3.95 \(\int x (d+e x^2)^2 (a+b \csc ^{-1}(c x)) \, dx\)

Optimal. Leaf size=195 \[ \frac{\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac{b x \sqrt{c^2 x^2-1} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{6 c^5 \sqrt{c^2 x^2}}+\frac{b c d^3 x \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{6 e \sqrt{c^2 x^2}}+\frac{b e x \left (c^2 x^2-1\right )^{3/2} \left (3 c^2 d+2 e\right )}{18 c^5 \sqrt{c^2 x^2}}+\frac{b e^2 x \left (c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt{c^2 x^2}} \]

[Out]

(b*(3*c^4*d^2 + 3*c^2*d*e + e^2)*x*Sqrt[-1 + c^2*x^2])/(6*c^5*Sqrt[c^2*x^2]) + (b*e*(3*c^2*d + 2*e)*x*(-1 + c^
2*x^2)^(3/2))/(18*c^5*Sqrt[c^2*x^2]) + (b*e^2*x*(-1 + c^2*x^2)^(5/2))/(30*c^5*Sqrt[c^2*x^2]) + ((d + e*x^2)^3*
(a + b*ArcCsc[c*x]))/(6*e) + (b*c*d^3*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*e*Sqrt[c^2*x^2])

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Rubi [A]  time = 0.144222, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {5237, 446, 88, 63, 205} \[ \frac{\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac{b x \sqrt{c^2 x^2-1} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{6 c^5 \sqrt{c^2 x^2}}+\frac{b c d^3 x \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{6 e \sqrt{c^2 x^2}}+\frac{b e x \left (c^2 x^2-1\right )^{3/2} \left (3 c^2 d+2 e\right )}{18 c^5 \sqrt{c^2 x^2}}+\frac{b e^2 x \left (c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x^2)^2*(a + b*ArcCsc[c*x]),x]

[Out]

(b*(3*c^4*d^2 + 3*c^2*d*e + e^2)*x*Sqrt[-1 + c^2*x^2])/(6*c^5*Sqrt[c^2*x^2]) + (b*e*(3*c^2*d + 2*e)*x*(-1 + c^
2*x^2)^(3/2))/(18*c^5*Sqrt[c^2*x^2]) + (b*e^2*x*(-1 + c^2*x^2)^(5/2))/(30*c^5*Sqrt[c^2*x^2]) + ((d + e*x^2)^3*
(a + b*ArcCsc[c*x]))/(6*e) + (b*c*d^3*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*e*Sqrt[c^2*x^2])

Rule 5237

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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 205

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

Rubi steps

\begin{align*} \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac{(b c x) \int \frac{\left (d+e x^2\right )^3}{x \sqrt{-1+c^2 x^2}} \, dx}{6 e \sqrt{c^2 x^2}}\\ &=\frac{\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac{(b c x) \operatorname{Subst}\left (\int \frac{(d+e x)^3}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{12 e \sqrt{c^2 x^2}}\\ &=\frac{\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac{(b c x) \operatorname{Subst}\left (\int \left (\frac{e \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{c^4 \sqrt{-1+c^2 x}}+\frac{d^3}{x \sqrt{-1+c^2 x}}+\frac{e^2 \left (3 c^2 d+2 e\right ) \sqrt{-1+c^2 x}}{c^4}+\frac{e^3 \left (-1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{12 e \sqrt{c^2 x^2}}\\ &=\frac{b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt{-1+c^2 x^2}}{6 c^5 \sqrt{c^2 x^2}}+\frac{b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt{c^2 x^2}}+\frac{b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac{\left (b c d^3 x\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{12 e \sqrt{c^2 x^2}}\\ &=\frac{b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt{-1+c^2 x^2}}{6 c^5 \sqrt{c^2 x^2}}+\frac{b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt{c^2 x^2}}+\frac{b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac{\left (b d^3 x\right ) \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 c e \sqrt{c^2 x^2}}\\ &=\frac{b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt{-1+c^2 x^2}}{6 c^5 \sqrt{c^2 x^2}}+\frac{b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt{c^2 x^2}}+\frac{b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac{b c d^3 x \tan ^{-1}\left (\sqrt{-1+c^2 x^2}\right )}{6 e \sqrt{c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.28052, size = 124, normalized size = 0.64 \[ \frac{1}{90} x \left (15 a x \left (3 d^2+3 d e x^2+e^2 x^4\right )+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} \left (3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )+2 c^2 e \left (15 d+2 e x^2\right )+8 e^2\right )}{c^5}+15 b x \csc ^{-1}(c x) \left (3 d^2+3 d e x^2+e^2 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(d + e*x^2)^2*(a + b*ArcCsc[c*x]),x]

[Out]

(x*(15*a*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) + (b*Sqrt[1 - 1/(c^2*x^2)]*(8*e^2 + 2*c^2*e*(15*d + 2*e*x^2) + 3*c^4*
(15*d^2 + 5*d*e*x^2 + e^2*x^4)))/c^5 + 15*b*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*ArcCsc[c*x]))/90

