Integrand size = 26, antiderivative size = 115 \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}+\frac {2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt {a \cos ^2(e+f x)}} \]
-1/3*cot(f*x+e)*csc(f*x+e)^2/a/f/(a*cos(f*x+e)^2)^(1/2)+2/5*cot(f*x+e)*csc (f*x+e)^4/a/f/(a*cos(f*x+e)^2)^(1/2)-1/7*cot(f*x+e)*csc(f*x+e)^6/a/f/(a*co s(f*x+e)^2)^(1/2)
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.44 \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot ^3(e+f x) \left (35-42 \csc ^2(e+f x)+15 \csc ^4(e+f x)\right )}{105 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
-1/105*(Cot[e + f*x]^3*(35 - 42*Csc[e + f*x]^2 + 15*Csc[e + f*x]^4))/(f*(a *Cos[e + f*x]^2)^(3/2))
Time = 0.43 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.57, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3042, 3655, 3042, 3686, 3042, 25, 3086, 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.
| \(\displaystyle \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^8 \left (a-a \sin (e+f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3655 |
| \(\displaystyle \int \frac {\cot ^8(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan \left (e+f x+\frac {\pi }{2}\right )^8}{\left (a \sin \left (e+f x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \frac {\cos (e+f x) \int \cot ^5(e+f x) \csc ^3(e+f x)dx}{a \sqrt {a \cos ^2(e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cos (e+f x) \int -\sec \left (e+f x-\frac {\pi }{2}\right )^3 \tan \left (e+f x-\frac {\pi }{2}\right )^5dx}{a \sqrt {a \cos ^2(e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\cos (e+f x) \int \sec \left (\frac {1}{2} (2 e-\pi )+f x\right )^3 \tan \left (\frac {1}{2} (2 e-\pi )+f x\right )^5dx}{a \sqrt {a \cos ^2(e+f x)}}\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle -\frac {\cos (e+f x) \int \csc ^2(e+f x) \left (1-\csc ^2(e+f x)\right )^2d\csc (e+f x)}{a f \sqrt {a \cos ^2(e+f x)}}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {\cos (e+f x) \int \left (\csc ^6(e+f x)-2 \csc ^4(e+f x)+\csc ^2(e+f x)\right )d\csc (e+f x)}{a f \sqrt {a \cos ^2(e+f x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\cos (e+f x) \left (\frac {1}{7} \csc ^7(e+f x)-\frac {2}{5} \csc ^5(e+f x)+\frac {1}{3} \csc ^3(e+f x)\right )}{a f \sqrt {a \cos ^2(e+f x)}}\) |
-((Cos[e + f*x]*(Csc[e + f*x]^3/3 - (2*Csc[e + f*x]^5)/5 + Csc[e + f*x]^7/ 7))/(a*f*Sqrt[a*Cos[e + f*x]^2]))
3.5.87.3.1 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[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
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 = 0.66 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.50
| method | result | size |
| default | \(-\frac {\cos \left (f x +e \right ) \left (35 \left (\cos ^{4}\left (f x +e \right )\right )-28 \left (\cos ^{2}\left (f x +e \right )\right )+8\right )}{105 a \sin \left (f x +e \right )^{7} \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\) | \(57\) |
| risch | \(\frac {8 i \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left (35 \,{\mathrm e}^{10 i \left (f x +e \right )}+28 \,{\mathrm e}^{8 i \left (f x +e \right )}+114 \,{\mathrm e}^{6 i \left (f x +e \right )}+28 \,{\mathrm e}^{4 i \left (f x +e \right )}+35 \,{\mathrm e}^{2 i \left (f x +e \right )}\right )}{105 \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{7} f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}\) | \(116\) |
-1/105/a*cos(f*x+e)*(35*cos(f*x+e)^4-28*cos(f*x+e)^2+8)/sin(f*x+e)^7/(a*co s(f*x+e)^2)^(1/2)/f
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {{\left (35 \, \cos \left (f x + e\right )^{4} - 28 \, \cos \left (f x + e\right )^{2} + 8\right )} \sqrt {a \cos \left (f x + e\right )^{2}}}{105 \, {\left (a^{2} f \cos \left (f x + e\right )^{7} - 3 \, a^{2} f \cos \left (f x + e\right )^{5} + 3 \, a^{2} f \cos \left (f x + e\right )^{3} - a^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]
1/105*(35*cos(f*x + e)^4 - 28*cos(f*x + e)^2 + 8)*sqrt(a*cos(f*x + e)^2)/( (a^2*f*cos(f*x + e)^7 - 3*a^2*f*cos(f*x + e)^5 + 3*a^2*f*cos(f*x + e)^3 - a^2*f*cos(f*x + e))*sin(f*x + e))
Timed out. \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2026 vs. \(2 (103) = 206\).
