\(\int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx\) [98]

Optimal result

Integrand size = 25, antiderivative size = 374 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{b d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 b^{3/2} d}+\frac {3 e^4 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 b^{3/2} d}+\frac {e^4 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 b^{3/2} d}+\frac {e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{8 b^{3/2} d}+\frac {3 e^4 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 b^{3/2} d}+\frac {e^4 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 b^{3/2} d} \] Output:

-2*e^4*(d*x+c-1)^(1/2)*(d*x+c)^4*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^ 
(1/2)+1/8*e^4*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^ 
(3/2)/d+3/16*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^(3/2)/d+1/16*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^(3/2)/d+1/8*e^4*Pi^(1/2)*erfi((a 
+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(3/2)/d/exp(a/b)+3/16*e^4*3^(1/2)*Pi^( 
1/2)*erfi(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(3/2)/d/exp(3*a/b) 
+1/16*e^4*5^(1/2)*Pi^(1/2)*erfi(5^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2) 
)/b^(3/2)/d/exp(5*a/b)
 

Mathematica [A] (warning: unable to verify)

Time = 1.06 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.06 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\frac {e^4 e^{-\frac {5 a}{b}} \left (-4 e^{\frac {5 a}{b}} \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)-2 e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )+\sqrt {5} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )+3 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )-3 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )-\sqrt {5} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )-6 e^{\frac {5 a}{b}} \sinh (3 \text {arccosh}(c+d x))-2 e^{\frac {5 a}{b}} \sinh (5 \text {arccosh}(c+d x))\right )}{16 b d \sqrt {a+b \text {arccosh}(c+d x)}} \] Input:

Integrate[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(3/2),x]
 

Output:

(e^4*(-4*E^((5*a)/b)*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) - 2* 
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*ArcC 
osh[c + d*x]))/b] + 3*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] + 2*E^((4*a)/b)*Sqrt[-((a 
+ b*ArcCosh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)] - 3*Sq 
rt[3]*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[1/2, (3*(a + b*ArcCos 
h[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] - 6*E^((5*a)/b)*Sinh[3*ArcCosh[c + d* 
x]] - 2*E^((5*a)/b)*Sinh[5*ArcCosh[c + d*x]]))/(16*b*d*E^((5*a)/b)*Sqrt[a 
+ b*ArcCosh[c + d*x]])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6411, 27, 6300, 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.

\(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 6300

\(\displaystyle \frac {e^4 \left (-\frac {2 \int \left (-\frac {5 \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {9 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 \sqrt {a+b \text {arccosh}(c+d x)}}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^4 \left (-\frac {2 \left (-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{32} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {5 \pi } \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{32} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {5 \pi } \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{d}\)

Input:

Int[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(3/2),x]
 

Output:

(e^4*((-2*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(b*Sqrt[a + b* 
ArcCosh[c + d*x]]) - (2*(-1/16*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*Ar 
cCosh[c + d*x]]/Sqrt[b]]) - (3*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[5* 
Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/32 - (Sqrt[b]*Sqr 
t[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(16*E^(a/b)) - (3*Sqrt[b 
]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*E^( 
(3*a)/b)) - (Sqrt[b]*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]] 
)/Sqrt[b]])/(32*E^((5*a)/b))))/b^2))/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 6300

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) 
)), x] + Simp[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 
 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, 
a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] 
&& LtQ[n, -1]
 

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]

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

Input:

int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(3/2),x)
 

Output:

int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((d*e*x+c*e)**4/(a+b*acosh(d*x+c))**(3/2),x)
 

Output:

e**4*(Integral(c**4/(a*sqrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + 
 d*x))*acosh(c + d*x)), x) + Integral(d**4*x**4/(a*sqrt(a + b*acosh(c + d* 
x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)), x) + Integral(4*c*d**3 
*x**3/(a*sqrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c 
 + d*x)), x) + Integral(6*c**2*d**2*x**2/(a*sqrt(a + b*acosh(c + d*x)) + b 
*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)), x) + Integral(4*c**3*d*x/(a*s 
qrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)), 
x))
 

Maxima [F]

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

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*e*x + c*e)^4/(b*arccosh(d*x + c) + a)^(3/2), x)
 

Giac [F]

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

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate((d*e*x + c*e)^4/(b*arccosh(d*x + c) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(c e+d e x)^4}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\text {too large to display} \] Input:

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

Output:

(e**4*(acosh(c + d*x)*int((sqrt(acosh(c + d*x)*b + a)*x**6)/(acosh(c + d*x 
)**2*b**2*c**2 + 2*acosh(c + d*x)**2*b**2*c*d*x + acosh(c + d*x)**2*b**2*d 
**2*x**2 - acosh(c + d*x)**2*b**2 + 2*acosh(c + d*x)*a*b*c**2 + 4*acosh(c 
+ d*x)*a*b*c*d*x + 2*acosh(c + d*x)*a*b*d**2*x**2 - 2*acosh(c + d*x)*a*b + 
 a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x**2 - a**2),x)*b**2*d**7 + 6*acosh( 
c + d*x)*int((sqrt(acosh(c + d*x)*b + a)*x**5)/(acosh(c + d*x)**2*b**2*c** 
2 + 2*acosh(c + d*x)**2*b**2*c*d*x + acosh(c + d*x)**2*b**2*d**2*x**2 - ac 
osh(c + d*x)**2*b**2 + 2*acosh(c + d*x)*a*b*c**2 + 4*acosh(c + d*x)*a*b*c* 
d*x + 2*acosh(c + d*x)*a*b*d**2*x**2 - 2*acosh(c + d*x)*a*b + a**2*c**2 + 
2*a**2*c*d*x + a**2*d**2*x**2 - a**2),x)*b**2*c*d**6 + 15*acosh(c + d*x)*i 
nt((sqrt(acosh(c + d*x)*b + a)*x**4)/(acosh(c + d*x)**2*b**2*c**2 + 2*acos 
h(c + d*x)**2*b**2*c*d*x + acosh(c + d*x)**2*b**2*d**2*x**2 - acosh(c + d* 
x)**2*b**2 + 2*acosh(c + d*x)*a*b*c**2 + 4*acosh(c + d*x)*a*b*c*d*x + 2*ac 
osh(c + d*x)*a*b*d**2*x**2 - 2*acosh(c + d*x)*a*b + a**2*c**2 + 2*a**2*c*d 
*x + a**2*d**2*x**2 - a**2),x)*b**2*c**2*d**5 - acosh(c + d*x)*int((sqrt(a 
cosh(c + d*x)*b + a)*x**4)/(acosh(c + d*x)**2*b**2*c**2 + 2*acosh(c + d*x) 
**2*b**2*c*d*x + acosh(c + d*x)**2*b**2*d**2*x**2 - acosh(c + d*x)**2*b**2 
 + 2*acosh(c + d*x)*a*b*c**2 + 4*acosh(c + d*x)*a*b*c*d*x + 2*acosh(c + d* 
x)*a*b*d**2*x**2 - 2*acosh(c + d*x)*a*b + a**2*c**2 + 2*a**2*c*d*x + a**2* 
d**2*x**2 - a**2),x)*b**2*d**5 + 20*acosh(c + d*x)*int((sqrt(acosh(c + ...