3.36 \(\int \frac{a+b \sin ^{-1}(c x)}{x^4 (d-c^2 d x^2)} \, dx\)

Optimal. Leaf size=173 \[ \frac{i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac{2 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{7 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d} \]

[Out]

-(b*c*Sqrt[1 - c^2*x^2])/(6*d*x^2) - (a + b*ArcSin[c*x])/(3*d*x^3) - (c^2*(a + b*ArcSin[c*x]))/(d*x) - ((2*I)*
c^3*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/d - (7*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d) + (I*b*c^3*P
olyLog[2, (-I)*E^(I*ArcSin[c*x])])/d - (I*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/d

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Rubi [A]  time = 0.243657, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {4701, 4657, 4181, 2279, 2391, 266, 63, 208, 51} \[ \frac{i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac{2 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{7 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)),x]

[Out]

-(b*c*Sqrt[1 - c^2*x^2])/(6*d*x^2) - (a + b*ArcSin[c*x])/(3*d*x^3) - (c^2*(a + b*ArcSin[c*x]))/(d*x) - ((2*I)*
c^3*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/d - (7*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d) + (I*b*c^3*P
olyLog[2, (-I)*E^(I*ArcSin[c*x])])/d - (I*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/d

Rule 4701

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 + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m
+ 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^F
racPart[p])/(f*(m + 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] && LtQ[m, -1] && Inte
gerQ[m]

Rule 4657

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

Rule 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 51

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

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x^4 \left (d-c^2 d x^2\right )} \, dx &=-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+c^2 \int \frac{a+b \sin ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx+\frac{(b c) \int \frac{1}{x^3 \sqrt{1-c^2 x^2}} \, dx}{3 d}\\ &=-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )}{d x}+c^4 \int \frac{a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{6 d}+\frac{\left (b c^3\right ) \int \frac{1}{x \sqrt{1-c^2 x^2}} \, dx}{d}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )}{d x}+\frac{c^3 \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{12 d}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 d}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac{2 i c^3 \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{6 d}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{d}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac{2 i c^3 \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{7 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d}+\frac{\left (i b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{\left (i b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac{2 i c^3 \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{7 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d}+\frac{i b c^3 \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{i b c^3 \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d}\\ \end{align*}

Mathematica [B]  time = 0.147467, size = 350, normalized size = 2.02 \[ -\frac{-6 i b c^3 x^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+6 i b c^3 x^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+6 a c^2 x^2+3 a c^3 x^3 \log (1-c x)-3 a c^3 x^3 \log (c x+1)+2 a+b c x \sqrt{1-c^2 x^2}+3 i \pi b c^3 x^3 \sin ^{-1}(c x)+6 b c^2 x^2 \sin ^{-1}(c x)+7 b c^3 x^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-6 b c^3 x^3 \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )-3 \pi b c^3 x^3 \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+6 b c^3 x^3 \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-3 \pi b c^3 x^3 \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+3 \pi b c^3 x^3 \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+3 \pi b c^3 x^3 \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+2 b \sin ^{-1}(c x)}{6 d x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)),x]

[Out]

-(2*a + 6*a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2] + 2*b*ArcSin[c*x] + 6*b*c^2*x^2*ArcSin[c*x] + (3*I)*b*c^3*Pi*x^3
*ArcSin[c*x] + 7*b*c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 3*b*c^3*Pi*x^3*Log[1 - I*E^(I*ArcSin[c*x])] - 6*b*c^3*
x^3*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 3*b*c^3*Pi*x^3*Log[1 + I*E^(I*ArcSin[c*x])] + 6*b*c^3*x^3*ArcSi
n[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 3*a*c^3*x^3*Log[1 - c*x] - 3*a*c^3*x^3*Log[1 + c*x] + 3*b*c^3*Pi*x^3*Log
[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 3*b*c^3*Pi*x^3*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (6*I)*b*c^3*x^3*PolyLog[2,
(-I)*E^(I*ArcSin[c*x])] + (6*I)*b*c^3*x^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/(6*d*x^3)

