\(\int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx\) [100]

Optimal result

Integrand size = 25, antiderivative size = 262 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}+\frac {e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}+\frac {e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.03 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.01 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\frac {e^2 e^{-\frac {3 a}{b}} \left (-e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )+\sqrt {3} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )-2 e^{\frac {3 a}{b}} \left (\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)+\sinh (3 \text {arccosh}(c+d x))\right )\right )}{4 b d \sqrt {a+b \text {arccosh}(c+d x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.94, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6300

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \left (-\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{d}\)

Input:

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

Output:

(e^2*((-2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*Sqrt[a + b* 
ArcCosh[c + d*x]]) - (2*(-1/8*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*Arc 
Cosh[c + d*x]]/Sqrt[b]]) - (Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sq 
rt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/8 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + 
b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*E^(a/b)) - (Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt 
[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*E^((3*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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((d*e*x+c*e)^2/(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]

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

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

Output:

e**2*(Integral(c**2/(a*sqrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + 
 d*x))*acosh(c + d*x)), x) + Integral(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(2*c*d*x/ 
(a*sqrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x 
)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e**2*(acosh(c + d*x)*int((sqrt(acosh(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 + 4*acosh( 
c + d*x)*int((sqrt(acosh(c + d*x)*b + a)*x**3)/(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**4 + 5*acosh(c + d*x)*in 
t((sqrt(acosh(c + d*x)*b + a)*x**2)/(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*aco 
sh(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**3 - acosh(c + d*x)*int((sqrt(ac 
osh(c + d*x)*b + a)*x**2)/(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**3 + 2*acosh(c + d*x)*int((sqrt(acosh(c + d*...