\(\int \frac {(d+e x^2) (a+b \arccos (c x))}{x^3} \, dx\) [22]

Optimal result

Integrand size = 19, antiderivative size = 119 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^3} \, dx=\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {d (a+b \arccos (c x))}{2 x^2}+\frac {1}{2} i b e \arcsin (c x)^2-b e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+e (a+b \arccos (c x)) \log (x)+b e \arcsin (c x) \log (x)+\frac {1}{2} i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {14, 4816, 6874, 270, 2363, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^3} \, dx=-\frac {d (a+b \arccos (c x))}{2 x^2}+e \log (x) (a+b \arccos (c x))+\frac {1}{2} i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+\frac {1}{2} i b e \arcsin (c x)^2-b e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+b e \log (x) \arcsin (c x)+\frac {b c d \sqrt {1-c^2 x^2}}{2 x} \]

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

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 2363

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-e, 2]*(x/Sqr
t[d])]*((a + b*Log[c*x^n])/Rt[-e, 2]), x] - Dist[b*(n/Rt[-e, 2]), Int[ArcSin[Rt[-e, 2]*(x/Sqrt[d])]/x, x], x]
/; FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && NegQ[e]

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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4721

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 4816

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] ||
(IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.91 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^3} \, dx=-\frac {a d}{2 x^2}+\frac {b c d \sqrt {1-c^2 x^2}}{2 x}-\frac {b d \arccos (c x)}{2 x^2}-\frac {1}{2} i b e \arccos (c x)^2+b e \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+a e \log (x)-\frac {1}{2} i b e \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.99 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a d}{2 c^{2} x^{2}}+\frac {a e \ln \left (c x \right )}{c^{2}}+\frac {b \left (-\frac {i e \arccos \left (c x \right )^{2}}{2}-\frac {d \left (-i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2 x^{2}}+\ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) e \arccos \left (c x \right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) e}{2}\right )}{c^{2}}\right )\)	\(137\)
default	\(c^{2} \left (-\frac {a d}{2 c^{2} x^{2}}+\frac {a e \ln \left (c x \right )}{c^{2}}+\frac {b \left (-\frac {i e \arccos \left (c x \right )^{2}}{2}-\frac {d \left (-i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2 x^{2}}+\ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) e \arccos \left (c x \right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) e}{2}\right )}{c^{2}}\right )\)	\(137\)
parts	\(-\frac {a d}{2 x^{2}}+a e \ln \left (x \right )+b \,c^{2} \left (-\frac {i \arccos \left (c x \right )^{2} e}{2 c^{2}}-\frac {d \left (-i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2 c^{2} x^{2}}+\frac {e \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{c^{2}}-\frac {i e \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 c^{2}}\right )\)	\(137\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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