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

Optimal. Leaf size=177 \[ -\frac{3 i b c \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac{3 i b c \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac{3 c^2 x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cos ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac{3 c \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d^2}+\frac{b c}{2 d^2 \sqrt{1-c^2 x^2}}+\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2} \]

[Out]

(b*c)/(2*d^2*Sqrt[1 - c^2*x^2]) - (a + b*ArcCos[c*x])/(d^2*x*(1 - c^2*x^2)) + (3*c^2*x*(a + b*ArcCos[c*x]))/(2
*d^2*(1 - c^2*x^2)) + (3*c*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/d^2 + (b*c*ArcTanh[Sqrt[1 - c^2*x^2
]])/d^2 - (((3*I)/2)*b*c*PolyLog[2, -E^(I*ArcCos[c*x])])/d^2 + (((3*I)/2)*b*c*PolyLog[2, E^(I*ArcCos[c*x])])/d
^2

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Rubi [A]  time = 0.181843, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {4702, 4656, 4658, 4183, 2279, 2391, 261, 266, 51, 63, 208} \[ -\frac{3 i b c \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac{3 i b c \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac{3 c^2 x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{a+b \cos ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac{3 c \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d^2}+\frac{b c}{2 d^2 \sqrt{1-c^2 x^2}}+\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c)/(2*d^2*Sqrt[1 - c^2*x^2]) - (a + b*ArcCos[c*x])/(d^2*x*(1 - c^2*x^2)) + (3*c^2*x*(a + b*ArcCos[c*x]))/(2
*d^2*(1 - c^2*x^2)) + (3*c*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/d^2 + (b*c*ArcTanh[Sqrt[1 - c^2*x^2
]])/d^2 - (((3*I)/2)*b*c*PolyLog[2, -E^(I*ArcCos[c*x])])/d^2 + (((3*I)/2)*b*c*PolyLog[2, E^(I*ArcCos[c*x])])/d
^2

Rule 4702

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(d*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m
+ 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^F
racPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[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] && LtQ[m, -1] && Inte
gerQ[m]

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]

Rule 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.464617, size = 251, normalized size = 1.42 \[ \frac{-6 i b c \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )+6 i b c \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )-\frac{2 a c^2 x}{c^2 x^2-1}-3 a c \log (1-c x)+3 a c \log (c x+1)-\frac{4 a}{x}+\frac{b c \sqrt{1-c^2 x^2}}{1-c x}+\frac{b c \sqrt{1-c^2 x^2}}{c x+1}+4 b c \log \left (\sqrt{1-c^2 x^2}+1\right )-4 b c \log (x)+\frac{b c \cos ^{-1}(c x)}{1-c x}-\frac{b c \cos ^{-1}(c x)}{c x+1}-\frac{4 b \cos ^{-1}(c x)}{x}-6 b c \cos ^{-1}(c x) \log \left (1-e^{i \cos ^{-1}(c x)}\right )+6 b c \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{4 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-4*a)/x + (b*c*Sqrt[1 - c^2*x^2])/(1 - c*x) + (b*c*Sqrt[1 - c^2*x^2])/(1 + c*x) - (2*a*c^2*x)/(-1 + c^2*x^2)
- (4*b*ArcCos[c*x])/x + (b*c*ArcCos[c*x])/(1 - c*x) - (b*c*ArcCos[c*x])/(1 + c*x) - 6*b*c*ArcCos[c*x]*Log[1 -
E^(I*ArcCos[c*x])] + 6*b*c*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] - 4*b*c*Log[x] - 3*a*c*Log[1 - c*x] + 3*a*c
*Log[1 + c*x] + 4*b*c*Log[1 + Sqrt[1 - c^2*x^2]] - (6*I)*b*c*PolyLog[2, -E^(I*ArcCos[c*x])] + (6*I)*b*c*PolyLo
g[2, E^(I*ArcCos[c*x])])/(4*d^2)

