\(\int \frac {c e+d e x}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx\) [95]

Optimal result

Integrand size = 23, antiderivative size = 113 \[ \int \frac {c e+d e x}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=-\frac {e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d}+\frac {e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {c e+d e x}{\sqrt {a+b \text {arccosh}(c+d x)}} \, dx=-\frac {e \sqrt {\frac {\pi }{2}} \left (\text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (-\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+\text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )\right )}{4 \sqrt {b} d} \] Input:

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

Output:

-1/4*(e*Sqrt[Pi/2]*(Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*( 
-Cosh[(2*a)/b] + Sinh[(2*a)/b]) + Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x] 
])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b])))/(Sqrt[b]*d)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.61 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6411, 27, 6302, 25, 5971, 27, 3042, 26, 3789, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

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

Output:

((I/2)*e*((I/2)*Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*Arc 
Cosh[c + d*x]])/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a 
+ b*ArcCosh[c + d*x]])/Sqrt[b]])/E^((2*a)/b)))/(b*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 3789

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

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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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