\(\int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx\) [101]

Optimal result

Integrand size = 23, antiderivative size = 155 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(314\) vs. \(2(155)=310\).

Time = 4.39 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.03 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{3/2}} \, dx=\frac {e \left (-2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-2 c \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+\sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-\frac {2 \sqrt {b} e^{-\frac {a}{b}} \left (c e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )-c \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )+e^{a/b} \sinh (2 \text {arccosh}(c+d x))\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}\right )}{2 b^{3/2} d} \] Input:

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

Output:

(e*((-2*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/E^(a/b) + ( 
Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/E^((2*a)/ 
b) - 2*c*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + S 
inh[a/b]) + Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]] 
*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) - (2*Sqrt[b]*(c*E^((2*a)/b)*Sqrt[a/b + Ar 
cCosh[c + d*x]]*Gamma[1/2, a/b + ArcCosh[c + d*x]] - c*Sqrt[-((a + b*ArcCo 
sh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)] + E^(a/b)*Sinh[ 
2*ArcCosh[c + d*x]]))/(E^(a/b)*Sqrt[a + b*ArcCosh[c + d*x]])))/(2*b^(3/2)* 
d)
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6411, 27, 6300, 25, 3042, 3788, 26, 2611, 2633, 2634}

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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6300

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

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

Output:

(e*((-2*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(b*Sqrt[a + b*ArcC 
osh[c + d*x]]) - (2*(-1/2*(Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqr 
t[a + b*ArcCosh[c + d*x]])/Sqrt[b]]) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*S 
qrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(2*E^((2*a)/b))))/b^2))/d
 

Defintions of rubi rules used

rule 25

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

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

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

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

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

Output:

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

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e*(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 - 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**4 + 2*acosh(c + 
 d*x)*int((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*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**3 + acosh(c + d*x)*int((sq 
rt(acosh(c + d*x)*b + a)*x)/(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*c**2*d**2 - acosh(c + d*x)*int((sqrt(acosh(c + 
d*x)*b + a)*x)/(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**2 + 2*acosh(c + d*x)*int((sqrt(c + d*x + 1)*sqrt(c + d...