\(\int \frac {x^3 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [116]

Optimal result

Integrand size = 27, antiderivative size = 150 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b x \sqrt {d-c^2 d x^2}}{c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^4 d^2}-\frac {b \sqrt {d-c^2 d x^2} \text {arctanh}(c x)}{c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

(a+b*arccosh(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4/d^2-b*x*(-c^2*d*x^2+
d)^(1/2)/c^3/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*arctanh(c*x)*(-c^2*d*x^2+d)^(1/2)/c^4/d^2/(c*x-1)^(1/2)/(c*x+1)
^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {272, 45, 5922, 12, 396, 212} \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^4 d^2}+\frac {a+b \text {arccosh}(c x)}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {b \text {arctanh}(c x) \sqrt {d-c^2 d x^2}}{c^4 d^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b x \sqrt {d-c^2 d x^2}}{c^3 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

-((b*x*Sqrt[d - c^2*d*x^2])/(c^3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (a + b*ArcCosh[c*x])/(c^4*d*Sqrt[d - c^2
*d*x^2]) + (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(c^4*d^2) - (b*Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(c^4*d^
2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 5922

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 +
c*x])], Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arccosh}(c x)}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^4 d^2}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {2-c^2 x^2}{c^4 d^2 \left (1-c^2 x^2\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {a+b \text {arccosh}(c x)}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^4 d^2}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \frac {2-c^2 x^2}{1-c^2 x^2} \, dx}{c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b x \sqrt {d-c^2 d x^2}}{c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^4 d^2}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b x \sqrt {d-c^2 d x^2}}{c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^4 d^2}-\frac {b \sqrt {d-c^2 d x^2} \text {arctanh}(c x)}{c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.65 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 a-a c^2 x^2+b c x \sqrt {-1+c x} \sqrt {1+c x}+b \left (2-c^2 x^2\right ) \text {arccosh}(c x)+b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{c^4 d \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

(2*a - a*c^2*x^2 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*(2 - c^2*x^2)*ArcCosh[c*x] + b*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*ArcTanh[c*x])/(c^4*d*Sqrt[d - c^2*d*x^2])

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.70

method	result	size
default	\(a \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-c^{3} x^{3}+\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x -\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )+\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{\left (c^{2} x^{2}-1\right )^{2} d^{2} c^{4}}\)	\(255\)
parts	\(a \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-c^{3} x^{3}+\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x -\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )+\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{\left (c^{2} x^{2}-1\right )^{2} d^{2} c^{4}}\)	\(255\)

[In]

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

[Out]

a*(-x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2/d/c^4/(-c^2*d*x^2+d)^(1/2))+b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-d*(c^2*x^2-1)
)^(1/2)*((c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-c^3*x^3+ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*x^2*c^
2-ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2-2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-ln((c*x-1)^(1/2
)*(c*x+1)^(1/2)+c*x-1)+ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))/(c^2*x^2-1)^2/d^2/c^4

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.86 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\left [-\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x - 4 \, {\left (b c^{2} x^{2} - 2 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{2} x^{2} - b\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \, {\left (a c^{2} x^{2} - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{4 \, {\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )}}, -\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x + {\left (b c^{2} x^{2} - b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) - 2 \, {\left (b c^{2} x^{2} - 2 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, {\left (a c^{2} x^{2} - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{2 \, {\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )}}\right ] \]

[In]

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

[Out]

[-1/4*(4*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*b*c*x - 4*(b*c^2*x^2 - 2*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqr
t(c^2*x^2 - 1)) + (b*c^2*x^2 - b)*sqrt(-d)*log(-(c^6*d*x^6 + 5*c^4*d*x^4 - 5*c^2*d*x^2 - 4*(c^3*x^3 + c*x)*sqr
t(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*sqrt(-d) - d)/(c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - 1)) - 4*(a*c^2*x^2 - 2*a)
*sqrt(-c^2*d*x^2 + d))/(c^6*d^2*x^2 - c^4*d^2), -1/2*(2*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*b*c*x + (b*c^2*
x^2 - b)*sqrt(d)*arctan(2*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*c*sqrt(d)*x/(c^4*d*x^4 - d)) - 2*(b*c^2*x^2 -
2*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - 2*(a*c^2*x^2 - 2*a)*sqrt(-c^2*d*x^2 + d))/(c^6*d^2*x
^2 - c^4*d^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.05 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {1}{2} \, b c {\left (\frac {2 \, \sqrt {-d} x}{c^{4} d^{2}} + \frac {\sqrt {-d} \log \left (c x + 1\right )}{c^{5} d^{2}} - \frac {\sqrt {-d} \log \left (c x - 1\right )}{c^{5} d^{2}}\right )} - b {\left (\frac {x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {2}{\sqrt {-c^{2} d x^{2} + d} c^{4} d}\right )} \operatorname {arcosh}\left (c x\right ) - a {\left (\frac {x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {2}{\sqrt {-c^{2} d x^{2} + d} c^{4} d}\right )} \]

[In]

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

[Out]

-1/2*b*c*(2*sqrt(-d)*x/(c^4*d^2) + sqrt(-d)*log(c*x + 1)/(c^5*d^2) - sqrt(-d)*log(c*x - 1)/(c^5*d^2)) - b*(x^2
/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 2/(sqrt(-c^2*d*x^2 + d)*c^4*d))*arccosh(c*x) - a*(x^2/(sqrt(-c^2*d*x^2 + d)*c^
2*d) - 2/(sqrt(-c^2*d*x^2 + d)*c^4*d))

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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