3.2.55 \(\int (c e+d e x)^3 \sqrt {a+b \text {arccosh}(c+d x)} \, dx\) [155]

3.2.55.1 Optimal result

Integrand size = 25, antiderivative size = 272 \[ \int (c e+d e x)^3 \sqrt {a+b \text {arccosh}(c+d x)} \, dx=-\frac {3 e^3 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}}{4 d}-\frac {\sqrt {b} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {\sqrt {b} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {\sqrt {b} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {\sqrt {b} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 d} \]

output

-1/64*e^3*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*b^(1/ 
2)*2^(1/2)*Pi^(1/2)/d-1/64*e^3*erfi(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^( 
1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d/exp(2*a/b)-1/256*e^3*exp(4*a/b)*erf(2*(a+ 
b*arccosh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/d-1/256*e^3*erfi(2*(a+b* 
arccosh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/d/exp(4*a/b)-3/32*e^3*(a+b 
*arccosh(d*x+c))^(1/2)/d+1/4*e^3*(d*x+c)^4*(a+b*arccosh(d*x+c))^(1/2)/d
 

3.2.55.2 Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.82 \[ \int (c e+d e x)^3 \sqrt {a+b \text {arccosh}(c+d x)} \, dx=\frac {e^3 e^{-\frac {4 a}{b}} \sqrt {a+b \text {arccosh}(c+d x)} \left (\sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+4 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}} \left (4 \sqrt {2} \Gamma \left (\frac {3}{2},\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )\right )\right )}{128 d \sqrt {-\frac {(a+b \text {arccosh}(c+d x))^2}{b^2}}} \]

input

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

output

(e^3*Sqrt[a + b*ArcCosh[c + d*x]]*(Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, 
 (-4*(a + b*ArcCosh[c + d*x]))/b] + 4*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcCo 
sh[c + d*x]]*Gamma[3/2, (-2*(a + b*ArcCosh[c + d*x]))/b] + E^((6*a)/b)*Sqr 
t[-((a + b*ArcCosh[c + d*x])/b)]*(4*Sqrt[2]*Gamma[3/2, (2*(a + b*ArcCosh[c 
 + d*x]))/b] + E^((2*a)/b)*Gamma[3/2, (4*(a + b*ArcCosh[c + d*x]))/b])))/( 
128*d*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])^2/b^2)])
 

3.2.55.3 Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6411, 27, 6299, 6368, 3042, 3793, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{8} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )^4}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )}{d}\)

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^3 \left (\frac {1}{8} \left (-\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{4} \sqrt {a+b \text {arccosh}(c+d x)}\right )+\frac {1}{4} (c+d x)^4 \sqrt {a+b \text {arccosh}(c+d x)}\right )}{d}\)

input

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

output

(e^3*(((c + d*x)^4*Sqrt[a + b*ArcCosh[c + d*x]])/4 + ((-3*Sqrt[a + b*ArcCo 
sh[c + d*x]])/4 - (Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[ 
c + d*x]])/Sqrt[b]])/32 - (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqr 
t[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/4 - (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a 
+ b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*E^((4*a)/b)) - (Sqrt[b]*Sqrt[Pi/2]*Er 
fi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(4*E^((2*a)/b)))/8))/d
 

3.2.55.3.1 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 3042

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

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6299

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

rule 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && 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]
 

3.2.55.4 Maple [F]

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

input

int((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(1/2),x)
 

output

int((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(1/2),x)
 

3.2.55.5 Fricas [F(-2)]

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

input

integrate((d*e*x+c*e)^3*(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)
 

3.2.55.6 Sympy [F]

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

input

integrate((d*e*x+c*e)**3*(a+b*acosh(d*x+c))**(1/2),x)
 

output

e**3*(Integral(c**3*sqrt(a + b*acosh(c + d*x)), x) + Integral(d**3*x**3*sq 
rt(a + b*acosh(c + d*x)), x) + Integral(3*c*d**2*x**2*sqrt(a + b*acosh(c + 
 d*x)), x) + Integral(3*c**2*d*x*sqrt(a + b*acosh(c + d*x)), x))
 

3.2.55.7 Maxima [F]

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

input

integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(1/2),x, algorithm="maxima")
 

output

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

3.2.55.8 Giac [F]

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

input

integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^(1/2),x, algorithm="giac")
 

output

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

3.2.55.9 Mupad [F(-1)]

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

input

int((c*e + d*e*x)^3*(a + b*acosh(c + d*x))^(1/2),x)
 

output

int((c*e + d*e*x)^3*(a + b*acosh(c + d*x))^(1/2), x)