3.29 \(\int \frac{x^3 (a+b \sin ^{-1}(c x))}{d-c^2 d x^2} \, dx\)

Optimal. Leaf size=144 \[ \frac{i b \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{2 c^4 d}-\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d}+\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^4 d}-\frac{\log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d}-\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}+\frac{b \sin ^{-1}(c x)}{4 c^4 d} \]

[Out]

-(b*x*Sqrt[1 - c^2*x^2])/(4*c^3*d) + (b*ArcSin[c*x])/(4*c^4*d) - (x^2*(a + b*ArcSin[c*x]))/(2*c^2*d) + ((I/2)*
(a + b*ArcSin[c*x])^2)/(b*c^4*d) - ((a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^4*d) + ((I/2)*b*Pol
yLog[2, -E^((2*I)*ArcSin[c*x])])/(c^4*d)

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Rubi [A]  time = 0.188384, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4715, 4675, 3719, 2190, 2279, 2391, 321, 216} \[ \frac{i b \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{2 c^4 d}-\frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d}+\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^4 d}-\frac{\log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d}-\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}+\frac{b \sin ^{-1}(c x)}{4 c^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*x*Sqrt[1 - c^2*x^2])/(4*c^3*d) + (b*ArcSin[c*x])/(4*c^4*d) - (x^2*(a + b*ArcSin[c*x]))/(2*c^2*d) + ((I/2)*
(a + b*ArcSin[c*x])^2)/(b*c^4*d) - ((a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^4*d) + ((I/2)*b*Pol
yLog[2, -E^((2*I)*ArcSin[c*x])])/(c^4*d)

Rule 4715

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

Rule 4675

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

Rule 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

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

Rubi steps

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

Mathematica [B]  time = 0.124941, size = 294, normalized size = 2.04 \[ -\frac{-4 i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )-4 i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+2 a c^2 x^2+2 a \log \left (1-c^2 x^2\right )+b c x \sqrt{1-c^2 x^2}+2 b c^2 x^2 \sin ^{-1}(c x)-2 i b \sin ^{-1}(c x)^2-b \sin ^{-1}(c x)+4 i \pi b \sin ^{-1}(c x)+8 \pi b \log \left (1+e^{-i \sin ^{-1}(c x)}\right )+4 b \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+2 \pi b \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+4 b \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-2 \pi b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-2 \pi b \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-8 \pi b \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+2 \pi b \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{4 c^4 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(2*a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2] - b*ArcSin[c*x] + (4*I)*b*Pi*ArcSin[c*x] + 2*b*c^2*x^2*ArcSin[c*x] - (
2*I)*b*ArcSin[c*x]^2 + 8*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 4*b*ArcSin
[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 4*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcS
in[c*x])] + 2*a*Log[1 - c^2*x^2] - 8*b*Pi*Log[Cos[ArcSin[c*x]/2]] + 2*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] -
2*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (4*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (4*I)*b*PolyLog[2, I*E
^(I*ArcSin[c*x])])/(4*c^4*d)

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Maple [A]  time = 0.148, size = 181, normalized size = 1.3 \begin{align*} -{\frac{a{x}^{2}}{2\,{c}^{2}d}}-{\frac{a\ln \left ( cx-1 \right ) }{2\,{c}^{4}d}}-{\frac{a\ln \left ( cx+1 \right ) }{2\,{c}^{4}d}}+{\frac{{\frac{i}{2}}b \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{{c}^{4}d}}-{\frac{bx}{4\,{c}^{3}d}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arcsin \left ( cx \right ){x}^{2}}{2\,{c}^{2}d}}+{\frac{b\arcsin \left ( cx \right ) }{4\,{c}^{4}d}}-{\frac{b\arcsin \left ( cx \right ) }{{c}^{4}d}\ln \left ( 1+ \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }+{\frac{{\frac{i}{2}}b}{{c}^{4}d}{\it polylog} \left ( 2,- \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/c^2*a/d*x^2-1/2/c^4*a/d*ln(c*x-1)-1/2/c^4*a/d*ln(c*x+1)+1/2*I/c^4*b/d*arcsin(c*x)^2-1/4*b*x*(-c^2*x^2+1)^
(1/2)/c^3/d-1/2/c^2*b/d*arcsin(c*x)*x^2+1/4*b*arcsin(c*x)/c^4/d-1/c^4*b/d*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)
^(1/2))^2)+1/2*I*b*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c^4/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, a{\left (\frac{x^{2}}{c^{2} d} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4} d}\right )} - \frac{{\left (c^{4} d \int \frac{c^{2} x^{2} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} + e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) + e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right )}{c^{7} d x^{4} - c^{5} d x^{2} -{\left (c^{5} d x^{2} - c^{3} d\right )}{\left (c x + 1\right )}{\left (c x - 1\right )}}\,{d x} + c^{2} x^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) + \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right )\right )} b}{2 \, c^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*(x^2/(c^2*d) + log(c^2*x^2 - 1)/(c^4*d)) - 1/2*(2*c^4*d*integrate(1/2*(c^2*x^2*e^(1/2*log(c*x + 1) + 1/
2*log(-c*x + 1)) + e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1))*log(c*x + 1) + e^(1/2*log(c*x + 1) + 1/2*log(-c*x
+ 1))*log(-c*x + 1))/(c^7*d*x^4 - c^5*d*x^2 + (c^5*d*x^2 - c^3*d)*e^(log(c*x + 1) + log(-c*x + 1))), x) + c^2*
x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) + arc
tan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1))*b/(c^4*d)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b x^{3} \arcsin \left (c x\right ) + a x^{3}}{c^{2} d x^{2} - d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a x^{3}}{c^{2} x^{2} - 1}\, dx + \int \frac{b x^{3} \operatorname{asin}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(Integral(a*x**3/(c**2*x**2 - 1), x) + Integral(b*x**3*asin(c*x)/(c**2*x**2 - 1), x))/d

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b \arcsin \left (c x\right ) + a\right )} x^{3}}{c^{2} d x^{2} - d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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