\(\int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx\) [41]

Optimal result

Integrand size = 23, antiderivative size = 309 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {160}{27} b^4 e^2 x+\frac {8 b^4 e^2 (c+d x)^3}{81 d}-\frac {160 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{27 d}-\frac {8 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{27 d}+\frac {8 b^2 e^2 (c+d x) (a+b \text {arccosh}(c+d x))^2}{3 d}+\frac {4 b^2 e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^2}{9 d}-\frac {8 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}-\frac {4 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^4}{3 d} \] Output:

160/27*b^4*e^2*x+8/81*b^4*e^2*(d*x+c)^3/d-160/27*b^3*e^2*(d*x+c-1)^(1/2)*( 
d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))/d-8/27*b^3*e^2*(d*x+c-1)^(1/2)*(d*x+c) 
^2*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))/d+8/3*b^2*e^2*(d*x+c)*(a+b*arccosh 
(d*x+c))^2/d+4/9*b^2*e^2*(d*x+c)^3*(a+b*arccosh(d*x+c))^2/d-8/9*b*e^2*(d*x 
+c-1)^(1/2)*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))^3/d-4/9*b*e^2*(d*x+c-1)^( 
1/2)*(d*x+c)^2*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))^3/d+1/3*e^2*(d*x+c)^3* 
(a+b*arccosh(d*x+c))^4/d
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.54 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {e^2 \left (24 b^2 \left (9 a^2+20 b^2\right ) (c+d x)+\left (27 a^4+36 a^2 b^2+8 b^4\right ) (c+d x)^3+12 a b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-6 a^2-40 b^2-\left (3 a^2+2 b^2\right ) (c+d x)^2\right )+12 b \left (36 a b^2 (c+d x)+9 a^3 (c+d x)^3+6 a b^2 (c+d x)^3-18 a^2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}-40 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x}-9 a^2 b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}-2 b^3 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+18 b^2 \left (12 b^2 (c+d x)+9 a^2 (c+d x)^3+2 b^2 (c+d x)^3-12 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}-6 a b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2-36 b^3 \left (-3 a (c+d x)^3+2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}+b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^3+27 b^4 (c+d x)^3 \text {arccosh}(c+d x)^4\right )}{81 d} \] Input:

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

Output:

(e^2*(24*b^2*(9*a^2 + 20*b^2)*(c + d*x) + (27*a^4 + 36*a^2*b^2 + 8*b^4)*(c 
 + d*x)^3 + 12*a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(-6*a^2 - 40*b^2 - 
 (3*a^2 + 2*b^2)*(c + d*x)^2) + 12*b*(36*a*b^2*(c + d*x) + 9*a^3*(c + d*x) 
^3 + 6*a*b^2*(c + d*x)^3 - 18*a^2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 
 40*b^3*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 9*a^2*b*Sqrt[-1 + c + d*x]* 
(c + d*x)^2*Sqrt[1 + c + d*x] - 2*b^3*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[ 
1 + c + d*x])*ArcCosh[c + d*x] + 18*b^2*(12*b^2*(c + d*x) + 9*a^2*(c + d*x 
)^3 + 2*b^2*(c + d*x)^3 - 12*a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 6* 
a*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^2 - 
 36*b^3*(-3*a*(c + d*x)^3 + 2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + b*S 
qrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^3 + 27*b 
^4*(c + d*x)^3*ArcCosh[c + d*x]^4))/(81*d)
 

Rubi [A] (verified)

Time = 3.52 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6411, 27, 6298, 6354, 6298, 6330, 6294, 6330, 24, 6354, 15, 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.

