\(\int \frac {(c e+d e x)^4}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx\) [92]

Optimal result

Integrand size = 25, antiderivative size = 326 \[ \int \frac {(c e+d e x)^4}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=-\frac {e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 \sqrt {b} d}-\frac {e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}-\frac {e^4 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}+\frac {e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 \sqrt {b} d}+\frac {e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d}+\frac {e^4 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 \sqrt {b} d} \] Output:

-1/16*e^4*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(1/2 
)/d-1/32*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))/b^(1/2)/d-1/160*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))/b^(1/2)/d+1/16*e^4*Pi^(1/2)*erfi((a+b 
*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d/exp(a/b)+1/32*e^4*3^(1/2)*Pi^(1/ 
2)*erfi(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d/exp(3*a/b)+1 
/160*e^4*5^(1/2)*Pi^(1/2)*erfi(5^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2)) 
/b^(1/2)/d/exp(5*a/b)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.98 \[ \int \frac {(c e+d e x)^4}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=\frac {e^4 e^{-\frac {5 a}{b}} \left (10 e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )+\sqrt {5} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )+5 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+10 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )+5 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\sqrt {5} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )\right )}{160 d \sqrt {a+b \text {arccosh}(c+d x)}} \] Input:

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

Output:

(e^4*(10*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[1/2, a/b + ArcCosh 
[c + d*x]] + Sqrt[5]*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[1/2, (-5*(a 
 + b*ArcCosh[c + d*x]))/b] + 5*Sqrt[3]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c 
 + d*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c + d*x]))/b] + 10*E^((4*a)/b)* 
Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/ 
b)] + 5*Sqrt[3]*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[1/2, (3*(a 
+ b*ArcCosh[c + d*x]))/b] + Sqrt[5]*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c + d* 
x]]*Gamma[1/2, (5*(a + b*ArcCosh[c + d*x]))/b]))/(160*d*E^((5*a)/b)*Sqrt[a 
 + b*ArcCosh[c + d*x]])
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6411, 27, 6302, 25, 5971, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e^4*(-1/16*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqr 
t[b]]) - (Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*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 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b 
*ArcCosh[c + d*x]]/Sqrt[b]])/(16*E^(a/b)) + (Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt 
[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*E^((3*a)/b)) + (Sqrt[b]*Sq 
rt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*E^((5*a 
)/b))))/(b*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 2009

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

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6302

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

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

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

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

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

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

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