 
)) - 4*(a^3*b + 6*a*b^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d
 

Sympy [F]

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

integrate((a+b*acosh(d*x+c))**4,x)
 

Output:

Integral((a + b*acosh(c + d*x))**4, x)
 

Maxima [F]

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

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

Output:

b^4*x*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4 + a^4*x + 4*((d 
*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*a^3*b/d + integrate(2*(2 
*((c^3 - c)*a*b^3 + (a*b^3*d^3 - b^4*d^3)*x^3 + (3*a*b^3*c*d^2 - 2*b^4*c*d 
^2)*x^2 + ((c^2 - 1)*a*b^3 + (a*b^3*d^2 - b^4*d^2)*x^2 + (2*a*b^3*c*d - b^ 
4*c*d)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + ((3*c^2*d - d)*a*b^3 - (c^ 
2*d - d)*b^4)*x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3 + 3* 
(a^2*b^2*d^3*x^3 + 3*a^2*b^2*c*d^2*x^2 + (3*c^2*d - d)*a^2*b^2*x + (c^3 - 
c)*a^2*b^2 + (a^2*b^2*d^2*x^2 + 2*a^2*b^2*c*d*x + (c^2 - 1)*a^2*b^2)*sqrt( 
d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 
 1) + c)^2)/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*s 
qrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*c^2*d - d)*x - c), x)
 

Giac [F]

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

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

Output:

integrate((b*arccosh(d*x + c) + a)^4, x)
 

Mupad [F(-1)]

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

int((a + b*acosh(c + d*x))^4,x)
 

Output:

int((a + b*acosh(c + d*x))^4, x)
 

Reduce [F]

\[ \int (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {4 \mathit {acosh} \left (d x +c \right ) a^{3} b c +4 \mathit {acosh} \left (d x +c \right ) a^{3} b d x -4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, a^{3} b +\left (\int \mathit {acosh} \left (d x +c \right )^{4}d x \right ) b^{4} d +4 \left (\int \mathit {acosh} \left (d x +c \right )^{3}d x \right ) a \,b^{3} d +6 \left (\int \mathit {acosh} \left (d x +c \right )^{2}d x \right ) a^{2} b^{2} d +a^{4} d x}{d} \] Input:

int((a+b*acosh(d*x+c))^4,x)
 

Output:

(4*acosh(c + d*x)*a**3*b*c + 4*acosh(c + d*x)*a**3*b*d*x - 4*sqrt(c + d*x 
+ 1)*sqrt(c + d*x - 1)*a**3*b + int(acosh(c + d*x)**4,x)*b**4*d + 4*int(ac 
osh(c + d*x)**3,x)*a*b**3*d + 6*int(acosh(c + d*x)**2,x)*a**2*b**2*d + a** 
4*d*x)/d