3.1.11 \(\int \frac {x (a+b \text {ArcCos}(c x))}{(d-c^2 d x^2)^2} \, dx\) [11]

Optimal. Leaf size=57 \[ \frac {b x}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \text {ArcCos}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4768, 197} \begin {gather*} \frac {a+b \text {ArcCos}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c d^2 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 197

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

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]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 49, normalized size = 0.86 \begin {gather*} \frac {a+b c x \sqrt {1-c^2 x^2}+b \text {ArcCos}(c x)}{2 c^2 d^2-2 c^4 d^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.17, size = 98, normalized size = 1.72

method	result	size
derivativedivides	\(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 \left (c x +1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 \left (c x -1\right )}\right )}{d^{2}}}{c^{2}}\)	\(98\)
default	\(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \left (-\frac {\arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{4 \left (c x +1\right )}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{4 \left (c x -1\right )}\right )}{d^{2}}}{c^{2}}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (50) = 100\).
time = 0.48, size = 136, normalized size = 2.39 \begin {gather*} -\frac {1}{4} \, {\left ({\left (\frac {\sqrt {-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{7} d^{4} x + c^{6} d^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{7} d^{4} x - c^{6} d^{4}}\right )} c^{2} + \frac {2 \, \arccos \left (c x\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} b - \frac {a}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((sqrt(-c^2*x^2 + 1)*c^2*d^2/(c^7*d^4*x + c^6*d^4) + sqrt(-c^2*x^2 + 1)*c^2*d^2/(c^7*d^4*x - c^6*d^4))*c^
2 + 2*arccos(c*x)/(c^4*d^2*x^2 - c^2*d^2))*b - 1/2*a/(c^4*d^2*x^2 - c^2*d^2)

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Fricas [A]
time = 2.68, size = 54, normalized size = 0.95 \begin {gather*} -\frac {a c^{2} x^{2} + \sqrt {-c^{2} x^{2} + 1} b c x + b \arccos \left (c x\right )}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a x}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x \operatorname {acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a*x/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*x*acos(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1), x))/d*
*2

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Giac [A]
time = 0.44, size = 100, normalized size = 1.75 \begin {gather*} -\frac {b x^{2} \arccos \left (c x\right )}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac {a x^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac {\sqrt {-c^{2} x^{2} + 1} b x}{2 \, {\left (c^{2} x^{2} - 1\right )} c d^{2}} + \frac {b \arccos \left (c x\right )}{2 \, c^{2} d^{2}} + \frac {a}{2 \, c^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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