Time = 0.36 (sec) , antiderivative size = 2026, normalized size of antiderivative = 17.62 \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
-8/105*((35*sin(11*f*x + 11*e) + 28*sin(9*f*x + 9*e) + 114*sin(7*f*x + 7*e ) + 28*sin(5*f*x + 5*e) + 35*sin(3*f*x + 3*e))*cos(14*f*x + 14*e) - 7*(35* sin(11*f*x + 11*e) + 28*sin(9*f*x + 9*e) + 114*sin(7*f*x + 7*e) + 28*sin(5 *f*x + 5*e) + 35*sin(3*f*x + 3*e))*cos(12*f*x + 12*e) - 245*(3*sin(10*f*x + 10*e) - 5*sin(8*f*x + 8*e) + 5*sin(6*f*x + 6*e) - 3*sin(4*f*x + 4*e) + s in(2*f*x + 2*e))*cos(11*f*x + 11*e) + 21*(28*sin(9*f*x + 9*e) + 114*sin(7* f*x + 7*e) + 28*sin(5*f*x + 5*e) + 35*sin(3*f*x + 3*e))*cos(10*f*x + 10*e) + 196*(5*sin(8*f*x + 8*e) - 5*sin(6*f*x + 6*e) + 3*sin(4*f*x + 4*e) - sin (2*f*x + 2*e))*cos(9*f*x + 9*e) - 35*(114*sin(7*f*x + 7*e) + 28*sin(5*f*x + 5*e) + 35*sin(3*f*x + 3*e))*cos(8*f*x + 8*e) - 798*(5*sin(6*f*x + 6*e) - 3*sin(4*f*x + 4*e) + sin(2*f*x + 2*e))*cos(7*f*x + 7*e) + 245*(4*sin(5*f* x + 5*e) + 5*sin(3*f*x + 3*e))*cos(6*f*x + 6*e) + 196*(3*sin(4*f*x + 4*e) - sin(2*f*x + 2*e))*cos(5*f*x + 5*e) - (35*cos(11*f*x + 11*e) + 28*cos(9*f *x + 9*e) + 114*cos(7*f*x + 7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3* e))*sin(14*f*x + 14*e) + 7*(35*cos(11*f*x + 11*e) + 28*cos(9*f*x + 9*e) + 114*cos(7*f*x + 7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3*e))*sin(12*f *x + 12*e) + 35*(21*cos(10*f*x + 10*e) - 35*cos(8*f*x + 8*e) + 35*cos(6*f* x + 6*e) - 21*cos(4*f*x + 4*e) + 7*cos(2*f*x + 2*e) - 1)*sin(11*f*x + 11*e ) - 21*(28*cos(9*f*x + 9*e) + 114*cos(7*f*x + 7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3*e))*sin(10*f*x + 10*e) - 28*(35*cos(8*f*x + 8*e) - 35...
Exception generated. \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m operator + Error: Bad Argument Value
Time = 30.49 (sec) , antiderivative size = 589, normalized size of antiderivative = 5.12 \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\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}}{3\,a^2\,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}\,464{}\mathrm {i}}{15\,a^2\,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}\,3072{}\mathrm {i}}{35\,a^2\,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}\,4736{}\mathrm {i}}{35\,a^2\,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 )}+\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}\,768{}\mathrm {i}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^6\,\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}\,256{}\mathrm {i}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^7\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )} \]
(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)/(3*a^2*f*(exp(e*2i + f*x*2i) - 1)^2*(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)*464i)/(15*a^2*f*(exp(e *2i + f*x*2i) - 1)^3*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i))) + (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)*3072i)/(35*a^2*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*1 i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*4736i)/(35*a^2*f*(exp(e*2i + f*x*2i) - 1)^5*(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)*768i)/(7*a^2*f*(exp(e*2i + f*x*2i) - 1)^6*(exp(e*1i + f*x*1i) + ex p(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)*256i)/(7*a^2*f*(exp(e*2i + f*x*2i ) - 1)^7*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i)))