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

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arccosh}(c+d x))^4d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arccosh}(c+d x))^4d(c+d x)}{d}\)

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6330

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

\(\Big \downarrow \) 6294

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

\(\Big \downarrow \) 6330

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^4-\frac {4}{3} b \left (-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \left (\frac {2}{3} \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))-\frac {1}{9} b (c+d x)^3\right )\right )+\frac {1}{3} \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arccosh}(c+d x))^2-2 b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))-b (c+d x)\right )\right )\right )\right )\right )}{d}\)

\(\Big \downarrow \) 6330

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^4-\frac {4}{3} b \left (-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \left (\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))-b \int 1d(c+d x)\right )+\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))-\frac {1}{9} b (c+d x)^3\right )\right )+\frac {1}{3} \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arccosh}(c+d x))^2-2 b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))-b (c+d x)\right )\right )\right )\right )\right )}{d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^4-\frac {4}{3} b \left (\frac {1}{3} \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))^2-\frac {2}{3} b \left (\frac {1}{3} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 (a+b \text {arccosh}(c+d x))+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))-b (c+d x)\right )-\frac {1}{9} b (c+d x)^3\right )\right )+\frac {2}{3} \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arccosh}(c+d x))^2-2 b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))-b (c+d x)\right )\right )\right )\right )\right )}{d}\)

Input:

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

Output:

(e^2*(((c + d*x)^3*(a + b*ArcCosh[c + d*x])^4)/3 - (4*b*((Sqrt[-1 + c + d* 
x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/3 - b*(((c + 
d*x)^3*(a + b*ArcCosh[c + d*x])^2)/3 - (2*b*(-1/9*(b*(c + d*x)^3) + (Sqrt[ 
-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/3 + 
(2*(-(b*(c + d*x)) + Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c 
 + d*x])))/3))/3) + (2*(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCos 
h[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*ArcCosh[c + d*x])))))/ 
3))/3))/d
 

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 24

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 6298

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

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 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]
 

rule 6411

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

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.67

method	result	size
derivativedivides	\(\frac {\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3}+e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{4}}{3}-\frac {8 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {4 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {8 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {160 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {160 d x}{27}+\frac {160 c}{27}+\frac {4 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{9}-\frac {8 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )+4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}+\frac {4 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{9}-\frac {2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{27}\right )+6 e^{2} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+4 e^{2} a^{3} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\)	\(517\)
default	\(\frac {\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3}+e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{4}}{3}-\frac {8 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {4 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {8 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {160 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {160 d x}{27}+\frac {160 c}{27}+\frac {4 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{9}-\frac {8 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )+4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}+\frac {4 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{9}-\frac {2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{27}\right )+6 e^{2} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+4 e^{2} a^{3} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\)	\(517\)
parts	\(\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{4}}{3}-\frac {8 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {4 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {8 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {160 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {160 d x}{27}+\frac {160 c}{27}+\frac {4 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{9}-\frac {8 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )}{d}+\frac {4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}+\frac {4 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{9}-\frac {2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{27}\right )}{d}+\frac {6 e^{2} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )}{d}+\frac {4 e^{2} a^{3} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\)	\(528\)
orering	\(\text {Expression too large to display}\)	\(2217\)

Input:

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

Output:

1/d*(1/3*e^2*a^4*(d*x+c)^3+e^2*b^4*(1/3*(d*x+c)^3*arccosh(d*x+c)^4-8/9*arc 
cosh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-4/9*(d*x+c)^2*arccosh(d*x+c) 
^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+8/3*(d*x+c)*arccosh(d*x+c)^2-160/27*arc 
cosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+160/27*d*x+160/27*c+4/9*(d*x+c 
)^3*arccosh(d*x+c)^2-8/27*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*( 
d*x+c)^2+8/81*(d*x+c)^3)+4*e^2*a*b^3*(1/3*(d*x+c)^3*arccosh(d*x+c)^3-2/3*a 
rccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-1/3*(d*x+c)^2*arccosh(d*x+ 
c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4/3*(d*x+c)*arccosh(d*x+c)-40/27*(d*x 
+c-1)^(1/2)*(d*x+c+1)^(1/2)+2/9*(d*x+c)^3*arccosh(d*x+c)-2/27*(d*x+c-1)^(1 
/2)*(d*x+c+1)^(1/2)*(d*x+c)^2)+6*e^2*a^2*b^2*(1/3*(d*x+c)^3*arccosh(d*x+c) 
^2-4/9*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-2/9*arccosh(d*x+c)*( 
d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^2+4/9*d*x+4/9*c+2/27*(d*x+c)^3)+4*e 
^2*a^3*b*(1/3*(d*x+c)^3*arccosh(d*x+c)-1/9*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2) 
*((d*x+c)^2+2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 890 vs. \(2 (275) = 550\).

