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

Optimal. Leaf size=235 \[ -\frac{1}{2} i b d^3 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac{7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{19}{48} b c d^3 x \sqrt{1-c^2 x^2}-\frac{19}{48} b d^3 \sin ^{-1}(c x) \]

[Out]

(-19*b*c*d^3*x*Sqrt[1 - c^2*x^2])/48 - (7*b*c*d^3*x*(1 - c^2*x^2)^(3/2))/72 - (b*c*d^3*x*(1 - c^2*x^2)^(5/2))/
36 - (19*b*d^3*ArcSin[c*x])/48 + (d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/2 + (d^3*(1 - c^2*x^2)^2*(a + b*ArcSi
n[c*x]))/4 + (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/6 - ((I/2)*d^3*(a + b*ArcSin[c*x])^2)/b + d^3*(a + b*Ar
cSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])] - (I/2)*b*d^3*PolyLog[2, E^((2*I)*ArcSin[c*x])]

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Rubi [A]  time = 0.283025, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4683, 4625, 3717, 2190, 2279, 2391, 195, 216} \[ -\frac{1}{2} i b d^3 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac{7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{19}{48} b c d^3 x \sqrt{1-c^2 x^2}-\frac{19}{48} b d^3 \sin ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-19*b*c*d^3*x*Sqrt[1 - c^2*x^2])/48 - (7*b*c*d^3*x*(1 - c^2*x^2)^(3/2))/72 - (b*c*d^3*x*(1 - c^2*x^2)^(5/2))/
36 - (19*b*d^3*ArcSin[c*x])/48 + (d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/2 + (d^3*(1 - c^2*x^2)^2*(a + b*ArcSi
n[c*x]))/4 + (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/6 - ((I/2)*d^3*(a + b*ArcSin[c*x])^2)/b + d^3*(a + b*Ar
cSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])] - (I/2)*b*d^3*PolyLog[2, E^((2*I)*ArcSin[c*x])]

Rule 4683

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

Rule 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3717

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 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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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{\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx &=\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac{1}{6} \left (b c d^3\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx\\ &=-\frac{1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d^2 \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac{1}{36} \left (5 b c d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx-\frac{1}{4} \left (b c d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx\\ &=-\frac{7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 \int \frac{a+b \sin ^{-1}(c x)}{x} \, dx-\frac{1}{48} \left (5 b c d^3\right ) \int \sqrt{1-c^2 x^2} \, dx-\frac{1}{16} \left (3 b c d^3\right ) \int \sqrt{1-c^2 x^2} \, dx-\frac{1}{2} \left (b c d^3\right ) \int \sqrt{1-c^2 x^2} \, dx\\ &=-\frac{19}{48} b c d^3 x \sqrt{1-c^2 x^2}-\frac{7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 \operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{96} \left (5 b c d^3\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{32} \left (3 b c d^3\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{4} \left (b c d^3\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{19}{48} b c d^3 x \sqrt{1-c^2 x^2}-\frac{7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac{19}{48} b d^3 \sin ^{-1}(c x)+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-\left (2 i d^3\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{19}{48} b c d^3 x \sqrt{1-c^2 x^2}-\frac{7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac{19}{48} b d^3 \sin ^{-1}(c x)+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\left (b d^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{19}{48} b c d^3 x \sqrt{1-c^2 x^2}-\frac{7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac{19}{48} b d^3 \sin ^{-1}(c x)+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{2} \left (i b d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac{19}{48} b c d^3 x \sqrt{1-c^2 x^2}-\frac{7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac{19}{48} b d^3 \sin ^{-1}(c x)+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac{1}{2} i b d^3 \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.210419, size = 183, normalized size = 0.78 \[ -\frac{1}{144} d^3 \left (72 i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+24 a c^6 x^6-108 a c^4 x^4+216 a c^2 x^2-144 a \log (x)+4 b c^5 x^5 \sqrt{1-c^2 x^2}-22 b c^3 x^3 \sqrt{1-c^2 x^2}+75 b c x \sqrt{1-c^2 x^2}+3 b \sin ^{-1}(c x) \left (8 c^6 x^6-36 c^4 x^4+72 c^2 x^2-48 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-25\right )+72 i b \sin ^{-1}(c x)^2\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(d^3*(216*a*c^2*x^2 - 108*a*c^4*x^4 + 24*a*c^6*x^6 + 75*b*c*x*Sqrt[1 - c^2*x^2] - 22*b*c^3*x^3*Sqrt[1 - c^2*x
^2] + 4*b*c^5*x^5*Sqrt[1 - c^2*x^2] + (72*I)*b*ArcSin[c*x]^2 + 3*b*ArcSin[c*x]*(-25 + 72*c^2*x^2 - 36*c^4*x^4
+ 8*c^6*x^6 - 48*Log[1 - E^((2*I)*ArcSin[c*x])]) - 144*a*Log[x] + (72*I)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])]))
/144

