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

Optimal. Leaf size=76 \[ -\frac{i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{c d}+\frac{i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{c d}+\frac{2 \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c d} \]

[Out]

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

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Rubi [A]  time = 0.0619147, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4658, 4183, 2279, 2391} \[ -\frac{i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{c d}+\frac{i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{c d}+\frac{2 \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0855092, size = 107, normalized size = 1.41 \[ \frac{-2 i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )+2 i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )-a \log (1-c x)+a \log (c x+1)-2 b \cos ^{-1}(c x) \log \left (1-e^{i \cos ^{-1}(c x)}\right )+2 b \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{2 c d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.135, size = 426, normalized size = 5.6 \begin{align*}{\frac{a{\it Artanh} \left ( cx \right ) }{dc}}+{\frac{b{\it Artanh} \left ( cx \right ) \arccos \left ( cx \right ) }{dc}}+{\frac{ib}{dc}{\it dilog} \left ({-i{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}-{icx{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{ib{\it Artanh} \left ( cx \right ) }{dc}\ln \left ( \left ( 1-i \right ) \cosh \left ({\frac{{\it Artanh} \left ( cx \right ) }{2}} \right ) + \left ( 1+i \right ) \sinh \left ({\frac{{\it Artanh} \left ( cx \right ) }{2}} \right ) \right ) }+{\frac{ib}{dc}\ln \left ( \left ( 1-i \right ) \cosh \left ({\frac{{\it Artanh} \left ( cx \right ) }{2}} \right ) + \left ( 1+i \right ) \sinh \left ({\frac{{\it Artanh} \left ( cx \right ) }{2}} \right ) \right ) \ln \left ({-i{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}-{icx{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{ib}{dc}{\it dilog} \left ({i{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}+{icx{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) }+{\frac{ib{\it Artanh} \left ( cx \right ) }{dc}\ln \left ( \left ( 1+i \right ) \cosh \left ({\frac{{\it Artanh} \left ( cx \right ) }{2}} \right ) + \left ( 1-i \right ) \sinh \left ({\frac{{\it Artanh} \left ( cx \right ) }{2}} \right ) \right ) }-{\frac{ib}{dc}\ln \left ( \left ( 1+i \right ) \cosh \left ({\frac{{\it Artanh} \left ( cx \right ) }{2}} \right ) + \left ( 1-i \right ) \sinh \left ({\frac{{\it Artanh} \left ( cx \right ) }{2}} \right ) \right ) \ln \left ({i{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}+{icx{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{i}{2}}b{\it Artanh} \left ( cx \right ) }{dc}\ln \left ({-i{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}-{icx{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{\frac{i}{2}}b{\it Artanh} \left ( cx \right ) }{dc}\ln \left ({i{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}+{icx{\frac{1}{\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),x)

[Out]

1/c*a/d*arctanh(c*x)+1/c*b/d*arctanh(c*x)*arccos(c*x)+I/c*b/d*dilog(-I/(-c^2*x^2+1)^(1/2)-I*c*x/(-c^2*x^2+1)^(
1/2))-I/c*b/d*ln((1-I)*cosh(1/2*arctanh(c*x))+(1+I)*sinh(1/2*arctanh(c*x)))*arctanh(c*x)+I/c*b/d*ln((1-I)*cosh
(1/2*arctanh(c*x))+(1+I)*sinh(1/2*arctanh(c*x)))*ln(-I/(-c^2*x^2+1)^(1/2)-I*c*x/(-c^2*x^2+1)^(1/2))-I/c*b/d*di
log(I/(-c^2*x^2+1)^(1/2)+I*c*x/(-c^2*x^2+1)^(1/2))+I/c*b/d*ln((1+I)*cosh(1/2*arctanh(c*x))+(1-I)*sinh(1/2*arct
anh(c*x)))*arctanh(c*x)-I/c*b/d*ln((1+I)*cosh(1/2*arctanh(c*x))+(1-I)*sinh(1/2*arctanh(c*x)))*ln(I/(-c^2*x^2+1
)^(1/2)+I*c*x/(-c^2*x^2+1)^(1/2))+1/2*I/c*b/d*arctanh(c*x)*ln(-I/(-c^2*x^2+1)^(1/2)-I*c*x/(-c^2*x^2+1)^(1/2))-
1/2*I/c*b/d*arctanh(c*x)*ln(I/(-c^2*x^2+1)^(1/2)+I*c*x/(-c^2*x^2+1)^(1/2))

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

[Out]

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

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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^{2} d x^{2} - d}, 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),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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

[Out]

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