3.2.73 \(\int \sqrt {d-c^2 d x^2} (a+b \cosh ^{-1}(c x))^2 \, dx\) [173]

Optimal. Leaf size=204 \[ \frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/4*b^2*x*(-c^2*d*x^2+d)^(1/2)+1/2*x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)+1/4*b^2*arccosh(c*x)*(-c^2*d*x^
2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*b*c*x^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*
x+1)^(1/2)-1/6*(a+b*arccosh(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]
time = 0.15, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5895, 5893, 5883, 92, 54} \begin {gather*} -\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 c \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*x*Sqrt[d - c^2*d*x^2])/4 + (b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(4*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b
*c*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcCosh[c*x])^2)/2 - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(6*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 5883

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

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5895

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

Rubi steps

\begin {align*} \int \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 0.71, size = 235, normalized size = 1.15 \begin {gather*} \frac {1}{24} \left (12 a^2 x \sqrt {d-c^2 d x^2}-\frac {12 a^2 \sqrt {d} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{c}-\frac {6 a b \sqrt {d-c^2 d x^2} \left (2 \cosh ^{-1}(c x)^2+\cosh \left (2 \cosh ^{-1}(c x)\right )-2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {b^2 \sqrt {d-c^2 d x^2} \left (-4 \cosh ^{-1}(c x)^3-6 \cosh ^{-1}(c x) \cosh \left (2 \cosh ^{-1}(c x)\right )+\left (3+6 \cosh ^{-1}(c x)^2\right ) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(12*a^2*x*Sqrt[d - c^2*d*x^2] - (12*a^2*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/c
- (6*a*b*Sqrt[d - c^2*d*x^2]*(2*ArcCosh[c*x]^2 + Cosh[2*ArcCosh[c*x]] - 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]]))/
(c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b^2*Sqrt[d - c^2*d*x^2]*(-4*ArcCosh[c*x]^3 - 6*ArcCosh[c*x]*Cosh[2
*ArcCosh[c*x]] + (3 + 6*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c*x]]))/(c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/24

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(526\) vs. \(2(172)=344\).
time = 1.97, size = 527, normalized size = 2.58

method	result	size
default	\(\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{3}}{6 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (2 \mathrm {arccosh}\left (c x \right )^{2}-2 \,\mathrm {arccosh}\left (c x \right )+1\right )}{16 \left (c x +1\right ) \left (c x -1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (2 \mathrm {arccosh}\left (c x \right )^{2}+2 \,\mathrm {arccosh}\left (c x \right )+1\right )}{16 \left (c x +1\right ) \left (c x -1\right ) c}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\mathrm {arccosh}\left (c x \right )\right )}{16 \left (c x +1\right ) \left (c x -1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\mathrm {arccosh}\left (c x \right )\right )}{16 \left (c x +1\right ) \left (c x -1\right ) c}\right )\)	\(527\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*x*(-c^2*d*x^2+d)^(1/2)+1/2*a^2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-1/6*
(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^3+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c
*x+2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(2*arccosh(c*x)^2-2*arccosh(c*x)+1)/(c*x
+1)/(c*x-1)/c+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x
+1)^(1/2)-2*c*x)*(2*arccosh(c*x)^2+2*arccosh(c*x)+1)/(c*x+1)/(c*x-1)/c)+2*a*b*(-1/4*(-d*(c^2*x^2-1))^(1/2)/(c*
x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x+1)^(1/2)*(c*x-1)
^(1/2)*x^2*c^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+2*arccosh(c*x))/(c*x+1)/(c*x-1)/c+1/16*(-d*(c^2*x^2-1))^(1/2)*
(-2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*(1+2*arccosh(c*x))/(c*x+1
)/(c*x-1)/c)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2*(-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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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