Time = 0.13 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.88 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx =\text {Too large to display} \] Input:

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

Output:

1/81*((27*a^4 + 36*a^2*b^2 + 8*b^4)*d^3*e^2*x^3 + 3*(27*a^4 + 36*a^2*b^2 + 
 8*b^4)*c*d^2*e^2*x^2 + 3*(72*a^2*b^2 + 160*b^4 + (27*a^4 + 36*a^2*b^2 + 8 
*b^4)*c^2)*d*e^2*x + 27*(b^4*d^3*e^2*x^3 + 3*b^4*c*d^2*e^2*x^2 + 3*b^4*c^2 
*d*e^2*x + b^4*c^3*e^2)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^4 
 + 36*(3*a*b^3*d^3*e^2*x^3 + 9*a*b^3*c*d^2*e^2*x^2 + 9*a*b^3*c^2*d*e^2*x + 
 3*a*b^3*c^3*e^2 - (b^4*d^2*e^2*x^2 + 2*b^4*c*d*e^2*x + (b^4*c^2 + 2*b^4)* 
e^2)*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))^3 + 18*((9*a^2*b^2 + 2*b^4)*d^3*e^2*x^3 + 3*(9*a^2*b^2 + 2* 
b^4)*c*d^2*e^2*x^2 + 3*(4*b^4 + (9*a^2*b^2 + 2*b^4)*c^2)*d*e^2*x + (12*b^4 
*c + (9*a^2*b^2 + 2*b^4)*c^3)*e^2 - 6*(a*b^3*d^2*e^2*x^2 + 2*a*b^3*c*d*e^2 
*x + (a*b^3*c^2 + 2*a*b^3)*e^2)*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))^2 + 12*(3*(3*a^3*b + 2*a*b^3)*d^ 
3*e^2*x^3 + 9*(3*a^3*b + 2*a*b^3)*c*d^2*e^2*x^2 + 9*(4*a*b^3 + (3*a^3*b + 
2*a*b^3)*c^2)*d*e^2*x + 3*(12*a*b^3*c + (3*a^3*b + 2*a*b^3)*c^3)*e^2 - ((9 
*a^2*b^2 + 2*b^4)*d^2*e^2*x^2 + 2*(9*a^2*b^2 + 2*b^4)*c*d*e^2*x + (18*a^2* 
b^2 + 40*b^4 + (9*a^2*b^2 + 2*b^4)*c^2)*e^2)*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)) - 12*((3*a^3*b + 2* 
a*b^3)*d^2*e^2*x^2 + 2*(3*a^3*b + 2*a*b^3)*c*d*e^2*x + (6*a^3*b + 40*a*b^3 
 + (3*a^3*b + 2*a*b^3)*c^2)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d
 

Sympy [F]

\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx=e^{2} \left (\int a^{4} c^{2}\, dx + \int a^{4} d^{2} x^{2}\, dx + \int b^{4} c^{2} \operatorname {acosh}^{4}{\left (c + d x \right )}\, dx + \int 4 a b^{3} c^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 6 a^{2} b^{2} c^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 4 a^{3} b c^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 a^{4} c d x\, dx + \int b^{4} d^{2} x^{2} \operatorname {acosh}^{4}{\left (c + d x \right )}\, dx + \int 4 a b^{3} d^{2} x^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 6 a^{2} b^{2} d^{2} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 4 a^{3} b d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 b^{4} c d x \operatorname {acosh}^{4}{\left (c + d x \right )}\, dx + \int 8 a b^{3} c d x \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 12 a^{2} b^{2} c d x \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 8 a^{3} b c d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \] Input:

integrate((d*e*x+c*e)**2*(a+b*acosh(d*x+c))**4,x)
 