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Maple [A]  time = 0.275, size = 302, normalized size = 1.3 \begin{align*} -{\frac{{d}^{3}a{c}^{6}{x}^{6}}{6}}+{\frac{3\,{d}^{3}a{c}^{4}{x}^{4}}{4}}-{\frac{3\,{d}^{3}a{c}^{2}{x}^{2}}{2}}+{d}^{3}a\ln \left ( cx \right ) +{\frac{3\,{d}^{3}b\arcsin \left ( cx \right ){c}^{4}{x}^{4}}{4}}-{\frac{3\,{d}^{3}b\arcsin \left ( cx \right ){c}^{2}{x}^{2}}{2}}-{\frac{{d}^{3}b\arcsin \left ( cx \right ){c}^{6}{x}^{6}}{6}}+{\frac{25\,b{d}^{3}\arcsin \left ( cx \right ) }{48}}-i{d}^{3}b{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -i{d}^{3}b{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{i}{2}}b{d}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{2}+{d}^{3}b\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +{d}^{3}b\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{{d}^{3}b{c}^{5}{x}^{5}}{36}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{11\,{d}^{3}b{c}^{3}{x}^{3}}{72}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{25\,{d}^{3}bcx}{48}\sqrt{-{c}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*d^3*a*c^6*x^6+3/4*d^3*a*c^4*x^4-3/2*d^3*a*c^2*x^2+d^3*a*ln(c*x)+3/4*d^3*b*arcsin(c*x)*c^4*x^4-3/2*d^3*b*a
rcsin(c*x)*c^2*x^2-1/6*d^3*b*arcsin(c*x)*c^6*x^6+25/48*b*d^3*arcsin(c*x)-I*d^3*b*polylog(2,I*c*x+(-c^2*x^2+1)^
(1/2))-I*d^3*b*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-1/2*I*b*d^3*arcsin(c*x)^2+d^3*b*arcsin(c*x)*ln(1+I*c*x+(-c
^2*x^2+1)^(1/2))+d^3*b*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-1/36*d^3*b*(-c^2*x^2+1)^(1/2)*c^5*x^5+11/72*
d^3*b*(-c^2*x^2+1)^(1/2)*c^3*x^3-25/48*b*c*d^3*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} \, a c^{6} d^{3} x^{6} + \frac{3}{4} \, a c^{4} d^{3} x^{4} - \frac{3}{2} \, a c^{2} d^{3} x^{2} + a d^{3} \log \left (x\right ) - \int \frac{{\left (b c^{6} d^{3} x^{6} - 3 \, b c^{4} d^{3} x^{4} + 3 \, b c^{2} d^{3} x^{2} - b d^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{a c^{6} d^{3} x^{6} - 3 \, a c^{4} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2} - a d^{3} +{\left (b c^{6} d^{3} x^{6} - 3 \, b c^{4} d^{3} x^{4} + 3 \, b c^{2} d^{3} x^{2} - b d^{3}\right )} \arcsin \left (c x\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a*c^6*d^3*x^6 - 3*a*c^4*d^3*x^4 + 3*a*c^2*d^3*x^2 - a*d^3 + (b*c^6*d^3*x^6 - 3*b*c^4*d^3*x^4 + 3*b*
c^2*d^3*x^2 - b*d^3)*arcsin(c*x))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d**3*(Integral(-a/x, x) + Integral(3*a*c**2*x, x) + Integral(-3*a*c**4*x**3, x) + Integral(a*c**6*x**5, x) +
Integral(-b*asin(c*x)/x, x) + Integral(3*b*c**2*x*asin(c*x), x) + Integral(-3*b*c**4*x**3*asin(c*x), x) + Inte
gral(b*c**6*x**5*asin(c*x), x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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