Integrand size = 13, antiderivative size = 122 \[ \int \frac {\cot ^7(x)}{a+b \csc (x)} \, dx=-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \csc (x)}{b^5}+\frac {a \left (a^2-3 b^2\right ) \csc ^2(x)}{2 b^4}-\frac {\left (a^2-3 b^2\right ) \csc ^3(x)}{3 b^3}+\frac {a \csc ^4(x)}{4 b^2}-\frac {\csc ^5(x)}{5 b}+\frac {\left (a^2-b^2\right )^3 \log (a+b \csc (x))}{a b^6}-\frac {\log (\sin (x))}{a} \]
-(a^4-3*a^2*b^2+3*b^4)*csc(x)/b^5+1/2*a*(a^2-3*b^2)*csc(x)^2/b^4-1/3*(a^2- 3*b^2)*csc(x)^3/b^3+1/4*a*csc(x)^4/b^2-1/5*csc(x)^5/b+(a^2-b^2)^3*ln(a+b*c sc(x))/a/b^6-ln(sin(x))/a
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.08 \[ \int \frac {\cot ^7(x)}{a+b \csc (x)} \, dx=\frac {-60 b \left (a^4-3 a^2 b^2+3 b^4\right ) \csc (x)+30 a b^2 \left (a^2-3 b^2\right ) \csc ^2(x)-20 b^3 \left (a^2-3 b^2\right ) \csc ^3(x)+15 a b^4 \csc ^4(x)-12 b^5 \csc ^5(x)-60 a \left (a^4-3 a^2 b^2+3 b^4\right ) \log (\sin (x))+\frac {60 \left (a^2-b^2\right )^3 \log (b+a \sin (x))}{a}}{60 b^6} \]
(-60*b*(a^4 - 3*a^2*b^2 + 3*b^4)*Csc[x] + 30*a*b^2*(a^2 - 3*b^2)*Csc[x]^2 - 20*b^3*(a^2 - 3*b^2)*Csc[x]^3 + 15*a*b^4*Csc[x]^4 - 12*b^5*Csc[x]^5 - 60 *a*(a^4 - 3*a^2*b^2 + 3*b^4)*Log[Sin[x]] + (60*(a^2 - b^2)^3*Log[b + a*Sin [x]])/a)/(60*b^6)
Time = 0.32 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 4373, 522, 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.
| \(\displaystyle \int \frac {\cot ^7(x)}{a+b \csc (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (x)^7}{a+b \csc (x)}dx\) |
| \(\Big \downarrow \) 4373 |
| \(\displaystyle \frac {\int \frac {\left (b^2-b^2 \csc ^2(x)\right )^3 \sin (x)}{b (a+b \csc (x))}d(b \csc (x))}{b^6}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {\int \left (\frac {\sin (x) b^5}{a}-\csc ^4(x) b^4+a \csc ^3(x) b^3-\left (a^2-3 b^2\right ) \csc ^2(x) b^2+a \left (a^2-3 b^2\right ) \csc (x) b-a^4 \left (\frac {3 \left (b^2-a^2\right ) b^2}{a^4}+1\right )+\frac {\left (a^2-b^2\right )^3}{a (a+b \csc (x))}\right )d(b \csc (x))}{b^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} a b^2 \left (a^2-3 b^2\right ) \csc ^2(x)+\frac {\left (a^2-b^2\right )^3 \log (a+b \csc (x))}{a}-\frac {1}{3} b^3 \left (a^2-3 b^2\right ) \csc ^3(x)-b \left (a^4-3 a^2 b^2+3 b^4\right ) \csc (x)+\frac {b^6 \log (b \csc (x))}{a}+\frac {1}{4} a b^4 \csc ^4(x)-\frac {1}{5} b^5 \csc ^5(x)}{b^6}\) |
(-(b*(a^4 - 3*a^2*b^2 + 3*b^4)*Csc[x]) + (a*b^2*(a^2 - 3*b^2)*Csc[x]^2)/2 - (b^3*(a^2 - 3*b^2)*Csc[x]^3)/3 + (a*b^4*Csc[x]^4)/4 - (b^5*Csc[x]^5)/5 + (b^6*Log[b*Csc[x]])/a + ((a^2 - b^2)^3*Log[a + b*Csc[x]])/a)/b^6
3.1.18.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 4.69 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(\frac {\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \ln \left (a \sin \left (x \right )+b \right )}{b^{6} a}-\frac {1}{5 b \sin \left (x \right )^{5}}-\frac {a^{2}-3 b^{2}}{3 b^{3} \sin \left (x \right )^{3}}-\frac {a^{4}-3 a^{2} b^{2}+3 b^{4}}{b^{5} \sin \left (x \right )}+\frac {a}{4 b^{2} \sin \left (x \right )^{4}}+\frac {\left (a^{2}-3 b^{2}\right ) a}{2 b^{4} \sin \left (x \right )^{2}}-\frac {\left (a^{4}-3 a^{2} b^{2}+3 b^{4}\right ) a \ln \left (\sin \left (x \right )\right )}{b^{6}}\) | \(149\) |