Output:

e**2*(Integral(a**4*c**2, x) + Integral(a**4*d**2*x**2, x) + Integral(b**4 
*c**2*acosh(c + d*x)**4, x) + Integral(4*a*b**3*c**2*acosh(c + d*x)**3, x) 
 + Integral(6*a**2*b**2*c**2*acosh(c + d*x)**2, x) + Integral(4*a**3*b*c** 
2*acosh(c + d*x), x) + Integral(2*a**4*c*d*x, x) + Integral(b**4*d**2*x**2 
*acosh(c + d*x)**4, x) + Integral(4*a*b**3*d**2*x**2*acosh(c + d*x)**3, x) 
 + Integral(6*a**2*b**2*d**2*x**2*acosh(c + d*x)**2, x) + Integral(4*a**3* 
b*d**2*x**2*acosh(c + d*x), x) + Integral(2*b**4*c*d*x*acosh(c + d*x)**4, 
x) + Integral(8*a*b**3*c*d*x*acosh(c + d*x)**3, x) + Integral(12*a**2*b**2 
*c*d*x*acosh(c + d*x)**2, x) + Integral(8*a**3*b*c*d*x*acosh(c + d*x), x))
 

Maxima [F]

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

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

Output:

1/3*a^4*d^2*e^2*x^3 + a^4*c*d*e^2*x^2 + 2*(2*x^2*arccosh(d*x + c) - d*(3*c 
^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 + sqrt 
(d^2*x^2 + 2*c*d*x + c^2 - 1)*x/d^2 - (c^2 - 1)*log(2*d^2*x + 2*c*d + 2*sq 
rt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 
1)*c/d^3))*a^3*b*c*d*e^2 + 2/9*(6*x^3*arccosh(d*x + c) - d*(2*sqrt(d^2*x^2 
 + 2*c*d*x + c^2 - 1)*x^2/d^2 - 15*c^3*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^ 
2 + 2*c*d*x + c^2 - 1)*d)/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x/d^ 
3 + 9*(c^2 - 1)*c*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1 
)*d)/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 
 2*c*d*x + c^2 - 1)*(c^2 - 1)/d^4))*a^3*b*d^2*e^2 + a^4*c^2*e^2*x + 4*((d* 
x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*a^3*b*c^2*e^2/d + 1/3*(b^ 
4*d^2*e^2*x^3 + 3*b^4*c*d*e^2*x^2 + 3*b^4*c^2*e^2*x)*log(d*x + sqrt(d*x + 
c + 1)*sqrt(d*x + c - 1) + c)^4 + integrate(2/3*(2*((3*a*b^3*d^5*e^2 - b^4 
*d^5*e^2)*x^5 + 3*(c^5*e^2 - c^3*e^2)*a*b^3 + 5*(3*a*b^3*c*d^4*e^2 - b^4*c 
*d^4*e^2)*x^4 + (3*(10*c^2*d^3*e^2 - d^3*e^2)*a*b^3 - (10*c^2*d^3*e^2 - d^ 
3*e^2)*b^4)*x^3 + 3*((10*c^3*d^2*e^2 - 3*c*d^2*e^2)*a*b^3 - (3*c^3*d^2*e^2 
 - c*d^2*e^2)*b^4)*x^2 + (3*(c^4*e^2 - c^2*e^2)*a*b^3 + (3*a*b^3*d^4*e^2 - 
 b^4*d^4*e^2)*x^4 + 4*(3*a*b^3*c*d^3*e^2 - b^4*c*d^3*e^2)*x^3 - 3*(2*b^4*c 
^2*d^2*e^2 - (6*c^2*d^2*e^2 - d^2*e^2)*a*b^3)*x^2 - 3*(b^4*c^3*d*e^2 - 2*( 
2*c^3*d*e^2 - c*d*e^2)*a*b^3)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + ...
 

