\(\int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx\) [157]

Optimal result

Integrand size = 14, antiderivative size = 137 \[ \int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx=-\frac {(a+b \arcsin (c x))^3}{x}-6 b c (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )+6 i b^2 c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-6 i b^2 c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-6 b^3 c \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+6 b^3 c \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right ) \]

[Out]

-(a+b*arcsin(c*x))^3/x-6*b*c*(a+b*arcsin(c*x))^2*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))+6*I*b^2*c*(a+b*arcsin(c*x))
*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-6*I*b^2*c*(a+b*arcsin(c*x))*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-6*b^3*c*
polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+6*b^3*c*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4723, 4803, 4268, 2611, 2320, 6724} \[ \int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx=-6 b c \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+6 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-6 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {(a+b \arcsin (c x))^3}{x}-6 b^3 c \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+6 b^3 c \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right ) \]

[In]

Int[(a + b*ArcSin[c*x])^3/x^2,x]

[Out]

-((a + b*ArcSin[c*x])^3/x) - 6*b*c*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])] + (6*I)*b^2*c*(a + b*ArcSi
n[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])] - (6*I)*b^2*c*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])] - 6*b^3
*c*PolyLog[3, -E^(I*ArcSin[c*x])] + 6*b^3*c*PolyLog[3, E^(I*ArcSin[c*x])]

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 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 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4803

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; Free
Q[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

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}& = -\frac {(a+b \arcsin (c x))^3}{x}+(3 b c) \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {(a+b \arcsin (c x))^3}{x}+(3 b c) \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {(a+b \arcsin (c x))^3}{x}-6 b c (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )-\left (6 b^2 c\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )+\left (6 b^2 c\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {(a+b \arcsin (c x))^3}{x}-6 b c (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )+6 i b^2 c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-6 i b^2 c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-\left (6 i b^3 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )+\left (6 i b^3 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (c x)\right ) \\ & = -\frac {(a+b \arcsin (c x))^3}{x}-6 b c (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )+6 i b^2 c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-6 i b^2 c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-\left (6 b^3 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )+\left (6 b^3 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arcsin (c x)}\right ) \\ & = -\frac {(a+b \arcsin (c x))^3}{x}-6 b c (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )+6 i b^2 c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-6 i b^2 c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-6 b^3 c \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+6 b^3 c \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right ) \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(283\) vs. \(2(137)=274\).

Time = 0.24 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.07 \[ \int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx=-\frac {a^3}{x}-\frac {3 a^2 b \arcsin (c x)}{x}+3 a^2 b c \log (x)-3 a^2 b c \log \left (1+\sqrt {1-c^2 x^2}\right )+3 a b^2 c \left (-\arcsin (c x) \left (\frac {\arcsin (c x)}{c x}-2 \log \left (1-e^{i \arcsin (c x)}\right )+2 \log \left (1+e^{i \arcsin (c x)}\right )\right )+2 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )+b^3 c \left (-\frac {\arcsin (c x)^3}{c x}+3 \arcsin (c x)^2 \log \left (1-e^{i \arcsin (c x)}\right )-3 \arcsin (c x)^2 \log \left (1+e^{i \arcsin (c x)}\right )+6 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-6 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-6 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+6 \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right ) \]

[In]

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

[Out]

-(a^3/x) - (3*a^2*b*ArcSin[c*x])/x + 3*a^2*b*c*Log[x] - 3*a^2*b*c*Log[1 + Sqrt[1 - c^2*x^2]] + 3*a*b^2*c*(-(Ar
cSin[c*x]*(ArcSin[c*x]/(c*x) - 2*Log[1 - E^(I*ArcSin[c*x])] + 2*Log[1 + E^(I*ArcSin[c*x])])) + (2*I)*PolyLog[2
, -E^(I*ArcSin[c*x])] - (2*I)*PolyLog[2, E^(I*ArcSin[c*x])]) + b^3*c*(-(ArcSin[c*x]^3/(c*x)) + 3*ArcSin[c*x]^2
*Log[1 - E^(I*ArcSin[c*x])] - 3*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + (6*I)*ArcSin[c*x]*PolyLog[2, -E^(I*
ArcSin[c*x])] - (6*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] - 6*PolyLog[3, -E^(I*ArcSin[c*x])] + 6*PolyLog
[3, E^(I*ArcSin[c*x])])

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.55

method	result	size
parts	\(-\frac {a^{3}}{x}+b^{3} c \left (-\frac {\arcsin \left (c x \right )^{3}}{c x}-3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} c \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a^{2} b c \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(349\)
derivativedivides	\(c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\arcsin \left (c x \right )^{3}}{c x}-3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a^{2} b \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(351\)
default	\(c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\arcsin \left (c x \right )^{3}}{c x}-3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a^{2} b \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(351\)

[In]

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

[Out]

-a^3/x+b^3*c*(-1/c/x*arcsin(c*x)^3-3*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+6*I*arcsin(c*x)*polylog(2,-I
*c*x-(-c^2*x^2+1)^(1/2))-6*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+3*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))
-6*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+6*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2)))+3*a*b^2*c*(-1/c/x*
arcsin(c*x)^2+2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+2*I*di
log(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I*dilog(1-I*c*x-(-c^2*x^2+1)^(1/2)))+3*a^2*b*c*(-1/c/x*arcsin(c*x)-arctanh(1
/(-c^2*x^2+1)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

integral((b^3*arcsin(c*x)^3 + 3*a*b^2*arcsin(c*x)^2 + 3*a^2*b*arcsin(c*x) + a^3)/x^2, x)

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \]

[In]

integrate((a+b*asin(c*x))**3/x**2,x)

[Out]

Integral((a + b*asin(c*x))**3/x**2, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate((b*arcsin(c*x) + a)^3/x^2, x)

Mupad [F(-1)]

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

[In]

int((a + b*asin(c*x))^3/x^2,x)

[Out]

int((a + b*asin(c*x))^3/x^2, x)