Integrand size = 26, antiderivative size = 60 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\frac {\cot (e+f x)}{f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \cos ^2(e+f x)}} \] Output:
cot(f*x+e)/f/(a*cos(f*x+e)^2)^(1/2)-1/3*cot(f*x+e)*csc(f*x+e)^2/f/(a*cos(f *x+e)^2)^(1/2)
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.62 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\cot (e+f x) \left (-3+\csc ^2(e+f x)\right )}{3 f \sqrt {a \cos ^2(e+f x)}} \] Input:
Integrate[Cot[e + f*x]^4/Sqrt[a - a*Sin[e + f*x]^2],x]
Output:
-1/3*(Cot[e + f*x]*(-3 + Csc[e + f*x]^2))/(f*Sqrt[a*Cos[e + f*x]^2])
Time = 0.41 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.77, 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, 25, 3086, 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.
\(\displaystyle \int \frac {\cot ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (e+f x)^4 \sqrt {a-a \sin (e+f x)^2}}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle \int \frac {\cot ^4(e+f x)}{\sqrt {a \cos ^2(e+f x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan \left (e+f x+\frac {\pi }{2}\right )^4}{\sqrt {a \sin \left (e+f x+\frac {\pi }{2}\right )^2}}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {\cos (e+f x) \int \cot ^3(e+f x) \csc (e+f x)dx}{\sqrt {a \cos ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cos (e+f x) \int -\sec \left (e+f x-\frac {\pi }{2}\right ) \tan \left (e+f x-\frac {\pi }{2}\right )^3dx}{\sqrt {a \cos ^2(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\cos (e+f x) \int \sec \left (\frac {1}{2} (2 e-\pi )+f x\right ) \tan \left (\frac {1}{2} (2 e-\pi )+f x\right )^3dx}{\sqrt {a \cos ^2(e+f x)}}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {\cos (e+f x) \int \left (\csc ^2(e+f x)-1\right )d\csc (e+f x)}{f \sqrt {a \cos ^2(e+f x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\cos (e+f x) \left (\frac {1}{3} \csc ^3(e+f x)-\csc (e+f x)\right )}{f \sqrt {a \cos ^2(e+f x)}}\) |
Input:
Int[Cot[e + f*x]^4/Sqrt[a - a*Sin[e + f*x]^2],x]
Output:
-((Cos[e + f*x]*(-Csc[e + f*x] + Csc[e + f*x]^3/3))/(f*Sqrt[a*Cos[e + f*x] ^2]))
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 = 0.51 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {\cos \left (f x +e \right ) \left (3 \cos \left (f x +e \right )^{2}-2\right )}{3 \left (\cos \left (f x +e \right )-1\right ) \left (\cos \left (f x +e \right )+1\right ) \sin \left (f x +e \right ) \sqrt {a \cos \left (f x +e \right )^{2}}\, f}\) | \(64\) |
risch | \(\frac {2 i \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left (3 \,{\mathrm e}^{4 i \left (f x +e \right )}-2 \,{\mathrm e}^{2 i \left (f x +e \right )}+3\right )}{3 \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}\) | \(81\) |
Input:
int(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*cos(f*x+e)*(3*cos(f*x+e)^2-2)/(cos(f*x+e)-1)/(cos(f*x+e)+1)/sin(f*x+e) /(a*cos(f*x+e)^2)^(1/2)/f
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (3 \, \cos \left (f x + e\right )^{2} - 2\right )}}{3 \, {\left (a f \cos \left (f x + e\right )^{3} - a f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \] Input:
integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="fricas")
Output:
1/3*sqrt(a*cos(f*x + e)^2)*(3*cos(f*x + e)^2 - 2)/((a*f*cos(f*x + e)^3 - a *f*cos(f*x + e))*sin(f*x + e))
\[ \int \frac {\cot ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \] Input:
integrate(cot(f*x+e)**4/(a-a*sin(f*x+e)**2)**(1/2),x)
Output:
Integral(cot(e + f*x)**4/sqrt(-a*(sin(e + f*x) - 1)*(sin(e + f*x) + 1)), x )
Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (54) = 108\).
Time = 0.17 (sec) , antiderivative size = 525, normalized size of antiderivative = 8.75 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx =\text {Too large to display} \] Input:
integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="maxima")
Output:
-2/3*((3*sin(5*f*x + 5*e) - 2*sin(3*f*x + 3*e) + 3*sin(f*x + e))*cos(6*f*x + 6*e) + 9*(sin(4*f*x + 4*e) - sin(2*f*x + 2*e))*cos(5*f*x + 5*e) + 3*(2* sin(3*f*x + 3*e) - 3*sin(f*x + e))*cos(4*f*x + 4*e) - (3*cos(5*f*x + 5*e) - 2*cos(3*f*x + 3*e) + 3*cos(f*x + e))*sin(6*f*x + 6*e) - 3*(3*cos(4*f*x + 4*e) - 3*cos(2*f*x + 2*e) + 1)*sin(5*f*x + 5*e) - 3*(2*cos(3*f*x + 3*e) - 3*cos(f*x + e))*sin(4*f*x + 4*e) - 2*(3*cos(2*f*x + 2*e) - 1)*sin(3*f*x + 3*e) + 6*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) - 9*cos(f*x + e)*sin(2*f*x + 2 *e) + 9*cos(2*f*x + 2*e)*sin(f*x + e) - 3*sin(f*x + e))*sqrt(a)/((a*cos(6* f*x + 6*e)^2 + 9*a*cos(4*f*x + 4*e)^2 + 9*a*cos(2*f*x + 2*e)^2 + a*sin(6*f *x + 6*e)^2 + 9*a*sin(4*f*x + 4*e)^2 - 18*a*sin(4*f*x + 4*e)*sin(2*f*x + 2 *e) + 9*a*sin(2*f*x + 2*e)^2 - 2*(3*a*cos(4*f*x + 4*e) - 3*a*cos(2*f*x + 2 *e) + a)*cos(6*f*x + 6*e) - 6*(3*a*cos(2*f*x + 2*e) - a)*cos(4*f*x + 4*e) - 6*a*cos(2*f*x + 2*e) - 6*(a*sin(4*f*x + 4*e) - a*sin(2*f*x + 2*e))*sin(6 *f*x + 6*e) + a)*f)
Exception generated. \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
Output:
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 = 40.24 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.97 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\frac {4\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,\left (-{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,3{}\mathrm {i}+3{}\mathrm {i}\right )}{3\,a\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )} \] Input:
int(cot(e + f*x)^4/(a - a*sin(e + f*x)^2)^(1/2),x)
Output:
(4*exp(e*2i + f*x*2i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f* x*1i)*1i)/2)^2)^(1/2)*(exp(e*4i + f*x*4i)*3i - exp(e*2i + f*x*2i)*2i + 3i) )/(3*a*f*(exp(e*2i + f*x*2i) - 1)^3*(exp(e*2i + f*x*2i) + 1))
\[ \int \frac {\cot ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\sin \left (f x +e \right )^{2}+1}\, \cot \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{2}-1}d x \right )}{a} \] Input:
int(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x)
Output:
( - sqrt(a)*int((sqrt( - sin(e + f*x)**2 + 1)*cot(e + f*x)**4)/(sin(e + f* x)**2 - 1),x))/a