Integrand size = 19, antiderivative size = 196 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (4 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b \left (4 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right ) \]
1/6*d*x^6*(a+b*arccsc(c*x))+1/8*e*x^8*(a+b*arccsc(c*x))+1/72*b*(8*c^2*d+9* e)*x*(c^2*x^2-1)^(3/2)/c^7/(c^2*x^2)^(1/2)+1/120*b*(4*c^2*d+9*e)*x*(c^2*x^ 2-1)^(5/2)/c^7/(c^2*x^2)^(1/2)+1/56*b*e*x*(c^2*x^2-1)^(7/2)/c^7/(c^2*x^2)^ (1/2)+1/24*b*(4*c^2*d+3*e)*x*(c^2*x^2-1)^(1/2)/c^7/(c^2*x^2)^(1/2)
Time = 0.17 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.59 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {x \left (105 a x^5 \left (4 d+3 e x^2\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (144 e+8 c^2 \left (28 d+9 e x^2\right )+2 c^4 \left (56 d x^2+27 e x^4\right )+c^6 \left (84 d x^4+45 e x^6\right )\right )}{c^7}+105 b x^5 \left (4 d+3 e x^2\right ) \csc ^{-1}(c x)\right )}{2520} \]
(x*(105*a*x^5*(4*d + 3*e*x^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*(144*e + 8*c^2*(2 8*d + 9*e*x^2) + 2*c^4*(56*d*x^2 + 27*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
Time = 0.37 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.82, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5762, 27, 354, 86, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 5762 |
| \(\displaystyle \frac {b c x \int \frac {x^5 \left (3 e x^2+4 d\right )}{24 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {x^5 \left (3 e x^2+4 d\right )}{\sqrt {c^2 x^2-1}}dx}{24 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {b c x \int \frac {x^4 \left (3 e x^2+4 d\right )}{\sqrt {c^2 x^2-1}}dx^2}{48 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {b c x \int \left (\frac {3 e \left (c^2 x^2-1\right )^{5/2}}{c^6}+\frac {\left (4 d c^2+9 e\right ) \left (c^2 x^2-1\right )^{3/2}}{c^6}+\frac {\left (8 d c^2+9 e\right ) \sqrt {c^2 x^2-1}}{c^6}+\frac {4 d c^2+3 e}{c^6 \sqrt {c^2 x^2-1}}\right )dx^2}{48 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {b c x \left (\frac {2 \left (c^2 x^2-1\right )^{5/2} \left (4 c^2 d+9 e\right )}{5 c^8}+\frac {2 \left (c^2 x^2-1\right )^{3/2} \left (8 c^2 d+9 e\right )}{3 c^8}+\frac {2 \sqrt {c^2 x^2-1} \left (4 c^2 d+3 e\right )}{c^8}+\frac {6 e \left (c^2 x^2-1\right )^{7/2}}{7 c^8}\right )}{48 \sqrt {c^2 x^2}}\) |
(b*c*x*((2*(4*c^2*d + 3*e)*Sqrt[-1 + c^2*x^2])/c^8 + (2*(8*c^2*d + 9*e)*(- 1 + c^2*x^2)^(3/2))/(3*c^8) + (2*(4*c^2*d + 9*e)*(-1 + c^2*x^2)^(5/2))/(5* c^8) + (6*e*(-1 + c^2*x^2)^(7/2))/(7*c^8)))/(48*Sqrt[c^2*x^2]) + (d*x^6*(a + b*ArcCsc[c*x]))/6 + (e*x^8*(a + b*ArcCsc[c*x]))/8
3.1.83.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((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]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
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]}, Sim p[(a + b*ArcCsc[c*x]) u, x] + Simp[b*c*(x/Sqrt[c^2*x^2]) Int[SimplifyIn tegrand[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])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
Time = 0.64 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.71
| method | result | size |
| parts | \(a \left (\frac {1}{8} e \,x^{8}+\frac {1}{6} x^{6} d \right )+\frac {b \left (\frac {c^{6} \operatorname {arccsc}\left (c x \right ) e \,x^{8}}{8}+\frac {\operatorname {arccsc}\left (c x \right ) d \,x^{6} c^{6}}{6}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )}{c^{6}}\) | \(139\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccsc}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\operatorname {arccsc}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}}\) | \(152\) |
| default | \(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccsc}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\operatorname {arccsc}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}}\) | \(152\) |
a*(1/8*e*x^8+1/6*x^6*d)+b/c^6*(1/8*c^6*arccsc(c*x)*e*x^8+1/6*arccsc(c*x)*d *x^6*c^6+1/2520/c^3*(c^2*x^2-1)*(45*c^6*e*x^6+84*c^6*d*x^4+54*c^4*e*x^4+11 2*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)/x)
Time = 0.31 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.65 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {315 \, a c^{8} e x^{8} + 420 \, a c^{8} d x^{6} + 105 \, {\left (3 \, b c^{8} e x^{8} + 4 \, b c^{8} d x^{6}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (45 \, b c^{6} e x^{6} + 6 \, {\left (14 \, b c^{6} d + 9 \, b c^{4} e\right )} x^{4} + 224 \, b c^{2} d + 8 \, {\left (14 \, b c^{4} d + 9 \, b c^{2} e\right )} x^{2} + 144 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{2520 \, c^{8}} \]
