3.2.49 \(\int x (a+b \text {arccosh}(c x))^{5/2} \, dx\) [149]

3.2.49.1 Optimal result

Integrand size = 14, antiderivative size = 228 \[ \int x (a+b \text {arccosh}(c x))^{5/2} \, dx=-\frac {15 b^2 \sqrt {a+b \text {arccosh}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{8 c}-\frac {(a+b \text {arccosh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{5/2}-\frac {15 b^{5/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{256 c^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{256 c^2} \]

output

-1/4*(a+b*arccosh(c*x))^(5/2)/c^2+1/2*x^2*(a+b*arccosh(c*x))^(5/2)-15/512* 
b^(5/2)*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*2^(1/2)*P 
i^(1/2)/c^2-15/512*b^(5/2)*erfi(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))* 
2^(1/2)*Pi^(1/2)/c^2/exp(2*a/b)-5/8*b*x*(a+b*arccosh(c*x))^(3/2)*(c*x-1)^( 
1/2)*(c*x+1)^(1/2)/c-15/64*b^2*(a+b*arccosh(c*x))^(1/2)/c^2+15/32*b^2*x^2* 
(a+b*arccosh(c*x))^(1/2)
 

3.2.49.2 Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.91 \[ \int x (a+b \text {arccosh}(c x))^{5/2} \, dx=\frac {-15 b^{5/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )-15 b^{5/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+8 \sqrt {a+b \text {arccosh}(c x)} \left (\left (16 a^2+15 b^2\right ) \cosh (2 \text {arccosh}(c x))+16 b^2 \text {arccosh}(c x)^2 \cosh (2 \text {arccosh}(c x))-20 a b \sinh (2 \text {arccosh}(c x))+4 b \text {arccosh}(c x) (8 a \cosh (2 \text {arccosh}(c x))-5 b \sinh (2 \text {arccosh}(c x)))\right )}{512 c^2} \]

input

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

output

(-15*b^(5/2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*( 
Cosh[(2*a)/b] - Sinh[(2*a)/b]) - 15*b^(5/2)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a 
 + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 8*Sqrt[a + 
b*ArcCosh[c*x]]*((16*a^2 + 15*b^2)*Cosh[2*ArcCosh[c*x]] + 16*b^2*ArcCosh[c 
*x]^2*Cosh[2*ArcCosh[c*x]] - 20*a*b*Sinh[2*ArcCosh[c*x]] + 4*b*ArcCosh[c*x 
]*(8*a*Cosh[2*ArcCosh[c*x]] - 5*b*Sinh[2*ArcCosh[c*x]])))/(512*c^2)
 

3.2.49.3 Rubi [A] (verified)

Time = 2.42 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {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.

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6308

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.2.49.3.1 Defintions of rubi rules used

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]
 

3.2.49.4 Maple [F]

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

input

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

output

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

3.2.49.5 Fricas [F(-2)]

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

input

integrate(x*(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)
 

3.2.49.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.2.49.7 Maxima [F]

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

input

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

output

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

3.2.49.8 Giac [F(-2)]

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

input

integrate(x*(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
 

3.2.49.9 Mupad [F(-1)]

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

input

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

output

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