Integrand size = 22, antiderivative size = 85 \[ \int \frac {(e x)^{-1+n}}{a+b \csc \left (c+d x^n\right )} \, dx=\frac {(e x)^n}{a e n}+\frac {2 b x^{-n} (e x)^n \text {arctanh}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d e n} \]
(e*x)^n/a/e/n+2*b*(e*x)^n*arctanh((a+b*tan(1/2*c+1/2*d*x^n))/(a^2-b^2)^(1/ 2))/a/d/e/n/(x^n)/(a^2-b^2)^(1/2)
Time = 0.92 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93 \[ \int \frac {(e x)^{-1+n}}{a+b \csc \left (c+d x^n\right )} \, dx=\frac {(e x)^n \left (d+c x^{-n}-\frac {2 b x^{-n} \arctan \left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}\right )}{a d e n} \]
((e*x)^n*(d + c/x^n - (2*b*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b ^2]])/(Sqrt[-a^2 + b^2]*x^n)))/(a*d*e*n)
Time = 0.42 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4697, 4693, 3042, 4270, 3042, 3139, 1083, 219}
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.
\(\displaystyle \int \frac {(e x)^{n-1}}{a+b \csc \left (c+d x^n\right )} \, dx\) |
\(\Big \downarrow \) 4697 |
\(\displaystyle \frac {x^{-n} (e x)^n \int \frac {x^{n-1}}{a+b \csc \left (d x^n+c\right )}dx}{e}\) |
\(\Big \downarrow \) 4693 |
\(\displaystyle \frac {x^{-n} (e x)^n \int \frac {1}{a+b \csc \left (d x^n+c\right )}dx^n}{e n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x^{-n} (e x)^n \int \frac {1}{a+b \csc \left (d x^n+c\right )}dx^n}{e n}\) |
\(\Big \downarrow \) 4270 |
\(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {x^n}{a}-\frac {\int \frac {1}{\frac {a \sin \left (d x^n+c\right )}{b}+1}dx^n}{a}\right )}{e n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {x^n}{a}-\frac {\int \frac {1}{\frac {a \sin \left (d x^n+c\right )}{b}+1}dx^n}{a}\right )}{e n}\) |
\(\Big \downarrow \) 3139 |
\(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {x^n}{a}-\frac {2 \int \frac {1}{x^{2 n}+\frac {2 a \tan \left (\frac {1}{2} \left (d x^n+c\right )\right )}{b}+1}d\tan \left (\frac {1}{2} \left (d x^n+c\right )\right )}{a d}\right )}{e n}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {4 \int \frac {1}{-x^{2 n}-4 \left (1-\frac {a^2}{b^2}\right )}d\left (\frac {2 a}{b}+2 \tan \left (\frac {1}{2} \left (d x^n+c\right )\right )\right )}{a d}+\frac {x^n}{a}\right )}{e n}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {2 b \text {arctanh}\left (\frac {b \left (\frac {2 a}{b}+2 \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}+\frac {x^n}{a}\right )}{e n}\) |
((e*x)^n*(x^n/a + (2*b*ArcTanh[(b*((2*a)/b + 2*Tan[(c + d*x^n)/2]))/(2*Sqr t[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d)))/(e*n*x^n)
3.1.79.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(-1), x_Symbol] :> Simp[x/a, x] - Simp[1/a Int[1/(1 + (a/b)*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_.), x _Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m]) Int[x^m*( a + b*Csc[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.09 (sec) , antiderivative size = 315, normalized size of antiderivative = 3.71
method | result | size |
risch | \(\frac {x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{a n}-\frac {2 i \arctan \left (\frac {2 i a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}-2 \,{\mathrm e}^{i c} b}{2 \sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}\right ) e^{n} b \,{\mathrm e}^{\frac {i \left (-\pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+\pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+\pi n \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-\pi n \operatorname {csgn}\left (i e x \right )^{3}+\pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )-\pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}+\pi \operatorname {csgn}\left (i e x \right )^{3}+2 c \right )}{2}}}{\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}\, d e n a}\) | \(315\) |
1/a/n*x*exp(1/2*(-1+n)*(-I*csgn(I*e)*csgn(I*x)*csgn(I*e*x)*Pi+I*csgn(I*e)* csgn(I*e*x)^2*Pi+I*csgn(I*x)*csgn(I*e*x)^2*Pi-I*csgn(I*e*x)^3*Pi+2*ln(e)+2 *ln(x)))-2*I*arctan(1/2*(2*I*a*exp(I*(d*x^n+2*c))-2*exp(I*c)*b)/(a^2*exp(2 *I*c)-exp(2*I*c)*b^2)^(1/2))/(a^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2)/d/e*e^n /n*b/a*exp(1/2*I*(-Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*n*csgn(I*e)*csg n(I*e*x)^2+Pi*n*csgn(I*x)*csgn(I*e*x)^2-Pi*n*csgn(I*e*x)^3+Pi*csgn(I*e)*cs gn(I*x)*csgn(I*e*x)-Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*x)*csgn(I*e*x)^2+ Pi*csgn(I*e*x)^3+2*c))