| risch | \(\frac {i x}{a}-\frac {2 i \left (-15 i a^{3} b \,{\mathrm e}^{8 i x}+45 i a^{3} b \,{\mathrm e}^{6 i x}+15 a^{4} {\mathrm e}^{9 i x}-45 a^{2} b^{2} {\mathrm e}^{9 i x}+45 b^{4} {\mathrm e}^{9 i x}-105 i a \,b^{3} {\mathrm e}^{6 i x}-45 i a^{3} b \,{\mathrm e}^{4 i x}-60 a^{4} {\mathrm e}^{7 i x}+160 a^{2} b^{2} {\mathrm e}^{7 i x}-120 b^{4} {\mathrm e}^{7 i x}+45 i a \,b^{3} {\mathrm e}^{8 i x}+105 i a \,b^{3} {\mathrm e}^{4 i x}+90 a^{4} {\mathrm e}^{5 i x}-230 a^{2} b^{2} {\mathrm e}^{5 i x}+198 b^{4} {\mathrm e}^{5 i x}+15 i a^{3} b \,{\mathrm e}^{2 i x}-45 i a \,b^{3} {\mathrm e}^{2 i x}-60 a^{4} {\mathrm e}^{3 i x}+160 a^{2} b^{2} {\mathrm e}^{3 i x}-120 b^{4} {\mathrm e}^{3 i x}+15 \,{\mathrm e}^{i x} a^{4}-45 \,{\mathrm e}^{i x} a^{2} b^{2}+45 \,{\mathrm e}^{i x} b^{4}\right )}{15 b^{5} \left ({\mathrm e}^{2 i x}-1\right )^{5}}-\frac {a^{5} \ln \left ({\mathrm e}^{2 i x}-1\right )}{b^{6}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i x}-1\right )}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{2 i x}-1\right )}{b^{2}}+\frac {a^{5} \ln \left ({\mathrm e}^{2 i x}-1+\frac {2 i b \,{\mathrm e}^{i x}}{a}\right )}{b^{6}}-\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i x}-1+\frac {2 i b \,{\mathrm e}^{i x}}{a}\right )}{b^{4}}+\frac {3 a \ln \left ({\mathrm e}^{2 i x}-1+\frac {2 i b \,{\mathrm e}^{i x}}{a}\right )}{b^{2}}-\frac {\ln \left ({\mathrm e}^{2 i x}-1+\frac {2 i b \,{\mathrm e}^{i x}}{a}\right )}{a}\) | \(438\) |
(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/b^6/a*ln(a*sin(x)+b)-1/5/b/sin(x)^5-1/3*(a^2 -3*b^2)/b^3/sin(x)^3-(a^4-3*a^2*b^2+3*b^4)/b^5/sin(x)+1/4*a/b^2/sin(x)^4+1 /2*(a^2-3*b^2)/b^4*a/sin(x)^2-(a^4-3*a^2*b^2+3*b^4)/b^6*a*ln(sin(x))
Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (114) = 228\).
Time = 0.30 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.68 \[ \int \frac {\cot ^7(x)}{a+b \csc (x)} \, dx=-\frac {60 \, a^{5} b - 160 \, a^{3} b^{3} + 132 \, a b^{5} + 60 \, {\left (a^{5} b - 3 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (x\right )^{4} - 20 \, {\left (6 \, a^{5} b - 17 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \cos \left (x\right )^{2} - 60 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6} + {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cos \left (x\right )^{4} - 2 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cos \left (x\right )^{2}\right )} \log \left (a \sin \left (x\right ) + b\right ) \sin \left (x\right ) + 60 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (x\right )^{4} - 2 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac {1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) - 15 \, {\left (2 \, a^{4} b^{2} - 5 \, a^{2} b^{4} - 2 \, {\left (a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{60 \, {\left (a b^{6} \cos \left (x\right )^{4} - 2 \, a b^{6} \cos \left (x\right )^{2} + a b^{6}\right )} \sin \left (x\right )} \]
-1/60*(60*a^5*b - 160*a^3*b^3 + 132*a*b^5 + 60*(a^5*b - 3*a^3*b^3 + 3*a*b^ 5)*cos(x)^4 - 20*(6*a^5*b - 17*a^3*b^3 + 15*a*b^5)*cos(x)^2 - 60*(a^6 - 3* a^4*b^2 + 3*a^2*b^4 - b^6 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(x)^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(x)^2)*log(a*sin(x) + b)*sin(x) + 60*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cos(x)^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cos(x)^2)*log(-1/2*sin(x))*sin(x) - 15* (2*a^4*b^2 - 5*a^2*b^4 - 2*(a^4*b^2 - 3*a^2*b^4)*cos(x)^2)*sin(x))/((a*b^6 *cos(x)^4 - 2*a*b^6*cos(x)^2 + a*b^6)*sin(x))
\[ \int \frac {\cot ^7(x)}{a+b \csc (x)} \, dx=\int \frac {\cot ^{7}{\left (x \right )}}{a + b \csc {\left (x \right )}}\, dx \]
