3.2.17 \(\int \frac {x^2 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [117]

3.2.17.1 Optimal result

Integrand size = 27, antiderivative size = 143 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c^3 d \sqrt {d-c^2 d x^2}} \]

output

x*(a+b*arccosh(c*x))/c^2/d/(-c^2*d*x^2+d)^(1/2)-1/2*(a+b*arccosh(c*x))^2*( 
c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c^3/d/(-c^2*d*x^2+d)^(1/2)-1/2*b*ln(-c^2*x^2+ 
1)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)
 

3.2.17.2 Mathematica [A] (warning: unable to verify)

Time = 0.61 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.11 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 a c d x+2 a \sqrt {d} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+b d \left (2 c x \text {arccosh}(c x)-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\text {arccosh}(c x)^2+2 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )}{2 c^3 d^2 \sqrt {d-c^2 d x^2}} \]

input

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

output

(2*a*c*d*x + 2*a*Sqrt[d]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^ 
2])/(Sqrt[d]*(-1 + c^2*x^2))] + b*d*(2*c*x*ArcCosh[c*x] - Sqrt[(-1 + c*x)/ 
(1 + c*x)]*(1 + c*x)*(ArcCosh[c*x]^2 + 2*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 
 + c*x)])))/(2*c^3*d^2*Sqrt[d - c^2*d*x^2])
 

3.2.17.3 Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6349, 25, 82, 240, 6307}

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.

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

\(\Big \downarrow \) 6349

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 6307

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

input

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

output

(x*(a + b*ArcCosh[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2]) - (Sqrt[-1 + c*x]*Sqr 
t[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) - (b*Sq 
rt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(2*c^3*d*Sqrt[d - c^2*d*x^2])
 

3.2.17.3.1 Defintions of rubi rules used

rule 25

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

rule 82

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) 
)^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, 
 e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
 

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 6307

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d 
 + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x 
] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
 

rule 6349

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

3.2.17.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(278\) vs. \(2(125)=250\).

Time = 1.07 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.95

method	result	size
default	\(\frac {a x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{2 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\)	\(279\)
parts	\(\frac {a x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{2 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\)	\(279\)

input

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

output

a*x/c^2/d/(-c^2*d*x^2+d)^(1/2)-a/c^2/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)* 
x/(-c^2*d*x^2+d)^(1/2))+1/2*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1) 
^(1/2)/d^2/c^3/(c^2*x^2-1)*arccosh(c*x)^2-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1) 
^(1/2)*(c*x+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*arccosh(c*x)-b*(-d*(c^2*x^2-1))^( 
1/2)*arccosh(c*x)/d^2/c^2/(c^2*x^2-1)*x+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^( 
1/2)*(c*x+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2) 
)^2-1)
 

3.2.17.5 Fricas [F]

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

input

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

output

integral(sqrt(-c^2*d*x^2 + d)*(b*x^2*arccosh(c*x) + a*x^2)/(c^4*d^2*x^4 - 
2*c^2*d^2*x^2 + d^2), x)
 

3.2.17.6 Sympy [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

input

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

output

Integral(x**2*(a + b*acosh(c*x))/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)
 

3.2.17.7 Maxima [F]

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

input

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

output

a*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + b*integra 
te(x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2), x)
 

3.2.17.8 Giac [F]

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

input

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

output

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

3.2.17.9 Mupad [F(-1)]

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

input

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

output

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