3.83 \(\int x^5 (d+e x^2) (a+b \csc ^{-1}(c x)) \, dx\)

Optimal. Leaf size=196 \[ \frac{1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x \left (c^2 x^2-1\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^7 \sqrt{c^2 x^2}}+\frac{b x \left (c^2 x^2-1\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^7 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left (4 c^2 d+3 e\right )}{24 c^7 \sqrt{c^2 x^2}}+\frac{b e x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt{c^2 x^2}} \]

[Out]

(b*(4*c^2*d + 3*e)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2]) + (b*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(3/2))/(
72*c^7*Sqrt[c^2*x^2]) + (b*(4*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*Sqrt[c^2*x^2]) + (b*e*x*(-1 + c^2*
x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d*x^6*(a + b*ArcCsc[c*x]))/6 + (e*x^8*(a + b*ArcCsc[c*x]))/8

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Rubi [A]  time = 0.150333, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {14, 5239, 12, 446, 77} \[ \frac{1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x \left (c^2 x^2-1\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^7 \sqrt{c^2 x^2}}+\frac{b x \left (c^2 x^2-1\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^7 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left (4 c^2 d+3 e\right )}{24 c^7 \sqrt{c^2 x^2}}+\frac{b e x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(d + e*x^2)*(a + b*ArcCsc[c*x]),x]

[Out]

(b*(4*c^2*d + 3*e)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2]) + (b*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(3/2))/(
72*c^7*Sqrt[c^2*x^2]) + (b*(4*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*Sqrt[c^2*x^2]) + (b*e*x*(-1 + c^2*
x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d*x^6*(a + b*ArcCsc[c*x]))/6 + (e*x^8*(a + b*ArcCsc[c*x]))/8

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5239

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsc[c*x], u, x] + Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac{1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \int \frac{x^5 \left (4 d+3 e x^2\right )}{24 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=\frac{1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \int \frac{x^5 \left (4 d+3 e x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{24 \sqrt{c^2 x^2}}\\ &=\frac{1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \operatorname{Subst}\left (\int \frac{x^2 (4 d+3 e x)}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{48 \sqrt{c^2 x^2}}\\ &=\frac{1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \operatorname{Subst}\left (\int \left (\frac{4 c^2 d+3 e}{c^6 \sqrt{-1+c^2 x}}+\frac{\left (8 c^2 d+9 e\right ) \sqrt{-1+c^2 x}}{c^6}+\frac{\left (4 c^2 d+9 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac{3 e \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{48 \sqrt{c^2 x^2}}\\ &=\frac{b \left (4 c^2 d+3 e\right ) x \sqrt{-1+c^2 x^2}}{24 c^7 \sqrt{c^2 x^2}}+\frac{b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt{c^2 x^2}}+\frac{b \left (4 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt{c^2 x^2}}+\frac{b e x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt{c^2 x^2}}+\frac{1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.226948, size = 115, normalized size = 0.59 \[ \frac{x \left (105 a x^5 \left (4 d+3 e x^2\right )+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} \left (c^6 \left (84 d x^4+45 e x^6\right )+2 c^4 \left (56 d x^2+27 e x^4\right )+8 c^2 \left (28 d+9 e x^2\right )+144 e\right )}{c^7}+105 b x^5 \csc ^{-1}(c x) \left (4 d+3 e x^2\right )\right )}{2520} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(d + e*x^2)*(a + b*ArcCsc[c*x]),x]

[Out]

(x*(105*a*x^5*(4*d + 3*e*x^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*(144*e + 8*c^2*(28*d + 9*e*x^2) + 2*c^4*(56*d*x^2 + 2
7*e*x^4) + c^6*(84*d*x^4 + 45*e*x^6)))/c^7 + 105*b*x^5*(4*d + 3*e*x^2)*ArcCsc[c*x]))/2520

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Maple [A]  time = 0.182, size = 152, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{6}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{8}{x}^{6}d}{6}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arccsc} \left (cx\right )e{c}^{8}{x}^{8}}{8}}+{\frac{{\rm arccsc} \left (cx\right ){c}^{8}{x}^{6}d}{6}}+{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 45\,{c}^{6}e{x}^{6}+84\,{c}^{6}d{x}^{4}+54\,{c}^{4}e{x}^{4}+112\,{c}^{4}d{x}^{2}+72\,{c}^{2}e{x}^{2}+224\,{c}^{2}d+144\,e \right ) }{2520\,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^5*(e*x^2+d)*(a+b*arccsc(c*x)),x)

[Out]

1/c^6*(a/c^2*(1/8*e*c^8*x^8+1/6*c^8*x^6*d)+b/c^2*(1/8*arccsc(c*x)*e*c^8*x^8+1/6*arccsc(c*x)*c^8*x^6*d+1/2520*(
c^2*x^2-1)*(45*c^6*e*x^6+84*c^6*d*x^4+54*c^4*e*x^4+112*c^4*d*x^2+72*c^2*e*x^2+224*c^2*d+144*e)/((c^2*x^2-1)/c^
2/x^2)^(1/2)/c/x))

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Maxima [A]  time = 0.979867, size = 247, normalized size = 1.26 \begin{align*} \frac{1}{8} \, a e x^{8} + \frac{1}{6} \, a d x^{6} + \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 d + \frac{1}{280} \,{\left (35 \, x^{8} \operatorname{arccsc}\left (c x\right ) + \frac{5 \, c^{6} x^{7}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{7}{2}} + 21 \, c^{4} x^{5}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + 35 \, c^{2} x^{3}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 35 \, x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*arccsc(c*x)),x, algorithm="maxima")

[Out]

1/8*a*e*x^8 + 1/6*a*d*x^6 + 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*d + 1/280*(35*x^8*arccsc(c*x) + (5*c^6*x^7*(-1/(c^2*x
^2) + 1)^(7/2) + 21*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 35*c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 35*x*sqrt(-1/(c^2
*x^2) + 1))/c^7)*b*e

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Fricas [A]  time = 3.00844, size = 304, normalized size = 1.55 \begin{align*} \frac{315 \, a c^{8} e x^{8} + 420 \, a c^{8} d x^{6} + 105 \,{\left (3 \, b c^{8} e x^{8} + 4 \, b c^{8} d x^{6}\right )} \operatorname{arccsc}\left (c x\right ) +{\left (45 \, b c^{6} e x^{6} + 6 \,{\left (14 \, b c^{6} d + 9 \, b c^{4} e\right )} x^{4} + 224 \, b c^{2} d + 8 \,{\left (14 \, b c^{4} d + 9 \, b c^{2} e\right )} x^{2} + 144 \, b e\right )} \sqrt{c^{2} x^{2} - 1}}{2520 \, c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*arccsc(c*x)),x, algorithm="fricas")

[Out]

1/2520*(315*a*c^8*e*x^8 + 420*a*c^8*d*x^6 + 105*(3*b*c^8*e*x^8 + 4*b*c^8*d*x^6)*arccsc(c*x) + (45*b*c^6*e*x^6
+ 6*(14*b*c^6*d + 9*b*c^4*e)*x^4 + 224*b*c^2*d + 8*(14*b*c^4*d + 9*b*c^2*e)*x^2 + 144*b*e)*sqrt(c^2*x^2 - 1))/
c^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \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**5*(e*x**2+d)*(a+b*acsc(c*x)),x)

[Out]

Integral(x**5*(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^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*arccsc(c*x)),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*(b*arccsc(c*x) + a)*x^5, x)