\(\int (c e+d e x)^4 \sqrt {a+b \text {arccosh}(c+d x)} \, dx\) [72]

Optimal result

Integrand size = 25, antiderivative size = 361 \[ \int (c e+d e x)^4 \sqrt {a+b \text {arccosh}(c+d x)} \, dx=\frac {e^4 (c+d x)^5 \sqrt {a+b \text {arccosh}(c+d x)}}{5 d}-\frac {\sqrt {b} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {\sqrt {b} e^4 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {\sqrt {b} e^4 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{320 d}-\frac {\sqrt {b} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {\sqrt {b} e^4 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {\sqrt {b} e^4 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{320 d} \] Output:

1/5*e^4*(d*x+c)^5*(a+b*arccosh(d*x+c))^(1/2)/d-1/32*b^(1/2)*e^4*exp(a/b)*P 
i^(1/2)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d-1/192*b^(1/2)*e^4*exp(3* 
a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d-1/ 
1600*b^(1/2)*e^4*exp(5*a/b)*5^(1/2)*Pi^(1/2)*erf(5^(1/2)*(a+b*arccosh(d*x+ 
c))^(1/2)/b^(1/2))/d-1/32*b^(1/2)*e^4*Pi^(1/2)*erfi((a+b*arccosh(d*x+c))^( 
1/2)/b^(1/2))/d/exp(a/b)-1/192*b^(1/2)*e^4*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*( 
a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d/exp(3*a/b)-1/1600*b^(1/2)*e^4*5^(1/2) 
*Pi^(1/2)*erfi(5^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d/exp(5*a/b)
 

Mathematica [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.95 \[ \int (c e+d e x)^4 \sqrt {a+b \text {arccosh}(c+d x)} \, dx=\frac {e^4 e^{-\frac {5 a}{b}} \sqrt {a+b \text {arccosh}(c+d x)} \left (150 e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )+3 \sqrt {5} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )+25 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+150 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )+25 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+3 \sqrt {5} e^{\frac {10 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )\right )}{2400 d \sqrt {-\frac {(a+b \text {arccosh}(c+d x))^2}{b^2}}} \] Input:

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

Output:

(e^4*Sqrt[a + b*ArcCosh[c + d*x]]*(150*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c 
 + d*x])/b)]*Gamma[3/2, a/b + ArcCosh[c + d*x]] + 3*Sqrt[5]*Sqrt[a/b + Arc 
Cosh[c + d*x]]*Gamma[3/2, (-5*(a + b*ArcCosh[c + d*x]))/b] + 25*Sqrt[3]*E^ 
((2*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c + d 
*x]))/b] + 150*E^((4*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, -((a + 
b*ArcCosh[c + d*x])/b)] + 25*Sqrt[3]*E^((8*a)/b)*Sqrt[-((a + b*ArcCosh[c + 
 d*x])/b)]*Gamma[3/2, (3*(a + b*ArcCosh[c + d*x]))/b] + 3*Sqrt[5]*E^((10*a 
)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[3/2, (5*(a + b*ArcCosh[c + 
d*x]))/b]))/(2400*d*E^((5*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])^2/b^2)])
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6411, 27, 6299, 6368, 3042, 3793, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6299

\(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{10} b \int \frac {(c+d x)^5}{\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)\right )}{d}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{10} \int \frac {\cosh ^5\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{10} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )^5}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )}{d}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{10} \int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {5 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {5 \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^4 \left (\frac {1}{10} \left (-\frac {5}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {5}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {5}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {5}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{5} (c+d x)^5 \sqrt {a+b \text {arccosh}(c+d x)}\right )}{d}\)

Input:

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

Output:

(e^4*(((c + d*x)^5*Sqrt[a + b*ArcCosh[c + d*x]])/5 + ((-5*Sqrt[b]*E^(a/b)* 
Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/16 - (5*Sqrt[b]*E^((3* 
a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/32 - 
 (Sqrt[b]*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]] 
)/Sqrt[b]])/32 - (5*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqr 
t[b]])/(16*E^(a/b)) - (5*Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCo 
sh[c + d*x]])/Sqrt[b]])/(32*E^((3*a)/b)) - (Sqrt[b]*Sqrt[Pi/5]*Erfi[(Sqrt[ 
5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*E^((5*a)/b)))/10))/d
 

Defintions of rubi rules used

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 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6299

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

rule 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 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 [F]

\[\int \left (d e x +c e \right )^{4} \sqrt {a +b \,\operatorname {arccosh}\left (d x +c \right )}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int (c e+d e x)^4 \sqrt {a+b \text {arccosh}(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

e**4*(Integral(c**4*sqrt(a + b*acosh(c + d*x)), x) + Integral(d**4*x**4*sq 
rt(a + b*acosh(c + d*x)), x) + Integral(4*c*d**3*x**3*sqrt(a + b*acosh(c + 
 d*x)), x) + Integral(6*c**2*d**2*x**2*sqrt(a + b*acosh(c + d*x)), x) + In 
tegral(4*c**3*d*x*sqrt(a + b*acosh(c + d*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

e**4*(int(sqrt(acosh(c + d*x)*b + a),x)*c**4 + int(sqrt(acosh(c + d*x)*b + 
 a)*x**4,x)*d**4 + 4*int(sqrt(acosh(c + d*x)*b + a)*x**3,x)*c*d**3 + 6*int 
(sqrt(acosh(c + d*x)*b + a)*x**2,x)*c**2*d**2 + 4*int(sqrt(acosh(c + d*x)* 
b + a)*x,x)*c**3*d)