Integrand size = 44, antiderivative size = 42 \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-x \csc \left (a+b \log \left (c x^n\right )\right )-b n x \cot \left (a+b \log \left (c x^n\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right ) \]
Time = 0.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-x \left (1+b n \cot \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right ) \]
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.32 (sec) , antiderivative size = 172, normalized size of antiderivative = 4.10, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {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.
\(\displaystyle \int \left (2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )-\left (b^2 n^2+1\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 e^{i a} x (b n+i) \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {i}{b n}\right ),\frac {1}{2} \left (3-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )-\frac {16 e^{3 i a} b^2 n^2 x \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {i}{b n}\right ),\frac {1}{2} \left (5-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-3 b n+i}\) |
2*E^(I*a)*(I + b*n)*x*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1 - I/(b*n))/2, (3 - I/(b*n))/2, E^((2*I)*a)*(c*x^n)^((2*I)*b)] - (16*b^2*E^((3*I)*a)*n^2* x*(c*x^n)^((3*I)*b)*Hypergeometric2F1[3, (3 - I/(b*n))/2, (5 - I/(b*n))/2, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/(I - 3*b*n)
3.4.1.3.1 Defintions of rubi rules used
Time = 14.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.95
method | result | size |
parallelrisch | \(\frac {x \left ({\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{4} b n -2 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}-b n -2 \tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )\right )}{4 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}}\) | \(82\) |
risch | \(\frac {2 c^{i b} \left (x^{n}\right )^{i b} x \left (n b \,c^{2 i b} \left (x^{n}\right )^{2 i b} {\mathrm e}^{\frac {3 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {3 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {3 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{\frac {3 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{3 i a}+b n \,{\mathrm e}^{\frac {b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{\frac {b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{i a}-i c^{2 i b} \left (x^{n}\right )^{2 i b} {\mathrm e}^{\frac {3 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {3 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {3 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{\frac {3 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{3 i a}+i {\mathrm e}^{\frac {b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{\frac {b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{i a}\right )}{{\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}-1\right )}^{2}}\) | \(523\) |
1/4*x*(tan(1/2*a+b*ln((c*x^n)^(1/2)))^4*b*n-2*tan(1/2*a+b*ln((c*x^n)^(1/2) ))^3-b*n-2*tan(1/2*a+b*ln((c*x^n)^(1/2))))/tan(1/2*a+b*ln((c*x^n)^(1/2)))^ 2
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19 \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {b n x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + x \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1} \]
integrate(-(b^2*n^2+1)*csc(a+b*log(c*x^n))+2*b^2*n^2*csc(a+b*log(c*x^n))^3 ,x, algorithm="fricas")
(b*n*x*cos(b*n*log(x) + b*log(c) + a) + x*sin(b*n*log(x) + b*log(c) + a))/ (cos(b*n*log(x) + b*log(c) + a)^2 - 1)
\[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (2 b^{2} n^{2} \csc ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} - b^{2} n^{2} - 1\right ) \csc {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1701 vs. \(2 (42) = 84\).
Time = 0.50 (sec) , antiderivative size = 1701, normalized size of antiderivative = 40.50 \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Too large to display} \]
integrate(-(b^2*n^2+1)*csc(a+b*log(c*x^n))+2*b^2*n^2*csc(a+b*log(c*x^n))^3 ,x, algorithm="maxima")
2*((b*n*cos(b*log(c)) - sin(b*log(c)))*x*cos(b*log(x^n) + a) - (b*n*sin(b* log(c)) + cos(b*log(c)))*x*sin(b*log(x^n) + a) + (((b*cos(4*b*log(c))*cos( 3*b*log(c)) + b*sin(4*b*log(c))*sin(3*b*log(c)))*n - cos(3*b*log(c))*sin(4 *b*log(c)) + cos(4*b*log(c))*sin(3*b*log(c)))*x*cos(3*b*log(x^n) + 3*a) + ((b*cos(4*b*log(c))*cos(b*log(c)) + b*sin(4*b*log(c))*sin(b*log(c)))*n + c os(b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(b*log(c)))*x*cos(b*log( x^n) + a) + ((b*cos(3*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(3* b*log(c)))*n + cos(4*b*log(c))*cos(3*b*log(c)) + sin(4*b*log(c))*sin(3*b*l og(c)))*x*sin(3*b*log(x^n) + 3*a) + ((b*cos(b*log(c))*sin(4*b*log(c)) - b* cos(4*b*log(c))*sin(b*log(c)))*n - cos(4*b*log(c))*cos(b*log(c)) - sin(4*b *log(c))*sin(b*log(c)))*x*sin(b*log(x^n) + a))*cos(4*b*log(x^n) + 4*a) - ( 2*((b*cos(3*b*log(c))*cos(2*b*log(c)) + b*sin(3*b*log(c))*sin(2*b*log(c))) *n + cos(2*b*log(c))*sin(3*b*log(c)) - cos(3*b*log(c))*sin(2*b*log(c)))*x* cos(2*b*log(x^n) + 2*a) + 2*((b*cos(2*b*log(c))*sin(3*b*log(c)) - b*cos(3* b*log(c))*sin(2*b*log(c)))*n - cos(3*b*log(c))*cos(2*b*log(c)) - sin(3*b*l og(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) - (b*n*cos(3*b*log(c)) + sin(3*b*log(c)))*x)*cos(3*b*log(x^n) + 3*a) - 2*(((b*cos(2*b*log(c))*cos( b*log(c)) + b*sin(2*b*log(c))*sin(b*log(c)))*n + cos(b*log(c))*sin(2*b*log (c)) - cos(2*b*log(c))*sin(b*log(c)))*x*cos(b*log(x^n) + a) + ((b*cos(b*lo g(c))*sin(2*b*log(c)) - b*cos(2*b*log(c))*sin(b*log(c)))*n - cos(2*b*lo...
\[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { 2 \, b^{2} n^{2} \csc \left (b \log \left (c x^{n}\right ) + a\right )^{3} - {\left (b^{2} n^{2} + 1\right )} \csc \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
integrate(-(b^2*n^2+1)*csc(a+b*log(c*x^n))+2*b^2*n^2*csc(a+b*log(c*x^n))^3 ,x, algorithm="giac")
Time = 28.65 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.02 \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,\left (b\,n+1{}\mathrm {i}\right )+2\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\,\left (b\,n-\mathrm {i}\right )}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}^2} \]