\(\int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx\) [85]

Optimal result

Integrand size = 23, antiderivative size = 269 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=-\frac {15 b^2 e \sqrt {a+b \text {arccosh}(c+d x)}}{64 d}+\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \text {arccosh}(c+d x)}}{32 d}-\frac {5 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{8 d}-\frac {e (a+b \text {arccosh}(c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}-\frac {15 b^{5/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{256 d}-\frac {15 b^{5/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{256 d} \] Output:

-15/64*b^2*e*(a+b*arccosh(d*x+c))^(1/2)/d+15/32*b^2*e*(d*x+c)^2*(a+b*arcco 
sh(d*x+c))^(1/2)/d-5/8*b*e*(d*x+c-1)^(1/2)*(d*x+c)*(d*x+c+1)^(1/2)*(a+b*ar 
ccosh(d*x+c))^(3/2)/d-1/4*e*(a+b*arccosh(d*x+c))^(5/2)/d+1/2*e*(d*x+c)^2*( 
a+b*arccosh(d*x+c))^(5/2)/d-15/512*b^(5/2)*e*exp(2*a/b)*2^(1/2)*Pi^(1/2)*e 
rf(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d-15/512*b^(5/2)*e*2^(1/2)* 
Pi^(1/2)*erfi(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))/d/exp(2*a/b)
 

Mathematica [A] (verified)

Time = 1.17 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.85 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\frac {e \left (-15 b^{5/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )-15 b^{5/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+8 \sqrt {a+b \text {arccosh}(c+d x)} \left (\left (16 a^2+15 b^2\right ) \cosh (2 \text {arccosh}(c+d x))+16 b^2 \text {arccosh}(c+d x)^2 \cosh (2 \text {arccosh}(c+d x))-20 a b \sinh (2 \text {arccosh}(c+d x))+4 b \text {arccosh}(c+d x) (8 a \cosh (2 \text {arccosh}(c+d x))-5 b \sinh (2 \text {arccosh}(c+d x)))\right )\right )}{512 d} \] Input:

Integrate[(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^(5/2),x]
 

Output:

(e*(-15*b^(5/2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqr 
t[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) - 15*b^(5/2)*Sqrt[2*Pi]*Erf[(Sqrt[2] 
*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 
8*Sqrt[a + b*ArcCosh[c + d*x]]*((16*a^2 + 15*b^2)*Cosh[2*ArcCosh[c + d*x]] 
 + 16*b^2*ArcCosh[c + d*x]^2*Cosh[2*ArcCosh[c + d*x]] - 20*a*b*Sinh[2*ArcC 
osh[c + d*x]] + 4*b*ArcCosh[c + d*x]*(8*a*Cosh[2*ArcCosh[c + d*x]] - 5*b*S 
inh[2*ArcCosh[c + d*x]]))))/(512*d)
 

Rubi [A] (verified)

Time = 2.15 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6411, 27, 6299, 6354, 6299, 6308, 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.

\(\displaystyle \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int e (c+d x) (a+b \text {arccosh}(c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e \int (c+d x) (a+b \text {arccosh}(c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6308

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

Int[(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^(5/2),x]
 

Output:

(e*(((c + d*x)^2*(a + b*ArcCosh[c + d*x])^(5/2))/2 - (5*b*((Sqrt[-1 + c + 
d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/2 + (a + 
b*ArcCosh[c + d*x])^(5/2)/(5*b) - (3*b*(((c + d*x)^2*Sqrt[a + b*ArcCosh[c 
+ d*x]])/2 + (-Sqrt[a + b*ArcCosh[c + d*x]] - (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi 
/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/4 - (Sqrt[b]*Sqrt 
[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(4*E^((2*a)/b 
)))/4))/4))/4))/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 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 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 
1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( 
m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1)))   Int[(f*x)^(m 
 - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f 
*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( 
-1 + c*x)^p]   Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( 
a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, 
p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 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]
 

Maple [F]

\[\int \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]

Input:

int((d*e*x+c*e)*(a+b*arccosh(d*x+c))^(5/2),x)
 

Output:

int((d*e*x+c*e)*(a+b*arccosh(d*x+c))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*e*x+c*e)*(a+b*arccosh(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate((d*e*x+c*e)*(a+b*acosh(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((d*e*x+c*e)*(a+b*arccosh(d*x+c))^(5/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((d*e*x + c*e)*(b*arccosh(d*x + c) + a)^(5/2), x)
 

Giac [F]

\[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((d*e*x+c*e)*(a+b*arccosh(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate((d*e*x + c*e)*(b*arccosh(d*x + c) + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:

int((c*e + d*e*x)*(a + b*acosh(c + d*x))^(5/2),x)
 

Output:

int((c*e + d*e*x)*(a + b*acosh(c + d*x))^(5/2), x)
 

Reduce [F]

\[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^{5/2} \, dx=e \left (\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}d x \right ) a^{2} c +2 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right ) x d x \right ) a b d +2 \left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )d x \right ) a b c +\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) b^{2} d +\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, \mathit {acosh} \left (d x +c \right )^{2}d x \right ) b^{2} c +\left (\int \sqrt {\mathit {acosh} \left (d x +c \right ) b +a}\, x d x \right ) a^{2} d \right ) \] Input:

int((d*e*x+c*e)*(a+b*acosh(d*x+c))^(5/2),x)
 

Output:

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