Integrand size = 26, antiderivative size = 124 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=-\frac {3 \sqrt {a \cos ^2(e+f x)} \csc (e+f x) \sec (e+f x)}{f}+\frac {\sqrt {a \cos ^2(e+f x)} \csc ^3(e+f x) \sec (e+f x)}{f}-\frac {\sqrt {a \cos ^2(e+f x)} \csc ^5(e+f x) \sec (e+f x)}{5 f}-\frac {\sqrt {a \cos ^2(e+f x)} \tan (e+f x)}{f} \]
-3*csc(f*x+e)*sec(f*x+e)*(a*cos(f*x+e)^2)^(1/2)/f+csc(f*x+e)^3*sec(f*x+e)* (a*cos(f*x+e)^2)^(1/2)/f-1/5*csc(f*x+e)^5*sec(f*x+e)*(a*cos(f*x+e)^2)^(1/2 )/f-(a*cos(f*x+e)^2)^(1/2)*tan(f*x+e)/f
Time = 0.21 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.54 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\frac {\sqrt {a \cos ^2(e+f x)} (-182+235 \cos (2 (e+f x))-90 \cos (4 (e+f x))+5 \cos (6 (e+f x))) \csc ^5(e+f x) \sec (e+f x)}{160 f} \]
(Sqrt[a*Cos[e + f*x]^2]*(-182 + 235*Cos[2*(e + f*x)] - 90*Cos[4*(e + f*x)] + 5*Cos[6*(e + f*x)])*Csc[e + f*x]^5*Sec[e + f*x])/(160*f)
Time = 0.40 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.50, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 3655, 3042, 3686, 3042, 3070, 244, 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.
| \(\displaystyle \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a-a \sin (e+f x)^2}}{\tan (e+f x)^6}dx\) |
| \(\Big \downarrow \) 3655 |
| \(\displaystyle \int \cot ^6(e+f x) \sqrt {a \cos ^2(e+f x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan \left (e+f x+\frac {\pi }{2}\right )^6 \sqrt {a \sin \left (e+f x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \sec (e+f x) \sqrt {a \cos ^2(e+f x)} \int \cos (e+f x) \cot ^6(e+f x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sec (e+f x) \sqrt {a \cos ^2(e+f x)} \int \sin \left (e+f x+\frac {\pi }{2}\right ) \tan \left (e+f x+\frac {\pi }{2}\right )^6dx\) |
| \(\Big \downarrow \) 3070 |
| \(\displaystyle -\frac {\sec (e+f x) \sqrt {a \cos ^2(e+f x)} \int \csc ^6(e+f x) \left (1-\sin ^2(e+f x)\right )^3d(-\sin (e+f x))}{f}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {\sec (e+f x) \sqrt {a \cos ^2(e+f x)} \int \left (\csc ^6(e+f x)-3 \csc ^4(e+f x)+3 \csc ^2(e+f x)-1\right )d(-\sin (e+f x))}{f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sec (e+f x) \sqrt {a \cos ^2(e+f x)} \left (\sin (e+f x)+\frac {1}{5} \csc ^5(e+f x)-\csc ^3(e+f x)+3 \csc (e+f x)\right )}{f}\) |
-((Sqrt[a*Cos[e + f*x]^2]*Sec[e + f*x]*(3*Csc[e + f*x] - Csc[e + f*x]^3 + Csc[e + f*x]^5/5 + Sin[e + f*x]))/f)
3.5.67.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m + n - 1)/2)/x^n, x], x, Cos[e + f *x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*cos[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a + b, 0]
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.70 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.69
| method | result | size |
| default | \(\frac {\cos \left (f x +e \right ) a \left (-5 \left (\sin ^{6}\left (f x +e \right )\right )-15 \left (\sin ^{4}\left (f x +e \right )\right )+5 \left (\sin ^{2}\left (f x +e \right )\right )-1\right )}{5 \left (\cos \left (f x +e \right )-1\right )^{2} \left (1+\cos \left (f x +e \right )\right )^{2} \sin \left (f x +e \right ) \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\) | \(85\) |
| risch | \(\frac {i \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, \left (5 \,{\mathrm e}^{12 i \left (f x +e \right )}-90 \,{\mathrm e}^{10 i \left (f x +e \right )}+235 \,{\mathrm e}^{8 i \left (f x +e \right )}-364 \,{\mathrm e}^{6 i \left (f x +e \right )}+235 \,{\mathrm e}^{4 i \left (f x +e \right )}-90 \,{\mathrm e}^{2 i \left (f x +e \right )}+5\right )}{10 \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{5} f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(127\) |
1/5*cos(f*x+e)*a*(-5*sin(f*x+e)^6-15*sin(f*x+e)^4+5*sin(f*x+e)^2-1)/(cos(f *x+e)-1)^2/(1+cos(f*x+e))^2/sin(f*x+e)/(a*cos(f*x+e)^2)^(1/2)/f
Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.69 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\frac {{\left (5 \, \cos \left (f x + e\right )^{6} - 30 \, \cos \left (f x + e\right )^{4} + 40 \, \cos \left (f x + e\right )^{2} - 16\right )} \sqrt {a \cos \left (f x + e\right )^{2}}}{5 \, {\left (f \cos \left (f x + e\right )^{5} - 2 \, f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]