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Maple [A]  time = 0.188, size = 182, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{6}{x}^{6}}{6}}+{\frac{{c}^{6}{x}^{4}de}{2}}+{\frac{{x}^{2}{c}^{6}{d}^{2}}{2}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{{\rm arccsc} \left (cx\right ){e}^{2}{c}^{6}{x}^{6}}{6}}+{\frac{{\rm arccsc} \left (cx\right ){c}^{6}{x}^{4}de}{2}}+{\frac{{\rm arccsc} \left (cx\right ){c}^{6}{x}^{2}{d}^{2}}{2}}+{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 3\,{c}^{4}{e}^{2}{x}^{4}+15\,{c}^{4}de{x}^{2}+45\,{d}^{2}{c}^{4}+4\,{c}^{2}{e}^{2}{x}^{2}+30\,{c}^{2}ed+8\,{e}^{2} \right ) }{90\,cx}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x^2+d)^2*(a+b*arccsc(c*x)),x)

[Out]

1/c^2*(a/c^4*(1/6*e^2*c^6*x^6+1/2*c^6*x^4*d*e+1/2*x^2*c^6*d^2)+b/c^4*(1/6*arccsc(c*x)*e^2*c^6*x^6+1/2*arccsc(c
*x)*c^6*x^4*d*e+1/2*arccsc(c*x)*c^6*x^2*d^2+1/90*(c^2*x^2-1)*(3*c^4*e^2*x^4+15*c^4*d*e*x^2+45*c^4*d^2+4*c^2*e^
2*x^2+30*c^2*d*e+8*e^2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x))

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Maxima [A]  time = 1.00152, size = 255, normalized size = 1.31 \begin{align*} \frac{1}{6} \, a e^{2} x^{6} + \frac{1}{2} \, a d e x^{4} + \frac{1}{2} \, a d^{2} x^{2} + \frac{1}{2} \,{\left (x^{2} \operatorname{arccsc}\left (c x\right ) + \frac{x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c}\right )} b d^{2} + \frac{1}{6} \,{\left (3 \, x^{4} \operatorname{arccsc}\left (c x\right ) + \frac{c^{2} x^{3}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 3 \, x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d e + \frac{1}{90} \,{\left (15 \, x^{6} \operatorname{arccsc}\left (c x\right ) + \frac{3 \, c^{4} x^{5}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + 10 \, c^{2} x^{3}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 15 \, x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x^2+d)^2*(a+b*arccsc(c*x)),x, algorithm="maxima")

[Out]

1/6*a*e^2*x^6 + 1/2*a*d*e*x^4 + 1/2*a*d^2*x^2 + 1/2*(x^2*arccsc(c*x) + x*sqrt(-1/(c^2*x^2) + 1)/c)*b*d^2 + 1/6
*(3*x^4*arccsc(c*x) + (c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 3*x*sqrt(-1/(c^2*x^2) + 1))/c^3)*b*d*e + 1/90*(15*x^
6*arccsc(c*x) + (3*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 10*c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 15*x*sqrt(-1/(c^2*
x^2) + 1))/c^5)*b*e^2

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Fricas [A]  time = 3.04265, size = 336, normalized size = 1.72 \begin{align*} \frac{15 \, a c^{6} e^{2} x^{6} + 45 \, a c^{6} d e x^{4} + 45 \, a c^{6} d^{2} x^{2} + 15 \,{\left (b c^{6} e^{2} x^{6} + 3 \, b c^{6} d e x^{4} + 3 \, b c^{6} d^{2} x^{2}\right )} \operatorname{arccsc}\left (c x\right ) +{\left (3 \, b c^{4} e^{2} x^{4} + 45 \, b c^{4} d^{2} + 30 \, b c^{2} d e + 8 \, b e^{2} +{\left (15 \, b c^{4} d e + 4 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{90 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x^2+d)^2*(a+b*arccsc(c*x)),x, algorithm="fricas")

[Out]

1/90*(15*a*c^6*e^2*x^6 + 45*a*c^6*d*e*x^4 + 45*a*c^6*d^2*x^2 + 15*(b*c^6*e^2*x^6 + 3*b*c^6*d*e*x^4 + 3*b*c^6*d
^2*x^2)*arccsc(c*x) + (3*b*c^4*e^2*x^4 + 45*b*c^4*d^2 + 30*b*c^2*d*e + 8*b*e^2 + (15*b*c^4*d*e + 4*b*c^2*e^2)*
x^2)*sqrt(c^2*x^2 - 1))/c^6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x**2+d)**2*(a+b*acsc(c*x)),x)

[Out]

Integral(x*(a + b*acsc(c*x))*(d + e*x**2)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x^2+d)^2*(a+b*arccsc(c*x)),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^2*(b*arccsc(c*x) + a)*x, x)