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
Time = 3.86 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.86 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d x^{6}}{6} + \frac {a e x^{8}}{8} + \frac {b d x^{6} \operatorname {acsc}{\left (c x \right )}}{6} + \frac {b e x^{8} \operatorname {acsc}{\left (c x \right )}}{8} + \frac {b d \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} + \frac {b e \left (\begin {cases} \frac {x^{6} \sqrt {c^{2} x^{2} - 1}}{7 c} + \frac {6 x^{4} \sqrt {c^{2} x^{2} - 1}}{35 c^{3}} + \frac {8 x^{2} \sqrt {c^{2} x^{2} - 1}}{35 c^{5}} + \frac {16 \sqrt {c^{2} x^{2} - 1}}{35 c^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{6} \sqrt {- c^{2} x^{2} + 1}}{7 c} + \frac {6 i x^{4} \sqrt {- c^{2} x^{2} + 1}}{35 c^{3}} + \frac {8 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{5}} + \frac {16 i \sqrt {- c^{2} x^{2} + 1}}{35 c^{7}} & \text {otherwise} \end {cases}\right )}{8 c} \]
a*d*x**6/6 + a*e*x**8/8 + b*d*x**6*acsc(c*x)/6 + b*e*x**8*acsc(c*x)/8 + b* d*Piecewise((x**4*sqrt(c**2*x**2 - 1)/(5*c) + 4*x**2*sqrt(c**2*x**2 - 1)/( 15*c**3) + 8*sqrt(c**2*x**2 - 1)/(15*c**5), Abs(c**2*x**2) > 1), (I*x**4*s qrt(-c**2*x**2 + 1)/(5*c) + 4*I*x**2*sqrt(-c**2*x**2 + 1)/(15*c**3) + 8*I* sqrt(-c**2*x**2 + 1)/(15*c**5), True))/(6*c) + b*e*Piecewise((x**6*sqrt(c* *2*x**2 - 1)/(7*c) + 6*x**4*sqrt(c**2*x**2 - 1)/(35*c**3) + 8*x**2*sqrt(c* *2*x**2 - 1)/(35*c**5) + 16*sqrt(c**2*x**2 - 1)/(35*c**7), Abs(c**2*x**2) > 1), (I*x**6*sqrt(-c**2*x**2 + 1)/(7*c) + 6*I*x**4*sqrt(-c**2*x**2 + 1)/( 35*c**3) + 8*I*x**2*sqrt(-c**2*x**2 + 1)/(35*c**5) + 16*I*sqrt(-c**2*x**2 + 1)/(35*c**7), True))/(8*c)
Time = 0.19 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.93 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{8} \, a e x^{8} + \frac {1}{6} \, a d x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arccsc}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} + 21 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 35 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e \]
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
Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (168) = 336\).
Time = 0.43 (sec) , antiderivative size = 1244, normalized size of antiderivative = 6.35 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
1/645120*(315*b*e*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8*arcsin(1/(c*x))/c + 3 15*a*e*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8/c + 90*b*e*x^7*(sqrt(-1/(c^2*x^2 ) + 1) + 1)^7/c^2 + 1680*b*d*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*arcsin(1/( c*x))/c + 1680*a*d*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6/c + 2520*b*e*x^6*(sq rt(-1/(c^2*x^2) + 1) + 1)^6*arcsin(1/(c*x))/c^3 + 2520*a*e*x^6*(sqrt(-1/(c ^2*x^2) + 1) + 1)^6/c^3 + 672*b*d*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c^2 + 882*b*e*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c^4 + 10080*b*d*x^4*(sqrt(-1/( c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))/c^3 + 10080*a*d*x^4*(sqrt(-1/(c^2*x^2 ) + 1) + 1)^4/c^3 + 8820*b*e*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/( c*x))/c^5 + 8820*a*e*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^5 + 5600*b*d*x^3 *(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^4 + 4410*b*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^6 + 25200*b*d*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x)) /c^5 + 25200*a*d*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^5 + 17640*b*e*x^2*(s qrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c^7 + 17640*a*e*x^2*(sqrt(-1/ (c^2*x^2) + 1) + 1)^2/c^7 + 33600*b*d*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 + 22050*b*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^8 + 33600*b*d*arcsin(1/(c*x))/ c^7 + 33600*a*d/c^7 + 22050*b*e*arcsin(1/(c*x))/c^9 + 22050*a*e/c^9 - 3360 0*b*d/(c^8*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) - 22050*b*e/(c^10*x*(sqrt(-1/(c ^2*x^2) + 1) + 1)) + 25200*b*d*arcsin(1/(c*x))/(c^9*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 25200*a*d/(c^9*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 17...
Timed out. \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^5\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]