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Maple [A]  time = 0.177, size = 303, normalized size = 1.8 \begin{align*} -{\frac{{c}^{3}a\ln \left ( cx-1 \right ) }{2\,d}}+{\frac{{c}^{3}a\ln \left ( cx+1 \right ) }{2\,d}}-{\frac{a}{3\,d{x}^{3}}}-{\frac{{c}^{2}a}{dx}}-{\frac{{c}^{2}b\arcsin \left ( cx \right ) }{dx}}-{\frac{bc}{6\,d{x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arcsin \left ( cx \right ) }{3\,d{x}^{3}}}+{\frac{7\,b{c}^{3}}{6\,d}\ln \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1}-1 \right ) }-{\frac{7\,b{c}^{3}}{6\,d}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{b{c}^{3}\arcsin \left ( cx \right ) }{d}\ln \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{b{c}^{3}\arcsin \left ( cx \right ) }{d}\ln \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{i{c}^{3}b}{d}{\it dilog} \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{i{c}^{3}b}{d}{\it dilog} \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d),x)

[Out]

-1/2*c^3*a/d*ln(c*x-1)+1/2*c^3*a/d*ln(c*x+1)-1/3*a/d/x^3-c^2*a/d/x-c^2*b/d*arcsin(c*x)/x-1/6*b*c*(-c^2*x^2+1)^
(1/2)/d/x^2-1/3*b/d*arcsin(c*x)/x^3+7/6*c^3*b/d*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-7/6*c^3*b/d*ln(1+I*c*x+(-c^2*x^
2+1)^(1/2))+c^3*b/d*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-c^3*b/d*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2
+1)^(1/2)))-I*c^3*b/d*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+I*c^3*b/d*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \,{\left (\frac{3 \, c^{3} \log \left (c x + 1\right )}{d} - \frac{3 \, c^{3} \log \left (c x - 1\right )}{d} - \frac{2 \,{\left (3 \, c^{2} x^{2} + 1\right )}}{d x^{3}}\right )} a + \frac{{\left (3 \, c^{3} x^{3} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) - 3 \, c^{3} x^{3} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right ) + d x^{3} \int \frac{{\left (3 \, c^{4} x^{3} \log \left (c x + 1\right ) - 3 \, c^{4} x^{3} \log \left (-c x + 1\right ) - 6 \, c^{3} x^{2} - 2 \, c\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{2} d x^{5} - d x^{3}}\,{d x} - 2 \,{\left (3 \, c^{2} x^{2} + 1\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )\right )} b}{6 \, d x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d),x, algorithm="maxima")

[Out]

1/6*(3*c^3*log(c*x + 1)/d - 3*c^3*log(c*x - 1)/d - 2*(3*c^2*x^2 + 1)/(d*x^3))*a + 1/6*(3*c^3*x^3*arctan2(c*x,
sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 3*c^3*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1
) + 6*d*x^3*integrate(1/6*(3*c^4*x^3*log(c*x + 1) - 3*c^4*x^3*log(-c*x + 1) - 6*c^3*x^2 - 2*c)*sqrt(c*x + 1)*s
qrt(-c*x + 1)/(c^2*d*x^5 - d*x^3), x) - 2*(3*c^2*x^2 + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*b/(d*x^3
)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b \arcsin \left (c x\right ) + a}{c^{2} d x^{6} - d x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d),x, algorithm="fricas")

[Out]

integral(-(b*arcsin(c*x) + a)/(c^2*d*x^6 - d*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a}{c^{2} x^{6} - x^{4}}\, dx + \int \frac{b \operatorname{asin}{\left (c x \right )}}{c^{2} x^{6} - x^{4}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))/x**4/(-c**2*d*x**2+d),x)

[Out]

-(Integral(a/(c**2*x**6 - x**4), x) + Integral(b*asin(c*x)/(c**2*x**6 - x**4), x))/d

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d),x, algorithm="giac")

[Out]

integrate(-(b*arcsin(c*x) + a)/((c^2*d*x^2 - d)*x^4), x)