Giac [F]

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

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

Output:

integrate((d*e*x + c*e)^2*(b*arccosh(d*x + c) + a)^4, x)
 

Mupad [F(-1)]

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

int((c*e + d*e*x)^2*(a + b*acosh(c + d*x))^4,x)
 

Output:

int((c*e + d*e*x)^2*(a + b*acosh(c + d*x))^4, x)
 

Reduce [F]

\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {e^{2} \left (36 \mathit {acosh} \left (d x +c \right ) a^{3} b \,c^{3}+36 \mathit {acosh} \left (d x +c \right ) a^{3} b \,c^{2} d x +36 \mathit {acosh} \left (d x +c \right ) a^{3} b c \,d^{2} x^{2}+12 \mathit {acosh} \left (d x +c \right ) a^{3} b \,d^{3} x^{3}+32 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{3} b \,c^{2}-8 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{3} b c d x -4 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{3} b \,d^{2} x^{2}-8 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a^{3} b -36 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, a^{3} b \,c^{2}+9 \left (\int \mathit {acosh} \left (d x +c \right )^{4}d x \right ) b^{4} c^{2} d +36 \left (\int \mathit {acosh} \left (d x +c \right )^{3}d x \right ) a \,b^{3} c^{2} d +54 \left (\int \mathit {acosh} \left (d x +c \right )^{2}d x \right ) a^{2} b^{2} c^{2} d +9 \left (\int \mathit {acosh} \left (d x +c \right )^{4} x^{2}d x \right ) b^{4} d^{3}+18 \left (\int \mathit {acosh} \left (d x +c \right )^{4} x d x \right ) b^{4} c \,d^{2}+36 \left (\int \mathit {acosh} \left (d x +c \right )^{3} x^{2}d x \right ) a \,b^{3} d^{3}+72 \left (\int \mathit {acosh} \left (d x +c \right )^{3} x d x \right ) a \,b^{3} c \,d^{2}+54 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x^{2}d x \right ) a^{2} b^{2} d^{3}+108 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) a^{2} b^{2} c \,d^{2}-24 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a^{3} b \,c^{3}+9 a^{4} c^{2} d x +9 a^{4} c \,d^{2} x^{2}+3 a^{4} d^{3} x^{3}\right )}{9 d} \] Input:

int((d*e*x+c*e)^2*(a+b*acosh(d*x+c))^4,x)
 

Output:

(e**2*(36*acosh(c + d*x)*a**3*b*c**3 + 36*acosh(c + d*x)*a**3*b*c**2*d*x + 
 36*acosh(c + d*x)*a**3*b*c*d**2*x**2 + 12*acosh(c + d*x)*a**3*b*d**3*x**3 
 + 32*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**3*b*c**2 - 8*sqrt(c**2 + 2*c 
*d*x + d**2*x**2 - 1)*a**3*b*c*d*x - 4*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1 
)*a**3*b*d**2*x**2 - 8*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**3*b - 36*sq 
rt(c + d*x + 1)*sqrt(c + d*x - 1)*a**3*b*c**2 + 9*int(acosh(c + d*x)**4,x) 
*b**4*c**2*d + 36*int(acosh(c + d*x)**3,x)*a*b**3*c**2*d + 54*int(acosh(c 
+ d*x)**2,x)*a**2*b**2*c**2*d + 9*int(acosh(c + d*x)**4*x**2,x)*b**4*d**3 
+ 18*int(acosh(c + d*x)**4*x,x)*b**4*c*d**2 + 36*int(acosh(c + d*x)**3*x** 
2,x)*a*b**3*d**3 + 72*int(acosh(c + d*x)**3*x,x)*a*b**3*c*d**2 + 54*int(ac 
osh(c + d*x)**2*x**2,x)*a**2*b**2*d**3 + 108*int(acosh(c + d*x)**2*x,x)*a* 
*2*b**2*c*d**2 - 24*log(sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1) + c + d*x)*a* 
*3*b*c**3 + 9*a**4*c**2*d*x + 9*a**4*c*d**2*x**2 + 3*a**4*d**3*x**3))/(9*d 
)