Time = 0.23 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.22 \[ \int \frac {\cot ^7(x)}{a+b \csc (x)} \, dx=-\frac {{\left (a^{5} - 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (\sin \left (x\right )\right )}{b^{6}} + \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (a \sin \left (x\right ) + b\right )}{a b^{6}} + \frac {15 \, a b^{3} \sin \left (x\right ) - 60 \, {\left (a^{4} - 3 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (x\right )^{4} - 12 \, b^{4} + 30 \, {\left (a^{3} b - 3 \, a b^{3}\right )} \sin \left (x\right )^{3} - 20 \, {\left (a^{2} b^{2} - 3 \, b^{4}\right )} \sin \left (x\right )^{2}}{60 \, b^{5} \sin \left (x\right )^{5}} \]
-(a^5 - 3*a^3*b^2 + 3*a*b^4)*log(sin(x))/b^6 + (a^6 - 3*a^4*b^2 + 3*a^2*b^ 4 - b^6)*log(a*sin(x) + b)/(a*b^6) + 1/60*(15*a*b^3*sin(x) - 60*(a^4 - 3*a ^2*b^2 + 3*b^4)*sin(x)^4 - 12*b^4 + 30*(a^3*b - 3*a*b^3)*sin(x)^3 - 20*(a^ 2*b^2 - 3*b^4)*sin(x)^2)/(b^5*sin(x)^5)
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.27 \[ \int \frac {\cot ^7(x)}{a+b \csc (x)} \, dx=-\frac {{\left (a^{5} - 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left ({\left | \sin \left (x\right ) \right |}\right )}{b^{6}} + \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left ({\left | a \sin \left (x\right ) + b \right |}\right )}{a b^{6}} + \frac {15 \, a b^{4} \sin \left (x\right ) - 12 \, b^{5} - 60 \, {\left (a^{4} b - 3 \, a^{2} b^{3} + 3 \, b^{5}\right )} \sin \left (x\right )^{4} + 30 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} \sin \left (x\right )^{3} - 20 \, {\left (a^{2} b^{3} - 3 \, b^{5}\right )} \sin \left (x\right )^{2}}{60 \, b^{6} \sin \left (x\right )^{5}} \]
-(a^5 - 3*a^3*b^2 + 3*a*b^4)*log(abs(sin(x)))/b^6 + (a^6 - 3*a^4*b^2 + 3*a ^2*b^4 - b^6)*log(abs(a*sin(x) + b))/(a*b^6) + 1/60*(15*a*b^4*sin(x) - 12* b^5 - 60*(a^4*b - 3*a^2*b^3 + 3*b^5)*sin(x)^4 + 30*(a^3*b^2 - 3*a*b^4)*sin (x)^3 - 20*(a^2*b^3 - 3*b^5)*sin(x)^2)/(b^6*sin(x)^5)
Time = 19.37 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.33 \[ \int \frac {\cot ^7(x)}{a+b \csc (x)} \, dx={\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (\frac {3}{32\,b}-\frac {a^2}{24\,b^3}\right )-\frac {19\,\mathrm {tan}\left (\frac {x}{2}\right )}{16\,b}+\frac {\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{a}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{160\,b}-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (\frac {a}{32\,b^2}+\frac {a\,\left (\frac {9}{32\,b}-\frac {a^2}{8\,b^3}\right )}{b}\right )+\frac {11\,a^2\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,b^3}+\frac {a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{64\,b^2}-\frac {a^4\,\mathrm {tan}\left (\frac {x}{2}\right )}{2\,b^5}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,\left (a^5-3\,a^3\,b^2+3\,a\,b^4\right )}{b^6}-\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^5\,\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (10\,a\,b^3-4\,a^3\,b\right )-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (3\,b^4-\frac {4\,a^2\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,\left (16\,a^4-44\,a^2\,b^2+38\,b^4\right )+\frac {b^4}{5}-\frac {a\,b^3\,\mathrm {tan}\left (\frac {x}{2}\right )}{2}\right )}{32\,b^5}+\frac {\ln \left (b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )+b\right )\,{\left (a^2-b^2\right )}^3}{a\,b^6} \]
tan(x/2)^3*(3/(32*b) - a^2/(24*b^3)) - (19*tan(x/2))/(16*b) + log(tan(x/2) ^2 + 1)/a - tan(x/2)^5/(160*b) - tan(x/2)^2*(a/(32*b^2) + (a*(9/(32*b) - a ^2/(8*b^3)))/b) + (11*a^2*tan(x/2))/(8*b^3) + (a*tan(x/2)^4)/(64*b^2) - (a ^4*tan(x/2))/(2*b^5) - (log(tan(x/2))*(3*a*b^4 + a^5 - 3*a^3*b^2))/b^6 - ( cot(x/2)^5*(tan(x/2)^3*(10*a*b^3 - 4*a^3*b) - tan(x/2)^2*(3*b^4 - (4*a^2*b ^2)/3) + tan(x/2)^4*(16*a^4 + 38*b^4 - 44*a^2*b^2) + b^4/5 - (a*b^3*tan(x/ 2))/2))/(32*b^5) + (log(b + 2*a*tan(x/2) + b*tan(x/2)^2)*(a^2 - b^2)^3)/(a *b^6)