Time = 0.29 (sec) , antiderivative size = 301, normalized size of antiderivative = 3.54 \[ \int \frac {(e x)^{-1+n}}{a+b \csc \left (c+d x^n\right )} \, dx=\left [\frac {2 \, {\left (a^{2} - b^{2}\right )} d e^{n - 1} x^{n} + \sqrt {a^{2} - b^{2}} b e^{n - 1} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x^{n} + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} a \cos \left (d x^{n} + c\right ) + a^{2} + b^{2} + 2 \, {\left (\sqrt {a^{2} - b^{2}} b \cos \left (d x^{n} + c\right ) + a b\right )} \sin \left (d x^{n} + c\right )}{a^{2} \cos \left (d x^{n} + c\right )^{2} - 2 \, a b \sin \left (d x^{n} + c\right ) - a^{2} - b^{2}}\right )}{2 \, {\left (a^{3} - a b^{2}\right )} d n}, \frac {{\left (a^{2} - b^{2}\right )} d e^{n - 1} x^{n} + \sqrt {-a^{2} + b^{2}} b e^{n - 1} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} b \sin \left (d x^{n} + c\right ) + \sqrt {-a^{2} + b^{2}} a}{{\left (a^{2} - b^{2}\right )} \cos \left (d x^{n} + c\right )}\right )}{{\left (a^{3} - a b^{2}\right )} d n}\right ] \]
[1/2*(2*(a^2 - b^2)*d*e^(n - 1)*x^n + sqrt(a^2 - b^2)*b*e^(n - 1)*log(((a^ 2 - 2*b^2)*cos(d*x^n + c)^2 + 2*sqrt(a^2 - b^2)*a*cos(d*x^n + c) + a^2 + b ^2 + 2*(sqrt(a^2 - b^2)*b*cos(d*x^n + c) + a*b)*sin(d*x^n + c))/(a^2*cos(d *x^n + c)^2 - 2*a*b*sin(d*x^n + c) - a^2 - b^2)))/((a^3 - a*b^2)*d*n), ((a ^2 - b^2)*d*e^(n - 1)*x^n + sqrt(-a^2 + b^2)*b*e^(n - 1)*arctan(-(sqrt(-a^ 2 + b^2)*b*sin(d*x^n + c) + sqrt(-a^2 + b^2)*a)/((a^2 - b^2)*cos(d*x^n + c ))))/((a^3 - a*b^2)*d*n)]
\[ \int \frac {(e x)^{-1+n}}{a+b \csc \left (c+d x^n\right )} \, dx=\int \frac {\left (e x\right )^{n - 1}}{a + b \csc {\left (c + d x^{n} \right )}}\, dx \]
\[ \int \frac {(e x)^{-1+n}}{a+b \csc \left (c+d x^n\right )} \, dx=\int { \frac {\left (e x\right )^{n - 1}}{b \csc \left (d x^{n} + c\right ) + a} \,d x } \]
-(2*a*b*e^(n + 1)*n*integrate((2*b*x^n*cos(d*x^n + c)^2 + a*x^n*cos(d*x^n + c)*sin(2*d*x^n + 2*c) - a*x^n*cos(2*d*x^n + 2*c)*sin(d*x^n + c) + 2*b*x^ n*sin(d*x^n + c)^2 + a*x^n*sin(d*x^n + c))/(a^3*e*x*cos(2*d*x^n + 2*c)^2 + 4*a*b^2*e*x*cos(d*x^n + c)^2 + 4*a^2*b*e*x*cos(d*x^n + c)*sin(2*d*x^n + 2 *c) + a^3*e*x*sin(2*d*x^n + 2*c)^2 + 4*a*b^2*e*x*sin(d*x^n + c)^2 + 4*a^2* b*e*x*sin(d*x^n + c) + a^3*e*x - 2*(2*a^2*b*e*x*sin(d*x^n + c) + a^3*e*x)* cos(2*d*x^n + 2*c)), x) - e^n*x^n)/(a*e*n)
\[ \int \frac {(e x)^{-1+n}}{a+b \csc \left (c+d x^n\right )} \, dx=\int { \frac {\left (e x\right )^{n - 1}}{b \csc \left (d x^{n} + c\right ) + a} \,d x } \]
Time = 19.77 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.69 \[ \int \frac {(e x)^{-1+n}}{a+b \csc \left (c+d x^n\right )} \, dx=\frac {x\,{\left (e\,x\right )}^{n-1}}{a\,n}-\frac {b\,x\,\ln \left (b\,x\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\,{\left (e\,x\right )}^{n-1}\,2{}\mathrm {i}-\frac {2\,b\,x\,{\left (e\,x\right )}^{n-1}\,\left (a\,1{}\mathrm {i}+b\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\right )}{\sqrt {a+b}\,\sqrt {a-b}}\right )\,{\left (e\,x\right )}^{n-1}}{a\,d\,n\,x^n\,\sqrt {a+b}\,\sqrt {a-b}}+\frac {b\,x\,\ln \left (b\,x\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\,{\left (e\,x\right )}^{n-1}\,2{}\mathrm {i}+\frac {2\,b\,x\,{\left (e\,x\right )}^{n-1}\,\left (a\,1{}\mathrm {i}+b\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\right )}{\sqrt {a+b}\,\sqrt {a-b}}\right )\,{\left (e\,x\right )}^{n-1}}{a\,d\,n\,x^n\,\sqrt {a+b}\,\sqrt {a-b}} \]
(x*(e*x)^(n - 1))/(a*n) - (b*x*log(b*x*exp(c*1i)*exp(d*x^n*1i)*(e*x)^(n - 1)*2i - (2*b*x*(e*x)^(n - 1)*(a*1i + b*exp(c*1i)*exp(d*x^n*1i)))/((a + b)^ (1/2)*(a - b)^(1/2)))*(e*x)^(n - 1))/(a*d*n*x^n*(a + b)^(1/2)*(a - b)^(1/2 )) + (b*x*log(b*x*exp(c*1i)*exp(d*x^n*1i)*(e*x)^(n - 1)*2i + (2*b*x*(e*x)^ (n - 1)*(a*1i + b*exp(c*1i)*exp(d*x^n*1i)))/((a + b)^(1/2)*(a - b)^(1/2))) *(e*x)^(n - 1))/(a*d*n*x^n*(a + b)^(1/2)*(a - b)^(1/2))