3.1.15 \(\int \frac {(f+g x)^2 (a+b \text {ArcCos}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\) [15]

Optimal. Leaf size=270 \[ -\frac {2 b f g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) (a+b \text {ArcCos}(c x))}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) (a+b \text {ArcCos}(c x))}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {f^2 \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}} \]

[Out]

-2*f*g*(-c^2*x^2+1)*(a+b*arccos(c*x))/c^2/(-c^2*d*x^2+d)^(1/2)-1/2*g^2*x*(-c^2*x^2+1)*(a+b*arccos(c*x))/c^2/(-
c^2*d*x^2+d)^(1/2)-2*b*f*g*x*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-1/4*b*g^2*x^2*(-c^2*x^2+1)^(1/2)/c/(-c^
2*d*x^2+d)^(1/2)-1/2*f^2*(a+b*arccos(c*x))^2*(-c^2*x^2+1)^(1/2)/b/c/(-c^2*d*x^2+d)^(1/2)-1/4*g^2*(a+b*arccos(c
*x))^2*(-c^2*x^2+1)^(1/2)/b/c^3/(-c^2*d*x^2+d)^(1/2)

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Rubi [A]
time = 0.29, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4862, 4848, 4738, 4768, 8, 4796, 30} \begin {gather*} -\frac {f^2 \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) (a+b \text {ArcCos}(c x))}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) (a+b \text {ArcCos}(c x))}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b f g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*f*g*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) - (b*g^2*x^2*Sqrt[1 - c^2*x^2])/(4*c*Sqrt[d - c^2*d*x^2
]) - (2*f*g*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) - (g^2*x*(1 - c^2*x^2)*(a + b*ArcCos[
c*x]))/(2*c^2*Sqrt[d - c^2*d*x^2]) - (f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2])
- (g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4738

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

Rule 4768

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

Rule 4796

Int[((a_.) + ArcCos[(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*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 4848

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g},
x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 ||
(n == 1 && p > -1) || (m == 2 && p < -2))

Rule 4862

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

Rubi steps

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

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Mathematica [A]
time = 0.44, size = 266, normalized size = 0.99 \begin {gather*} \frac {2 b \sqrt {d} \left (2 c^2 f^2+g^2\right ) \left (-1+c^2 x^2\right ) \text {ArcCos}(c x)^2-4 a \left (2 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d} g \left (-1+c^2 x^2\right ) \left (4 c \left (4 b c f x+a (4 f+g x) \sqrt {1-c^2 x^2}\right )+b g \cos (2 \text {ArcCos}(c x))\right )+2 b \sqrt {d} g \left (-1+c^2 x^2\right ) \text {ArcCos}(c x) \left (8 c f \sqrt {1-c^2 x^2}+g \sin (2 \text {ArcCos}(c x))\right )}{8 c^3 \sqrt {d} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*Sqrt[d]*(2*c^2*f^2 + g^2)*(-1 + c^2*x^2)*ArcCos[c*x]^2 - 4*a*(2*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*Sqrt[d -
c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d]*g*(-1 + c^2*x^2)*(4*c*(4*b*c*
f*x + a*(4*f + g*x)*Sqrt[1 - c^2*x^2]) + b*g*Cos[2*ArcCos[c*x]]) + 2*b*Sqrt[d]*g*(-1 + c^2*x^2)*ArcCos[c*x]*(8
*c*f*Sqrt[1 - c^2*x^2] + g*Sin[2*ArcCos[c*x]]))/(8*c^3*Sqrt[d]*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])

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Maple [C] Result contains complex when optimal does not.
time = 0.95, size = 507, normalized size = 1.88

method	result	size
default	\(-\frac {a \,g^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a \,g^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}-\frac {2 a f g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+\frac {a \,f^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} \left (2 c^{2} f^{2}+g^{2}\right )}{4 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) f g \left (\arccos \left (c x \right )+i\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) f g \left (\arccos \left (c x \right )-i\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) g^{2} x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2}}{16 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) g^{2} \cos \left (3 \arccos \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \sin \left (3 \arccos \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\)	\(507\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*g^2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2*a*g^2/c^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2
))-2*a*f*g/c^2/d*(-c^2*d*x^2+d)^(1/2)+a*f^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(1/4*
(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arccos(c*x)^2*(2*c^2*f^2+g^2)-(-d*(c^2*x^2-1))^(1/
2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*g*(arccos(c*x)+I)/c^2/d/(c^2*x^2-1)-(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-
I*(-c^2*x^2+1)^(1/2)*x*c-1)*f*g*(arccos(c*x)-I)/c^2/d/(c^2*x^2-1)+1/8*(-d*(c^2*x^2-1))^(1/2)/c^2/d/(c^2*x^2-1)
*arccos(c*x)*g^2*x-1/16*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*g^2-1/8*(-d*(c^2*x^2-1))^(
1/2)/c^3/d/(c^2*x^2-1)*arccos(c*x)*g^2*cos(3*arccos(c*x))+1/16*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*g^2*si
n(3*arccos(c*x)))

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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((g*x+f)^2*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

-1/2*a*g^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt(d))) + b*f^2*arccos(c*x)*arcsin(c*x)/(c*sqr
t(d)) + 1/2*b*f^2*arcsin(c*x)^2/(c*sqrt(d)) + b*g^2*integrate(x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/(
sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d) - 2*b*f*g*x/(c*sqrt(d)) + a*f^2*arcsin(c*x)/(c*sqrt(d)) - 2*sqrt(-c^
2*d*x^2 + d)*b*f*g*arccos(c*x)/(c^2*d) - 2*sqrt(-c^2*d*x^2 + d)*a*f*g/(c^2*d)

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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((g*x+f)^2*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(a*g^2*x^2 + 2*a*f*g*x + a*f^2 + (b*g^2*x^2 + 2*b*f*g*x + b*f^2)*arccos(c*x))/(
c^2*d*x^2 - d), x)

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Sympy [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((g*x+f)**2*(a+b*acos(c*x))/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

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Giac [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((g*x+f)^2*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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