3.12 \(\int \frac{a+b \cos ^{-1}(c x)}{(d-c^2 d x^2)^2} \, dx\)

Optimal. Leaf size=132 \[ -\frac{i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac{i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac{x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{\tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c d^2}+\frac{b}{2 c d^2 \sqrt{1-c^2 x^2}} \]

[Out]

b/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcCos[c*x]))/(2*d^2*(1 - c^2*x^2)) + ((a + b*ArcCos[c*x])*ArcTanh[E
^(I*ArcCos[c*x])])/(c*d^2) - ((I/2)*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c*d^2) + ((I/2)*b*PolyLog[2, E^(I*ArcCo
s[c*x])])/(c*d^2)

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Rubi [A]  time = 0.0915474, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4656, 4658, 4183, 2279, 2391, 261} \[ -\frac{i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac{i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac{x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{\tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c d^2}+\frac{b}{2 c d^2 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

b/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcCos[c*x]))/(2*d^2*(1 - c^2*x^2)) + ((a + b*ArcCos[c*x])*ArcTanh[E
^(I*ArcCos[c*x])])/(c*d^2) - ((I/2)*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c*d^2) + ((I/2)*b*PolyLog[2, E^(I*ArcCo
s[c*x])])/(c*d^2)

Rule 4656

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

Rule 4658

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(c*d)^(-1), Subst[Int[(
a + b*x)^n*Csc[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 261

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

Rubi steps

\begin{align*} \int \frac{a+b \cos ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{(b c) \int \frac{x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac{\int \frac{a+b \cos ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 d}\\ &=\frac{b}{2 c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{\operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\cos ^{-1}(c x)\right )}{2 c d^2}\\ &=\frac{b}{2 c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{\left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c d^2}+\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 c d^2}-\frac{b \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 c d^2}\\ &=\frac{b}{2 c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{\left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c d^2}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}\\ &=\frac{b}{2 c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{\left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c d^2}-\frac{i b \text{Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac{i b \text{Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}\\ \end{align*}

Mathematica [A]  time = 0.280222, size = 220, normalized size = 1.67 \[ \frac{-\frac{2 i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{c}+\frac{2 i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{c}-\frac{2 a x}{c^2 x^2-1}-\frac{a \log (1-c x)}{c}+\frac{a \log (c x+1)}{c}+\frac{b \sqrt{1-c^2 x^2}}{c-c^2 x}+\frac{b \sqrt{1-c^2 x^2}}{c^2 x+c}+\frac{b \cos ^{-1}(c x)}{c-c^2 x}-\frac{b \cos ^{-1}(c x)}{c^2 x+c}-\frac{2 b \cos ^{-1}(c x) \log \left (1-e^{i \cos ^{-1}(c x)}\right )}{c}+\frac{2 b \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{c}}{4 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((b*Sqrt[1 - c^2*x^2])/(c - c^2*x) + (b*Sqrt[1 - c^2*x^2])/(c + c^2*x) - (2*a*x)/(-1 + c^2*x^2) + (b*ArcCos[c*
x])/(c - c^2*x) - (b*ArcCos[c*x])/(c + c^2*x) - (2*b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])])/c + (2*b*ArcCos[c
*x]*Log[1 + E^(I*ArcCos[c*x])])/c - (a*Log[1 - c*x])/c + (a*Log[1 + c*x])/c - ((2*I)*b*PolyLog[2, -E^(I*ArcCos
[c*x])])/c + ((2*I)*b*PolyLog[2, E^(I*ArcCos[c*x])])/c)/(4*d^2)

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Maple [A]  time = 0.146, size = 250, normalized size = 1.9 \begin{align*} -{\frac{a}{4\,c{d}^{2} \left ( cx-1 \right ) }}-{\frac{a\ln \left ( cx-1 \right ) }{4\,c{d}^{2}}}-{\frac{a}{4\,c{d}^{2} \left ( cx+1 \right ) }}+{\frac{a\ln \left ( cx+1 \right ) }{4\,c{d}^{2}}}-{\frac{b\arccos \left ( cx \right ) x}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b}{2\,c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b\arccos \left ( cx \right ) }{2\,c{d}^{2}}\ln \left ( 1+cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{\frac{i}{2}}b}{c{d}^{2}}{\it polylog} \left ( 2,-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{b\arccos \left ( cx \right ) }{2\,c{d}^{2}}\ln \left ( 1-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{{\frac{i}{2}}b}{c{d}^{2}}{\it polylog} \left ( 2,cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/c*a/d^2/(c*x-1)-1/4/c*a/d^2*ln(c*x-1)-1/4/c*a/d^2/(c*x+1)+1/4/c*a/d^2*ln(c*x+1)-1/2*b/d^2/(c^2*x^2-1)*arc
cos(c*x)*x-1/2/c*b/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+1/2/c*b/d^2*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-1
/2*I*b*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))/c/d^2-1/2/c*b/d^2*arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))+1/2*I
*b*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))/c/d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, a{\left (\frac{2 \, x}{c^{2} d^{2} x^{2} - d^{2}} - \frac{\log \left (c x + 1\right )}{c d^{2}} + \frac{\log \left (c x - 1\right )}{c d^{2}}\right )} - \frac{{\left ({\left (2 \, c x -{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) +{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) -{\left (c^{3} d^{2} x^{2} - c d^{2}\right )} \int \frac{{\left (2 \, c x -{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) +{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}\,{d x}\right )} b}{4 \,{\left (c^{3} d^{2} x^{2} - c d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a*(2*x/(c^2*d^2*x^2 - d^2) - log(c*x + 1)/(c*d^2) + log(c*x - 1)/(c*d^2)) - 1/4*((2*c*x - (c^2*x^2 - 1)*l
og(c*x + 1) + (c^2*x^2 - 1)*log(-c*x + 1))*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + 4*(c^3*d^2*x^2 - c*d^2
)*integrate(-1/4*(2*c*x - (c^2*x^2 - 1)*log(c*x + 1) + (c^2*x^2 - 1)*log(-c*x + 1))*sqrt(c*x + 1)*sqrt(-c*x +
1)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x))*b/(c^3*d^2*x^2 - c*d^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \arccos \left (c x\right ) + a}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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