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

Optimal. Leaf size=138 \[ \frac{\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac{b c d^2 x \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{4 e \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left (2 c^2 d+e\right )}{4 c^3 \sqrt{c^2 x^2}}+\frac{b e x \left (c^2 x^2-1\right )^{3/2}}{12 c^3 \sqrt{c^2 x^2}} \]

[Out]

(b*(2*c^2*d + e)*x*Sqrt[-1 + c^2*x^2])/(4*c^3*Sqrt[c^2*x^2]) + (b*e*x*(-1 + c^2*x^2)^(3/2))/(12*c^3*Sqrt[c^2*x
^2]) + ((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/(4*e) + (b*c*d^2*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(4*e*Sqrt[c^2*x^2])

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Rubi [A]  time = 0.0957639, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {5237, 446, 88, 63, 205} \[ \frac{\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac{b c d^2 x \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{4 e \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left (2 c^2 d+e\right )}{4 c^3 \sqrt{c^2 x^2}}+\frac{b e x \left (c^2 x^2-1\right )^{3/2}}{12 c^3 \sqrt{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(2*c^2*d + e)*x*Sqrt[-1 + c^2*x^2])/(4*c^3*Sqrt[c^2*x^2]) + (b*e*x*(-1 + c^2*x^2)^(3/2))/(12*c^3*Sqrt[c^2*x
^2]) + ((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/(4*e) + (b*c*d^2*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(4*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 ) \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac{(b c x) \int \frac{\left (d+e x^2\right )^2}{x \sqrt{-1+c^2 x^2}} \, dx}{4 e \sqrt{c^2 x^2}}\\ &=\frac{\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac{(b c x) \operatorname{Subst}\left (\int \frac{(d+e x)^2}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{8 e \sqrt{c^2 x^2}}\\ &=\frac{\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac{(b c x) \operatorname{Subst}\left (\int \left (\frac{e \left (2 c^2 d+e\right )}{c^2 \sqrt{-1+c^2 x}}+\frac{d^2}{x \sqrt{-1+c^2 x}}+\frac{e^2 \sqrt{-1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{8 e \sqrt{c^2 x^2}}\\ &=\frac{b \left (2 c^2 d+e\right ) x \sqrt{-1+c^2 x^2}}{4 c^3 \sqrt{c^2 x^2}}+\frac{b e x \left (-1+c^2 x^2\right )^{3/2}}{12 c^3 \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac{\left (b c d^2 x\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{8 e \sqrt{c^2 x^2}}\\ &=\frac{b \left (2 c^2 d+e\right ) x \sqrt{-1+c^2 x^2}}{4 c^3 \sqrt{c^2 x^2}}+\frac{b e x \left (-1+c^2 x^2\right )^{3/2}}{12 c^3 \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac{\left (b d^2 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 )}{4 c e \sqrt{c^2 x^2}}\\ &=\frac{b \left (2 c^2 d+e\right ) x \sqrt{-1+c^2 x^2}}{4 c^3 \sqrt{c^2 x^2}}+\frac{b e x \left (-1+c^2 x^2\right )^{3/2}}{12 c^3 \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac{b c d^2 x \tan ^{-1}\left (\sqrt{-1+c^2 x^2}\right )}{4 e \sqrt{c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0952709, size = 78, normalized size = 0.57 \[ \frac{x \left (3 a c^3 x \left (2 d+e x^2\right )+b \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 \left (6 d+e x^2\right )+2 e\right )+3 b c^3 x \csc ^{-1}(c x) \left (2 d+e x^2\right )\right )}{12 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(3*a*c^3*x*(2*d + e*x^2) + b*Sqrt[1 - 1/(c^2*x^2)]*(2*e + c^2*(6*d + e*x^2)) + 3*b*c^3*x*(2*d + e*x^2)*ArcC
sc[c*x]))/(12*c^3)

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Maple [A]  time = 0.178, size = 115, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{4}{x}^{4}}{4}}+{\frac{{x}^{2}{c}^{4}d}{2}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arccsc} \left (cx\right )e{c}^{4}{x}^{4}}{4}}+{\frac{{\rm arccsc} \left (cx\right )d{c}^{4}{x}^{2}}{2}}+{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ({c}^{2}e{x}^{2}+6\,{c}^{2}d+2\,e \right ) }{12\,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)*(a+b*arccsc(c*x)),x)

[Out]

1/c^2*(a/c^2*(1/4*e*c^4*x^4+1/2*x^2*c^4*d)+b/c^2*(1/4*arccsc(c*x)*e*c^4*x^4+1/2*arccsc(c*x)*d*c^4*x^2+1/12*(c^
2*x^2-1)*(c^2*e*x^2+6*c^2*d+2*e)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x))

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Maxima [A]  time = 1.00324, size = 132, normalized size = 0.96 \begin{align*} \frac{1}{4} \, a e x^{4} + \frac{1}{2} \, a d 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 + \frac{1}{12} \,{\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 e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*e*x^4 + 1/2*a*d*x^2 + 1/2*(x^2*arccsc(c*x) + x*sqrt(-1/(c^2*x^2) + 1)/c)*b*d + 1/12*(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*e

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Fricas [A]  time = 2.78292, size = 192, normalized size = 1.39 \begin{align*} \frac{3 \, a c^{4} e x^{4} + 6 \, a c^{4} d x^{2} + 3 \,{\left (b c^{4} e x^{4} + 2 \, b c^{4} d x^{2}\right )} \operatorname{arccsc}\left (c x\right ) +{\left (b c^{2} e x^{2} + 6 \, b c^{2} d + 2 \, b e\right )} \sqrt{c^{2} x^{2} - 1}}{12 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*a*c^4*e*x^4 + 6*a*c^4*d*x^2 + 3*(b*c^4*e*x^4 + 2*b*c^4*d*x^2)*arccsc(c*x) + (b*c^2*e*x^2 + 6*b*c^2*d +
2*b*e)*sqrt(c^2*x^2 - 1))/c^4

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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 )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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