3.2.50 \(\int (a+b \text {arccosh}(c x))^{5/2} \, dx\) [150]

3.2.50.1 Optimal result

Integrand size = 12, antiderivative size = 160 \[ \int (a+b \text {arccosh}(c x))^{5/2} \, dx=\frac {15}{4} b^2 x \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{2 c}+x (a+b \text {arccosh}(c x))^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c} \]

output

x*(a+b*arccosh(c*x))^(5/2)-15/16*b^(5/2)*exp(a/b)*erf((a+b*arccosh(c*x))^( 
1/2)/b^(1/2))*Pi^(1/2)/c-15/16*b^(5/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/ 
2))*Pi^(1/2)/c/exp(a/b)-5/2*b*(a+b*arccosh(c*x))^(3/2)*(c*x-1)^(1/2)*(c*x+ 
1)^(1/2)/c+15/4*b^2*x*(a+b*arccosh(c*x))^(1/2)
 

3.2.50.2 Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(452\) vs. \(2(160)=320\).

Time = 1.67 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.82 \[ \int (a+b \text {arccosh}(c x))^{5/2} \, dx=\frac {4 b \sqrt {a+b \text {arccosh}(c x)} \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a-5 b \text {arccosh}(c x))+b c x \left (15+4 \text {arccosh}(c x)^2\right )\right )+8 a^2 e^{-\frac {a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arccosh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arccosh}(c x)}{b}}}\right )-\sqrt {b} \left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )-\sqrt {b} \left (4 a^2-12 a b+15 b^2\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+4 a b \left (-12 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {a+b \text {arccosh}(c x)}+8 c x \text {arccosh}(c x) \sqrt {a+b \text {arccosh}(c x)}+\frac {(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}+\frac {(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}\right )}{16 c} \]

input

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

output

(4*b*Sqrt[a + b*ArcCosh[c*x]]*(2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a - 
 5*b*ArcCosh[c*x]) + b*c*x*(15 + 4*ArcCosh[c*x]^2)) + (8*a^2*Sqrt[a + b*Ar 
cCosh[c*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c*x]])/Sqrt[a/b + ArcCo 
sh[c*x]] + Gamma[3/2, -((a + b*ArcCosh[c*x])/b)]/Sqrt[-((a + b*ArcCosh[c*x 
])/b)]))/E^(a/b) - Sqrt[b]*(4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a 
+ b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) - Sqrt[b]*(4*a^2 - 12*a 
*b + 15*b^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + S 
inh[a/b]) + 4*a*b*(-12*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a + b*Arc 
Cosh[c*x]] + 8*c*x*ArcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + ((2*a + 3*b)*Sq 
rt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/Sqr 
t[b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a 
/b] + Sinh[a/b]))/Sqrt[b]))/(16*c)
 

3.2.50.3 Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6294, 6330, 6294, 6368, 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.

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

\(\Big \downarrow \) 6294

\(\displaystyle x (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{2} b c \int \frac {x (a+b \text {arccosh}(c x))^{3/2}}{\sqrt {c x-1} \sqrt {c x+1}}dx\)

\(\Big \downarrow \) 6330

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

\(\Big \downarrow \) 6294

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

input

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

output

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

3.2.50.3.1 Defintions of rubi rules used

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 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 6294

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

rule 6330

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p 
_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + 
e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 
*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ 
p]   Int[(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, p}, x] && EqQ[e1, c*d1] && E 
qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
 

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]
 

3.2.50.4 Maple [F]

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

input

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

output

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

3.2.50.5 Fricas [F(-2)]

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

input

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

output

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

3.2.50.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.2.50.7 Maxima [F]

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

input

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

output

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

3.2.50.8 Giac [F(-2)]

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

input

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

output

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

3.2.50.9 Mupad [F(-1)]

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

input

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

output

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