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

Optimal. Leaf size=263 \[ \frac{3}{2} i b c^2 d^3 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{3 i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}-\frac{7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}+\frac{3}{32} b c^3 d^3 x \sqrt{1-c^2 x^2}+\frac{3}{32} b c^2 d^3 \sin ^{-1}(c x) \]

[Out]

(3*b*c^3*d^3*x*Sqrt[1 - c^2*x^2])/32 - (7*b*c^3*d^3*x*(1 - c^2*x^2)^(3/2))/16 - (b*c*d^3*(1 - c^2*x^2)^(5/2))/
(2*x) + (3*b*c^2*d^3*ArcSin[c*x])/32 - (3*c^2*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/2 - (3*c^2*d^3*(1 - c^2*x
^2)^2*(a + b*ArcSin[c*x]))/4 - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/(2*x^2) + (((3*I)/2)*c^2*d^3*(a + b*A
rcSin[c*x])^2)/b - 3*c^2*d^3*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])] + ((3*I)/2)*b*c^2*d^3*PolyLog[
2, E^((2*I)*ArcSin[c*x])]

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Rubi [A]  time = 0.298457, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4685, 277, 195, 216, 4683, 4625, 3717, 2190, 2279, 2391} \[ \frac{3}{2} i b c^2 d^3 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{3 i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}-\frac{7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}+\frac{3}{32} b c^3 d^3 x \sqrt{1-c^2 x^2}+\frac{3}{32} b c^2 d^3 \sin ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*c^3*d^3*x*Sqrt[1 - c^2*x^2])/32 - (7*b*c^3*d^3*x*(1 - c^2*x^2)^(3/2))/16 - (b*c*d^3*(1 - c^2*x^2)^(5/2))/
(2*x) + (3*b*c^2*d^3*ArcSin[c*x])/32 - (3*c^2*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/2 - (3*c^2*d^3*(1 - c^2*x
^2)^2*(a + b*ArcSin[c*x]))/4 - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/(2*x^2) + (((3*I)/2)*c^2*d^3*(a + b*A
rcSin[c*x])^2)/b - 3*c^2*d^3*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])] + ((3*I)/2)*b*c^2*d^3*PolyLog[
2, E^((2*I)*ArcSin[c*x])]

Rule 4685

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((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]))/(f*(m + 1)), x] + (-Dist[(b*c*d^p)/(f*(m + 1)), Int[(f*x)^(m + 1)*
(1 - c^2*x^2)^(p - 1/2), x], x] - Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*Arc
Sin[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]

Rule 277

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

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]

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]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx &=-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d\right ) \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx+\frac{1}{2} \left (b c d^3\right ) \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x^2} \, dx\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^2\right ) \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx+\frac{1}{4} \left (3 b c^3 d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx-\frac{1}{2} \left (5 b c^3 d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx\\ &=-\frac{7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{x} \, dx+\frac{1}{16} \left (9 b c^3 d^3\right ) \int \sqrt{1-c^2 x^2} \, dx+\frac{1}{2} \left (3 b c^3 d^3\right ) \int \sqrt{1-c^2 x^2} \, dx-\frac{1}{8} \left (15 b c^3 d^3\right ) \int \sqrt{1-c^2 x^2} \, dx\\ &=\frac{3}{32} b c^3 d^3 x \sqrt{1-c^2 x^2}-\frac{7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^3\right ) \operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )+\frac{1}{32} \left (9 b c^3 d^3\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx+\frac{1}{4} \left (3 b c^3 d^3\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{16} \left (15 b c^3 d^3\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{3}{32} b c^3 d^3 x \sqrt{1-c^2 x^2}-\frac{7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}+\frac{3}{32} b c^2 d^3 \sin ^{-1}(c x)-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{3 i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+\left (6 i c^2 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{3}{32} b c^3 d^3 x \sqrt{1-c^2 x^2}-\frac{7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}+\frac{3}{32} b c^2 d^3 \sin ^{-1}(c x)-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{3 i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\left (3 b c^2 d^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac{3}{32} b c^3 d^3 x \sqrt{1-c^2 x^2}-\frac{7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}+\frac{3}{32} b c^2 d^3 \sin ^{-1}(c x)-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{3 i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-3 c^2 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 (3 i b c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=\frac{3}{32} b c^3 d^3 x \sqrt{1-c^2 x^2}-\frac{7}{16} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2}}{2 x}+\frac{3}{32} b c^2 d^3 \sin ^{-1}(c x)-\frac{3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{3 i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\frac{3}{2} i b c^2 d^3 \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.178301, size = 203, normalized size = 0.77 \[ -\frac{d^3 \left (-48 i b c^2 x^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+8 a c^6 x^6-48 a c^4 x^4+96 a c^2 x^2 \log (x)+16 a+2 b c^5 x^5 \sqrt{1-c^2 x^2}-21 b c^3 x^3 \sqrt{1-c^2 x^2}+16 b c x \sqrt{1-c^2 x^2}-48 i b c^2 x^2 \sin ^{-1}(c x)^2+b \sin ^{-1}(c x) \left (8 c^6 x^6-48 c^4 x^4+21 c^2 x^2+96 c^2 x^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+16\right )\right )}{32 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(d^3*(16*a - 48*a*c^4*x^4 + 8*a*c^6*x^6 + 16*b*c*x*Sqrt[1 - c^2*x^2] - 21*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 2*b*c
^5*x^5*Sqrt[1 - c^2*x^2] - (48*I)*b*c^2*x^2*ArcSin[c*x]^2 + b*ArcSin[c*x]*(16 + 21*c^2*x^2 - 48*c^4*x^4 + 8*c^
6*x^6 + 96*c^2*x^2*Log[1 - E^((2*I)*ArcSin[c*x])]) + 96*a*c^2*x^2*Log[x] - (48*I)*b*c^2*x^2*PolyLog[2, E^((2*I
)*ArcSin[c*x])]))/(32*x^2)

