3.4.71 \(\int \frac {x (d-c^2 d x^2)}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [371]

3.4.71.1 Optimal result

Integrand size = 25, antiderivative size = 241 \[ \int \frac {x \left (d-c^2 d x^2\right )}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {2 d x (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {d e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}-\frac {d e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2} \]

output

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

3.4.71.2 Mathematica [F]

\[ \int \frac {x \left (d-c^2 d x^2\right )}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {x \left (d-c^2 d x^2\right )}{(a+b \text {arccosh}(c x))^{3/2}} \, dx \]

input

Integrate[(x*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2),x]
 

output

Integrate[(x*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2), x]
 

3.4.71.3 Rubi [A] (verified)

Time = 2.93 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6357, 6322, 3042, 25, 3793, 2009, 6368, 5971, 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.

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

\(\Big \downarrow \) 6357

\(\displaystyle -\frac {8 c d \int \frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b}+\frac {2 d \int \frac {\sqrt {c x-1} \sqrt {c x+1}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}+\frac {2 d x (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 6322

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

input

Int[(x*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2),x]
 

output

(2*d*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) - 
(8*d*(-1/4*Sqrt[a + b*ArcCosh[c*x]] + (Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2 
*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/32 + (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a 
 + b*ArcCosh[c*x]])/Sqrt[b]])/(32*E^((4*a)/b))))/(b^2*c^2) + (2*d*(-Sqrt[a 
 + b*ArcCosh[c*x]] + (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + 
 b*ArcCosh[c*x]])/Sqrt[b]])/4 + (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + 
 b*ArcCosh[c*x]])/Sqrt[b]])/(4*E^((2*a)/b))))/(b^2*c^2)
 

3.4.71.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 6322

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( 
(d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d1 + e1*x)^p/ 
(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[Int[x^n*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[2*p, 0]
 

rule 6357

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c* 
x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[ 
f*(m/(b*c*(n + 1)))*Simp[(d + e*x^2)^p/((1 + c*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] - Simp[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(( 
1 + c*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, d, e, 
f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 
2*p + 1, 0] && IGtQ[m, -3]
 

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]
 

3.4.71.4 Maple [F]

\[\int \frac {x \left (-c^{2} d \,x^{2}+d \right )}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]

input

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

output

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

3.4.71.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {x \left (d-c^2 d x^2\right )}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate(x*(-c^2*d*x^2+d)/(a+b*arccosh(c*x))^(3/2),x, algorithm="fricas")
 

output

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

3.4.71.6 Sympy [F]

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

input

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

output

-d*(Integral(-x/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh 
(c*x)), x) + Integral(c**2*x**3/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*a 
cosh(c*x))*acosh(c*x)), x))
 

3.4.71.7 Maxima [F]

\[ \int \frac {x \left (d-c^2 d x^2\right )}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

input

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

output

-integrate((c^2*d*x^2 - d)*x/(b*arccosh(c*x) + a)^(3/2), x)
 

3.4.71.8 Giac [F(-2)]

Exception generated. \[ \int \frac {x \left (d-c^2 d x^2\right )}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate(x*(-c^2*d*x^2+d)/(a+b*arccosh(c*x))^(3/2),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
 

3.4.71.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (d-c^2 d x^2\right )}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {x\,\left (d-c^2\,d\,x^2\right )}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]

input

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

output

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