3.7 \(\int \frac{a+b \cos ^{-1}(c x)}{x^3 (d-c^2 d x^2)} \, dx\)

Optimal. Leaf size=124 \[ -\frac{i b c^2 \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac{i b c^2 \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac{2 c^2 \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d}-\frac{a+b \cos ^{-1}(c x)}{2 d x^2}+\frac{b c \sqrt{1-c^2 x^2}}{2 d x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.185964, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {4702, 4680, 4419, 4183, 2279, 2391, 264} \[ -\frac{i b c^2 \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac{i b c^2 \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac{2 c^2 \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d}-\frac{a+b \cos ^{-1}(c x)}{2 d x^2}+\frac{b c \sqrt{1-c^2 x^2}}{2 d x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])/(x^3*(d - c^2*d*x^2)),x]

[Out]

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

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 4680

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

Rule 4419

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]

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 264

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

Rubi steps

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

Mathematica [A]  time = 0.42693, size = 150, normalized size = 1.21 \[ -\frac{b c^2 \left (i \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )-i \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )-\frac{\sqrt{1-c^2 x^2}}{c x}+\frac{\cos ^{-1}(c x)}{c^2 x^2}+2 \cos ^{-1}(c x) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )-2 \cos ^{-1}(c x) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )\right )+a c^2 \log \left (1-c^2 x^2\right )-2 a c^2 \log (x)+\frac{a}{x^2}}{2 d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCos[c*x])/(x^3*(d - c^2*d*x^2)),x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.177, size = 301, normalized size = 2.4 \begin{align*} -{\frac{{c}^{2}a\ln \left ( cx-1 \right ) }{2\,d}}-{\frac{{c}^{2}a\ln \left ( cx+1 \right ) }{2\,d}}-{\frac{a}{2\,d{x}^{2}}}+{\frac{{c}^{2}a\ln \left ( cx \right ) }{d}}+{\frac{{\frac{i}{2}}{c}^{2}b}{d}}+{\frac{bc}{2\,dx}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arccos \left ( cx \right ) }{2\,d{x}^{2}}}-{\frac{{c}^{2}b\arccos \left ( cx \right ) }{d}\ln \left ( 1+cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{i{c}^{2}b}{d}{\it polylog} \left ( 2,-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{c}^{2}b\arccos \left ( cx \right ) }{d}\ln \left ( 1-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{i{c}^{2}b}{d}{\it polylog} \left ( 2,cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{{c}^{2}b\arccos \left ( cx \right ) }{d}\ln \left ( 1+ \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }-{\frac{{\frac{i}{2}}{c}^{2}b}{d}{\it polylog} \left ( 2,- \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(c*x))/x^3/(-c^2*d*x^2+d),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/x^3/(-c^2*d*x^2+d),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b \arccos \left (c x\right ) + a}{c^{2} d x^{5} - d x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/x^3/(-c^2*d*x^2+d),x, algorithm="fricas")

[Out]

integral(-(b*arccos(c*x) + a)/(c^2*d*x^5 - d*x^3), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a}{c^{2} x^{5} - x^{3}}\, dx + \int \frac{b \operatorname{acos}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(c*x))/x**3/(-c**2*d*x**2+d),x)

[Out]

-(Integral(a/(c**2*x**5 - x**3), x) + Integral(b*acos(c*x)/(c**2*x**5 - x**3), x))/d

________________________________________________________________________________________

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 )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/x^3/(-c^2*d*x^2+d),x, algorithm="giac")

[Out]

integrate(-(b*arccos(c*x) + a)/((c^2*d*x^2 - d)*x^3), x)