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Maple [A]  time = 0.473, size = 330, normalized size = 1.3 \begin{align*} -{\frac{{c}^{6}{d}^{3}a{x}^{4}}{4}}+{\frac{3\,{c}^{4}{d}^{3}a{x}^{2}}{2}}-{\frac{{d}^{3}a}{2\,{x}^{2}}}-3\,{c}^{2}{d}^{3}a\ln \left ( cx \right ) -{\frac{b{d}^{3}\arcsin \left ( cx \right ) }{2\,{x}^{2}}}-{\frac{{c}^{6}{d}^{3}b\arcsin \left ( cx \right ){x}^{4}}{4}}+{\frac{3\,{c}^{4}{d}^{3}b\arcsin \left ( cx \right ){x}^{2}}{2}}-{\frac{21\,b{c}^{2}{d}^{3}\arcsin \left ( cx \right ) }{32}}+3\,i{c}^{2}{d}^{3}b{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{{d}^{3}bc}{2\,x}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{i}{2}}{c}^{2}{d}^{3}b-3\,{c}^{2}{d}^{3}b\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +3\,i{c}^{2}{d}^{3}b{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -3\,{c}^{2}{d}^{3}b\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{{c}^{5}{d}^{3}b{x}^{3}}{16}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{21\,{d}^{3}b{c}^{3}x}{32}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{3\,i}{2}}{c}^{2}{d}^{3}b \left ( \arcsin \left ( cx \right ) \right ) ^{2} \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^3,x)

[Out]

-1/4*c^6*d^3*a*x^4+3/2*c^4*d^3*a*x^2-1/2*d^3*a/x^2-3*c^2*d^3*a*ln(c*x)-1/2*d^3*b*arcsin(c*x)/x^2-1/4*c^6*d^3*b
*arcsin(c*x)*x^4+3/2*c^4*d^3*b*arcsin(c*x)*x^2-21/32*b*c^2*d^3*arcsin(c*x)+3*I*c^2*d^3*b*polylog(2,-I*c*x-(-c^
2*x^2+1)^(1/2))-1/2*b*c*d^3*(-c^2*x^2+1)^(1/2)/x+1/2*I*c^2*d^3*b-3*c^2*d^3*b*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+
1)^(1/2))+3*I*c^2*d^3*b*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-3*c^2*d^3*b*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1
/2))-1/16*c^5*d^3*b*(-c^2*x^2+1)^(1/2)*x^3+21/32*b*c^3*d^3*x*(-c^2*x^2+1)^(1/2)+3/2*I*c^2*d^3*b*arcsin(c*x)^2

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

[Out]

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

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

[Out]

-d**3*(Integral(-a/x**3, x) + Integral(3*a*c**2/x, x) + Integral(-3*a*c**4*x, x) + Integral(a*c**6*x**3, x) +
Integral(-b*asin(c*x)/x**3, x) + Integral(3*b*c**2*asin(c*x)/x, x) + Integral(-3*b*c**4*x*asin(c*x), x) + Inte
gral(b*c**6*x**3*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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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