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

Optimal. Leaf size=122 \[ \frac {a+b \cos ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d^2}+\frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}-\frac {i b \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac {i b \text {Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )}{2 d^2} \]

[Out]

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

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Rubi [A]  time = 0.17, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4706, 4680, 4419, 4183, 2279, 2391, 191} \[ -\frac {i b \text {PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac {i b \text {PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac {a+b \cos ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d^2}+\frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 191

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 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 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 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 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 4706

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)/(2*d*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d*(p +
1)), Int[(f*x)^m*(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)^Frac
Part[p])/(2*f*(p + 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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ
[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])

Rubi steps

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

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Mathematica [A]  time = 0.52, size = 152, normalized size = 1.25 \[ \frac {\frac {a}{1-c^2 x^2}-a \log \left (1-c^2 x^2\right )+2 a \log (x)+b \left (\frac {c x}{\sqrt {1-c^2 x^2}}+\frac {\cos ^{-1}(c x)}{1-c^2 x^2}-i \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )+i \text {Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )-2 \cos ^{-1}(c x) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )+2 \cos ^{-1}(c x) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )\right )}{2 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arccos \left (c x\right ) + a}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arccos(c*x) + a)/(c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))/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), x)

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maple [B]  time = 0.75, size = 340, normalized size = 2.79 \[ \frac {a}{4 d^{2} \left (c x +1\right )}-\frac {a \ln \left (c x +1\right )}{2 d^{2}}+\frac {a \ln \left (c x \right )}{d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{2 d^{2}}-\frac {i b \,c^{2} x^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b c x \sqrt {-c^{2} x^{2}+1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \arccos \left (c x \right )}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {i b}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {i b \polylog \left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {i b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d^{2}}-\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {i b \polylog \left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, a {\left (\frac {1}{c^{2} d^{2} x^{2} - d^{2}} + \frac {\log \left (c x + 1\right )}{d^{2}} + \frac {\log \left (c x - 1\right )}{d^{2}} - \frac {2 \, \log \relax (x)}{d^{2}}\right )} + b \int \frac {\arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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