3.2.62 \(\int \frac {(d-c^2 d x^2) (a+b \text {ArcSin}(c x))^2}{x^2} \, dx\) [162]

Optimal. Leaf size=149 \[ 2 b^2 c^2 d x-2 b c d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-2 c^2 d x (a+b \text {ArcSin}(c x))^2-\frac {d \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{x}-4 b c d (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )+2 i b^2 c d \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-2 i b^2 c d \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right ) \]

[Out]

2*b^2*c^2*d*x-2*c^2*d*x*(a+b*arcsin(c*x))^2-d*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/x-4*b*c*d*(a+b*arcsin(c*x))*arc
tanh(I*c*x+(-c^2*x^2+1)^(1/2))+2*I*b^2*c*d*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*b^2*c*d*polylog(2,I*c*x+(-
c^2*x^2+1)^(1/2))-2*b*c*d*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4785, 4715, 4767, 8, 4783, 4803, 4268, 2317, 2438} \begin {gather*} -2 b c d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {d \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{x}-2 c^2 d x (a+b \text {ArcSin}(c x))^2-4 b c d \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))+2 i b^2 c d \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right )-2 i b^2 c d \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right )+2 b^2 c^2 d x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*b^2*c^2*d*x - 2*b*c*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - 2*c^2*d*x*(a + b*ArcSin[c*x])^2 - (d*(1 - c^2*
x^2)*(a + b*ArcSin[c*x])^2)/x - 4*b*c*d*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + (2*I)*b^2*c*d*PolyLog
[2, -E^(I*ArcSin[c*x])] - (2*I)*b^2*c*d*PolyLog[2, E^(I*ArcSin[c*x])]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 4715

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

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4783

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f
*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/(f*(m + 2))), x] + (Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/S
qrt[1 - c^2*x^2]], Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x] - Dist[b*c*(n/(f*(m + 2)))*Si
mp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b,
c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 4785

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(f*(m + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x)^
(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c,
d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[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]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}+(2 b c d) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\left (2 c^2 d\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=2 b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-2 c^2 d x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}+(2 b c d) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx-\left (2 b^2 c^2 d\right ) \int 1 \, dx+\left (4 b c^3 d\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-2 b^2 c^2 d x-2 b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-2 c^2 d x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}+(2 b c d) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )+\left (4 b^2 c^2 d\right ) \int 1 \, dx\\ &=2 b^2 c^2 d x-2 b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-2 c^2 d x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\left (2 b^2 c d\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (2 b^2 c d\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=2 b^2 c^2 d x-2 b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-2 c^2 d x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+\left (2 i b^2 c d\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )-\left (2 i b^2 c d\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=2 b^2 c^2 d x-2 b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-2 c^2 d x \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+2 i b^2 c d \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.31, size = 203, normalized size = 1.36 \begin {gather*} -\frac {d \left (a^2+a^2 c^2 x^2+2 a b c x \left (\sqrt {1-c^2 x^2}+c x \text {ArcSin}(c x)\right )+b^2 c x \left (2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)+c x \left (-2+\text {ArcSin}(c x)^2\right )\right )+2 a b \left (\text {ArcSin}(c x)+c x \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\right )-i b^2 \left (i \text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 c x \left (-\log \left (1-e^{i \text {ArcSin}(c x)}\right )+\log \left (1+e^{i \text {ArcSin}(c x)}\right )\right )\right )+2 c x \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-2 c x \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((d*(a^2 + a^2*c^2*x^2 + 2*a*b*c*x*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]) + b^2*c*x*(2*Sqrt[1 - c^2*x^2]*ArcSi
n[c*x] + c*x*(-2 + ArcSin[c*x]^2)) + 2*a*b*(ArcSin[c*x] + c*x*ArcTanh[Sqrt[1 - c^2*x^2]]) - I*b^2*(I*ArcSin[c*
x]*(ArcSin[c*x] + 2*c*x*(-Log[1 - E^(I*ArcSin[c*x])] + Log[1 + E^(I*ArcSin[c*x])])) + 2*c*x*PolyLog[2, -E^(I*A
rcSin[c*x])] - 2*c*x*PolyLog[2, E^(I*ArcSin[c*x])])))/x)

________________________________________________________________________________________

Maple [A]
time = 0.31, size = 250, normalized size = 1.68

method	result	size
derivativedivides	\(c \left (-d \,a^{2} \left (c x +\frac {1}{c x}\right )-2 d \,b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-d \,b^{2} \arcsin \left (c x \right )^{2} c x +2 d \,b^{2} c x -\frac {d \,b^{2} \arcsin \left (c x \right )^{2}}{c x}+2 d \,b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 d \,b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d \,b^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d \,b^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 d a b \left (c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}+\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(250\)
default	\(c \left (-d \,a^{2} \left (c x +\frac {1}{c x}\right )-2 d \,b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-d \,b^{2} \arcsin \left (c x \right )^{2} c x +2 d \,b^{2} c x -\frac {d \,b^{2} \arcsin \left (c x \right )^{2}}{c x}+2 d \,b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 d \,b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d \,b^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d \,b^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 d a b \left (c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}+\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(-d*a^2*(c*x+1/c/x)-2*d*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-d*b^2*arcsin(c*x)^2*c*x+2*d*b^2*c*x-d*b^2/c/x*arc
sin(c*x)^2+2*d*b^2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*d*b^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2
))-2*I*d*b^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+2*I*d*b^2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-2*d*a*b*(c*x*a
rcsin(c*x)+1/c/x*arcsin(c*x)+(-c^2*x^2+1)^(1/2)+arctanh(1/(-c^2*x^2+1)^(1/2))))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - d \left (\int a^{2} c^{2}\, dx + \int \left (- \frac {a^{2}}{x^{2}}\right )\, dx + \int b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{2}}\right )\, dx + \int 2 a b c^{2} \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{2}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*(Integral(a**2*c**2, x) + Integral(-a**2/x**2, x) + Integral(b**2*c**2*asin(c*x)**2, x) + Integral(-b**2*as
in(c*x)**2/x**2, x) + Integral(2*a*b*c**2*asin(c*x), x) + Integral(-2*a*b*asin(c*x)/x**2, x))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________