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Maple [A]  time = 0.223, size = 260, normalized size = 1.5 \begin{align*} -{\frac{ca}{4\,{d}^{2} \left ( cx-1 \right ) }}-{\frac{3\,ca\ln \left ( cx-1 \right ) }{4\,{d}^{2}}}-{\frac{ca}{4\,{d}^{2} \left ( cx+1 \right ) }}+{\frac{3\,ca\ln \left ( cx+1 \right ) }{4\,{d}^{2}}}-{\frac{a}{{d}^{2}x}}-{\frac{3\,b\arccos \left ( cx \right ){c}^{2}x}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bc}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b\arccos \left ( cx \right ) }{{d}^{2}x \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{2\,icb}{{d}^{2}}\arctan \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{\frac{3\,i}{2}}cb}{{d}^{2}}{\it dilog} \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{\frac{3\,i}{2}}cb}{{d}^{2}}{\it dilog} \left ( 1+cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{3\,bc\arccos \left ( cx \right ) }{2\,{d}^{2}}\ln \left ( 1+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))/x^2/(-c^2*d*x^2+d)^2,x)

[Out]

-1/4*c*a/d^2/(c*x-1)-3/4*c*a/d^2*ln(c*x-1)-1/4*c*a/d^2/(c*x+1)+3/4*c*a/d^2*ln(c*x+1)-a/d^2/x-3/2*b/d^2/(c^2*x^
2-1)*arccos(c*x)*c^2*x-1/2*c*b/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+b/d^2/x/(c^2*x^2-1)*arccos(c*x)-2*I*c*b/d^2*
arctan(c*x+I*(-c^2*x^2+1)^(1/2))-3/2*I*c*b/d^2*dilog(c*x+I*(-c^2*x^2+1)^(1/2))-3/2*I*c*b/d^2*dilog(1+c*x+I*(-c
^2*x^2+1)^(1/2))+3/2*c*b/d^2*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, a{\left (\frac{2 \,{\left (3 \, c^{2} x^{2} - 2\right )}}{c^{2} d^{2} x^{3} - d^{2} x} - \frac{3 \, c \log \left (c x + 1\right )}{d^{2}} + \frac{3 \, c \log \left (c x - 1\right )}{d^{2}}\right )} - \frac{{\left ({\left (6 \, c^{2} x^{2} - 3 \,{\left (c^{3} x^{3} - c x\right )} \log \left (c x + 1\right ) + 3 \,{\left (c^{3} x^{3} - c x\right )} \log \left (-c x + 1\right ) - 4\right )} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) -{\left (c^{2} d^{2} x^{3} - d^{2} x\right )} \int \frac{{\left (6 \, c^{3} x^{2} - 3 \,{\left (c^{4} x^{3} - c^{2} x\right )} \log \left (c x + 1\right ) + 3 \,{\left (c^{4} x^{3} - c^{2} x\right )} \log \left (-c x + 1\right ) - 4 \, c\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x}\,{d x}\right )} b}{4 \,{\left (c^{2} d^{2} x^{3} - d^{2} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a*(2*(3*c^2*x^2 - 2)/(c^2*d^2*x^3 - d^2*x) - 3*c*log(c*x + 1)/d^2 + 3*c*log(c*x - 1)/d^2) - 1/4*((6*c^2*x
^2 - 3*(c^3*x^3 - c*x)*log(c*x + 1) + 3*(c^3*x^3 - c*x)*log(-c*x + 1) - 4)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1
), c*x) + 4*(c^2*d^2*x^3 - d^2*x)*integrate(-1/4*(6*c^3*x^2 - 3*(c^4*x^3 - c^2*x)*log(c*x + 1) + 3*(c^4*x^3 -
c^2*x)*log(-c*x + 1) - 4*c)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x), x))*b/(c^2*d^2
*x^3 - d^2*x)

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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^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arccos(c*x) + a)/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac{b \operatorname{acos}{\left (c x \right )}}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a/(c**4*x**6 - 2*c**2*x**4 + x**2), x) + Integral(b*acos(c*x)/(c**4*x**6 - 2*c**2*x**4 + x**2), x))/
d**2

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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} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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