1/5*(5*cos(f*x + e)^6 - 30*cos(f*x + e)^4 + 40*cos(f*x + e)^2 - 16)*sqrt(a *cos(f*x + e)^2)/((f*cos(f*x + e)^5 - 2*f*cos(f*x + e)^3 + f*cos(f*x + e)) *sin(f*x + e))
\[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\int \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \cot ^{6}{\left (e + f x \right )}\, dx \]
Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.55 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=-\frac {16 \, \sqrt {a} \tan \left (f x + e\right )^{6} + 8 \, \sqrt {a} \tan \left (f x + e\right )^{4} - 2 \, \sqrt {a} \tan \left (f x + e\right )^{2} + \sqrt {a}}{5 \, \sqrt {\tan \left (f x + e\right )^{2} + 1} f \tan \left (f x + e\right )^{5}} \]
-1/5*(16*sqrt(a)*tan(f*x + e)^6 + 8*sqrt(a)*tan(f*x + e)^4 - 2*sqrt(a)*tan (f*x + e)^2 + sqrt(a))/(sqrt(tan(f*x + e)^2 + 1)*f*tan(f*x + e)^5)
Leaf count of result is larger than twice the leaf count of optimal. 8626 vs. \(2 (114) = 228\).
Time = 4.03 (sec) , antiderivative size = 8626, normalized size of antiderivative = 69.56 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\text {Too large to display} \]
1/160*(320*(sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan (1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)* tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)* tan(1/2*e)^2 - sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3* tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f* x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f* x) - 2*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2* e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1 /2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*e))/((tan( 1/2*f*x)^2 + 1)*(tan(1/2*e)^2 + 1)) - (5*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/ 2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3 *tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)* tan(1/2*e) + 1)*tan(1/2*f*x)^9*tan(1/2*e)^16 + 10*sqrt(a)*sgn(tan(1/2*f*x) ^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1 /2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan( 1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^8*tan(1/2*e)^17 + 10*sqrt(a)*sgn(tan (1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^7*tan(1/2*e)^18 + 5*sqrt(a) *sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(...
Time = 24.39 (sec) , antiderivative size = 555, normalized size of antiderivative = 4.48 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=-\frac {\left (\frac {1{}\mathrm {i}}{f}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{f}\right )\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,12{}\mathrm {i}}{f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,16{}\mathrm {i}}{f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,144{}\mathrm {i}}{5\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,128{}\mathrm {i}}{5\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^4\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,64{}\mathrm {i}}{5\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^5\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )} \]
- ((1i/f - (exp(e*2i + f*x*2i)*1i)/f)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2))/(exp(e*2i + f*x*2i) + 1) - (exp(e*3 i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/ 2)^2)^(1/2)*12i)/(f*(exp(e*2i + f*x*2i) - 1)*(exp(e*1i + f*x*1i) + exp(e*3 i + f*x*3i))) - (exp(e*3i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*16i)/(f*(exp(e*2i + f*x*2i) - 1)^2*(ex p(e*1i + f*x*1i) + exp(e*3i + f*x*3i))) - (exp(e*3i + f*x*3i)*(a - a*((exp (- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*144i)/(5*f*( exp(e*2i + f*x*2i) - 1)^3*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i))) - (ex p(e*3i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i) *1i)/2)^2)^(1/2)*128i)/(5*f*(exp(e*2i + f*x*2i) - 1)^4*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i))) - (exp(e*3i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1i )*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*64i)/(5*f*(exp(e*2i + f*x*2i ) - 1)^5*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i)))