\(\int \frac {(a+b \arcsin (c x))^2}{x} \, dx\) [151]

Optimal result

Integrand size = 14, antiderivative size = 90 \[ \int \frac {(a+b \arcsin (c x))^2}{x} \, dx=-\frac {i (a+b \arcsin (c x))^3}{3 b}+(a+b \arcsin (c x))^2 \log \left (1-e^{2 i \arcsin (c x)}\right )-i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right ) \]

[Out]

-1/3*I*(a+b*arcsin(c*x))^3/b+(a+b*arcsin(c*x))^2*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-I*b*(a+b*arcsin(c*x))*poly
log(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)+1/2*b^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4721, 3798, 2221, 2611, 2320, 6724} \[ \int \frac {(a+b \arcsin (c x))^2}{x} \, dx=-i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^3}{3 b}+\log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right ) \]

[In]

Int[(a + b*ArcSin[c*x])^2/x,x]

[Out]

((-1/3*I)*(a + b*ArcSin[c*x])^3)/b + (a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] - I*b*(a + b*ArcSin[
c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] + (b^2*PolyLog[3, E^((2*I)*ArcSin[c*x])])/2

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

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 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {i (a+b \arcsin (c x))^3}{3 b}+(a+b \arcsin (c x))^2 \log \left (1-e^{2 i \arcsin (c x)}\right )-(2 b) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {i (a+b \arcsin (c x))^3}{3 b}+(a+b \arcsin (c x))^2 \log \left (1-e^{2 i \arcsin (c x)}\right )-i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {i (a+b \arcsin (c x))^3}{3 b}+(a+b \arcsin (c x))^2 \log \left (1-e^{2 i \arcsin (c x)}\right )-i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+\frac {1}{2} b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right ) \\ & = -\frac {i (a+b \arcsin (c x))^3}{3 b}+(a+b \arcsin (c x))^2 \log \left (1-e^{2 i \arcsin (c x)}\right )-i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b \arcsin (c x))^2}{x} \, dx=a^2 \log (c x)+2 a b \left (\arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-\frac {1}{2} i \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )\right )+b^2 \left (-\frac {i \pi ^3}{24}+\frac {1}{3} i \arcsin (c x)^3+\arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )+i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )\right ) \]

[In]

Integrate[(a + b*ArcSin[c*x])^2/x,x]

[Out]

a^2*Log[c*x] + 2*a*b*(ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - (I/2)*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*
ArcSin[c*x])])) + b^2*((-1/24*I)*Pi^3 + (I/3)*ArcSin[c*x]^3 + ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] +
I*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] + PolyLog[3, E^((-2*I)*ArcSin[c*x])]/2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (114 ) = 228\).

Time = 0.07 (sec) , antiderivative size = 294, normalized size of antiderivative = 3.27

method	result	size
parts	\(a^{2} \ln \left (x \right )+b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\)	\(294\)
derivativedivides	\(a^{2} \ln \left (c x \right )+b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\)	\(296\)
default	\(a^{2} \ln \left (c x \right )+b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\)	\(296\)

[In]

int((a+b*arcsin(c*x))^2/x,x,method=_RETURNVERBOSE)

[Out]

a^2*ln(x)+b^2*(-1/3*I*arcsin(c*x)^3+arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I*arcsin(c*x)*polylog(2,-I*
c*x-(-c^2*x^2+1)^(1/2))+2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*
I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+2*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2)))+2*a*b*(-1/2*I*arcsin(
c*x)^2+arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+arcsin(c*x)*ln(1-I*c*
x-(-c^2*x^2+1)^(1/2))-I*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

integral((b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/x, x)

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x}\, dx \]

[In]

integrate((a+b*asin(c*x))**2/x,x)

[Out]

Integral((a + b*asin(c*x))**2/x, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

integrate((a+b*arcsin(c*x))^2/x,x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^2/x, x)

Mupad [F(-1)]

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

[In]

int((a + b*asin(c*x))^2/x,x)

[Out]

int((a + b*asin(c*x))^2/x, x)