3.67 \(\int x \sqrt {d-c^2 d x^2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=118 \[ -\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d}+\frac {b x \sqrt {d-c^2 d x^2}}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/c^2/d+1/3*b*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-
1/9*b*c*x^3*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.21, antiderivative size = 126, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {5798, 5718} \[ -\frac {(1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac {b c x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \sqrt {d-c^2 d x^2}}{3 c \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) - ((1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*c^2)

Rule 5718

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] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(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, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rubi steps

\begin {align*} \int x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\sqrt {d-c^2 d x^2} \int x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {(1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{3 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b x \sqrt {d-c^2 d x^2}}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {(1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 98, normalized size = 0.83 \[ \frac {\sqrt {d-c^2 d x^2} \left (3 a \left (c^2 x^2-1\right )^2+b c x \sqrt {c x-1} \sqrt {c x+1} \left (3-c^2 x^2\right )+3 b \left (c^2 x^2-1\right )^2 \cosh ^{-1}(c x)\right )}{9 c^2 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - c^2*x^2) + 3*a*(-1 + c^2*x^2)^2 + 3*b*(-1 + c^2*
x^2)^2*ArcCosh[c*x]))/(9*c^2*(-1 + c^2*x^2))

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 142, normalized size = 1.20 \[ \frac {3 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{3} x^{3} - 3 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 3 \, {\left (a c^{4} x^{4} - 2 \, a c^{2} x^{2} + a\right )} \sqrt {-c^{2} d x^{2} + d}}{9 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

1/9*(3*(b*c^4*x^4 - 2*b*c^2*x^2 + b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^3*x^3 - 3*b*c*x)
*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 3*(a*c^4*x^4 - 2*a*c^2*x^2 + a)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2
)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [B]  time = 0.29, size = 356, normalized size = 3.02 \[ -\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +1\right ) \left (-1+3 \,\mathrm {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (-1+\mathrm {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (1+\mathrm {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -5 c^{2} x^{2}+1\right ) \left (1+3 \,\mathrm {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2),x)

[Out]

-1/3*a/c^2/d*(-c^2*d*x^2+d)^(3/2)+b*(1/72*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x+1)^(1/2)*(c*x-1)^
(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+3*arccosh(c*x))/(c*x+1)/c^2/(c*x-1)-1/8*(-d*(c^2*x^2-1)
)^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(-1+arccosh(c*x))/(c*x+1)/c^2/(c*x-1)-1/8*(-d*(c^2*x^2-1))
^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(1+arccosh(c*x))/(c*x+1)/c^2/(c*x-1)+1/72*(-d*(c^2*x^2-1))
^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(1+3*a
rccosh(c*x))/(c*x+1)/c^2/(c*x-1))

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 81, normalized size = 0.69 \[ -\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b \operatorname {arcosh}\left (c x\right )}{3 \, c^{2} d} - \frac {{\left (c^{2} \sqrt {-d} d x^{3} - 3 \, \sqrt {-d} d x\right )} b}{9 \, c d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

-1/3*(-c^2*d*x^2 + d)^(3/2)*b*arccosh(c*x)/(c^2*d) - 1/9*(c^2*sqrt(-d)*d*x^3 - 3*sqrt(-d)*d*x)*b/(c*d) - 1/3*(
-c^2*d*x^2 + d)^(3/2)*a/(c^2*d)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(1/2),x)

[Out]

int(x*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x)), x)

________________________________________________________________________________________