\(\int \frac {a+b \arccos (c x)}{x^3 (d-c^2 d x^2)} \, dx\) [7]

Optimal result

Integrand size = 25, antiderivative size = 124 \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \arccos (c x)}{2 d x^2}+\frac {2 c^2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d}-\frac {i b c^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d}+\frac {i b c^2 \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4790, 4770, 4504, 4268, 2317, 2438, 270} \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\frac {2 c^2 \text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))}{d}-\frac {a+b \arccos (c x)}{2 d x^2}-\frac {i b c^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d}+\frac {i b c^2 \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d}+\frac {b c \sqrt {1-c^2 x^2}}{2 d x} \]

[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 270

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]

Rule 2317

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 2438

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 4268

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 4504

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 4770

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 4790

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/(f*(m + 1)))*Simp[(d + e*x
^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arccos (c x)}{2 d x^2}+c^2 \int \frac {a+b \arccos (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 \arccos (c x)}{2 d x^2}-\frac {c^2 \text {Subst}(\int (a+b x) \csc (x) \sec (x) \, dx,x,\arccos (c x))}{d} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \arccos (c x)}{2 d x^2}-\frac {\left (2 c^2\right ) \text {Subst}(\int (a+b x) \csc (2 x) \, dx,x,\arccos (c x))}{d} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \arccos (c x)}{2 d x^2}+\frac {2 c^2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d}+\frac {\left (b c^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arccos (c x)\right )}{d}-\frac {\left (b c^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (c x)\right )}{d} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \arccos (c x)}{2 d x^2}+\frac {2 c^2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d}-\frac {\left (i b c^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arccos (c x)}\right )}{2 d}+\frac {\left (i b c^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arccos (c x)}\right )}{2 d} \\ & = \frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \arccos (c x)}{2 d x^2}+\frac {2 c^2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d}-\frac {i b c^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d}+\frac {i b c^2 \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.77 \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=-\frac {a-b c x \sqrt {1-c^2 x^2}+b \arccos (c x)+2 b c^2 x^2 \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+2 b c^2 x^2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )-2 b c^2 x^2 \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )-2 a c^2 x^2 \log (x)+a c^2 x^2 \log \left (1-c^2 x^2\right )-2 i b c^2 x^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-2 i b c^2 x^2 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )+i b c^2 x^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d x^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.25 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.00

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\frac {-i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )}{2 c^{2} x^{2}}+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\)	\(248\)
default	\(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\frac {-i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )}{2 c^{2} x^{2}}+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\)	\(248\)
parts	\(-\frac {a \left (\frac {1}{2 x^{2}}-c^{2} \ln \left (x \right )+\frac {c^{2} \ln \left (c x -1\right )}{2}+\frac {c^{2} \ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \,c^{2} \left (\frac {-i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )}{2 c^{2} x^{2}}+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\)	\(251\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{3}} \,d x } \]

[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]

\[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx}{d} \]

[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

Maxima [F]

\[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{3}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x^3\,\left (d-c^2\,d\,x^2\right )} \,d x \]

[In]

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

[Out]

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