\(\int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [198]

Optimal result

Integrand size = 27, antiderivative size = 155 \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {2 a b x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {2 b^2 (1-c x) (1+c x)}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5914, 5879, 75} \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 a b x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{c \sqrt {d-c^2 d x^2}}-\frac {2 b^2 (1-c x) (c x+1)}{c^2 \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Rule 75

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

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5914

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

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

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.96 \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (2 a b c x \sqrt {-1+c x} \sqrt {1+c x}+a^2 \left (1-c^2 x^2\right )-2 b^2 \left (-1+c^2 x^2\right )+2 b \left (a-a c^2 x^2+b c x \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {arccosh}(c x)+b^2 \left (1-c^2 x^2\right ) \text {arccosh}(c x)^2\right )}{c^2 d (-1+c x) (1+c x)} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs. \(2(139)=278\).

Time = 0.59 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.03

method	result	size
default	\(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\)	\(314\)
parts	\(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\)	\(314\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.41 \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} a b c x - {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 2 \, {\left (\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b^{2} c x - {\left (a b c^{2} x^{2} - a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left ({\left (a^{2} + 2 \, b^{2}\right )} c^{2} x^{2} - a^{2} - 2 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d x^{2} - c^{2} d} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.94 \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=2 \, b^{2} {\left (\frac {\sqrt {-d} x \operatorname {arcosh}\left (c x\right )}{c d} - \frac {\sqrt {c^{2} x^{2} - 1} \sqrt {-d}}{c^{2} d}\right )} + \frac {2 \, a b \sqrt {-d} x}{c d} - \frac {\sqrt {-c^{2} d x^{2} + d} b^{2} \operatorname {arcosh}\left (c x\right )^{2}}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} + d} a b \operatorname {arcosh}\left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a^{2}}{c^{2} d} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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