Integrand size = 26, antiderivative size = 38 \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}} \] Output:
-1/3*cot(f*x+e)*csc(f*x+e)^2/a/f/(a*cos(f*x+e)^2)^(1/2)
Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot ^3(e+f x)}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \] Input:
Integrate[Cot[e + f*x]^4/(a - a*Sin[e + f*x]^2)^(3/2),x]
Output:
-1/3*Cot[e + f*x]^3/(f*(a*Cos[e + f*x]^2)^(3/2))
Time = 0.40 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, 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, 15}
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.
\(\displaystyle \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (e+f x)^4 \left (a-a \sin (e+f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle \int \frac {\cot ^4(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan \left (e+f x+\frac {\pi }{2}\right )^4}{\left (a \sin \left (e+f x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {\cos (e+f x) \int \cot (e+f x) \csc ^3(e+f x)dx}{a \sqrt {a \cos ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cos (e+f x) \int -\sec \left (e+f x-\frac {\pi }{2}\right )^3 \tan \left (e+f x-\frac {\pi }{2}\right )dx}{a \sqrt {a \cos ^2(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\cos (e+f x) \int \sec \left (\frac {1}{2} (2 e-\pi )+f x\right )^3 \tan \left (\frac {1}{2} (2 e-\pi )+f x\right )dx}{a \sqrt {a \cos ^2(e+f x)}}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {\cos (e+f x) \int \csc ^2(e+f x)d\csc (e+f x)}{a f \sqrt {a \cos ^2(e+f x)}}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}\) |
Input:
Int[Cot[e + f*x]^4/(a - a*Sin[e + f*x]^2)^(3/2),x]
Output:
-1/3*(Cot[e + f*x]*Csc[e + f*x]^2)/(a*f*Sqrt[a*Cos[e + f*x]^2])
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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 = 0.56 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {\cos \left (f x +e \right )}{3 a \sin \left (f x +e \right )^{3} \sqrt {a \cos \left (f x +e \right )^{2}}\, f}\) | \(35\) |
risch | \(\frac {8 i \left ({\mathrm e}^{4 i \left (f x +e \right )}+{\mathrm e}^{2 i \left (f x +e \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3} f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}\) | \(68\) |
Input:
int(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/3/a*cos(f*x+e)/sin(f*x+e)^3/(a*cos(f*x+e)^2)^(1/2)/f
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\sqrt {a \cos \left (f x + e\right )^{2}}}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{3} - a^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \] Input:
integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="fricas")
Output:
1/3*sqrt(a*cos(f*x + e)^2)/((a^2*f*cos(f*x + e)^3 - a^2*f*cos(f*x + e))*si n(f*x + e))
\[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(f*x+e)**4/(a-a*sin(f*x+e)**2)**(3/2),x)
Output:
Integral(cot(e + f*x)**4/(-a*(sin(e + f*x) - 1)*(sin(e + f*x) + 1))**(3/2) , x)
Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (34) = 68\).
Time = 0.16 (sec) , antiderivative size = 382, normalized size of antiderivative = 10.05 \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {8 \, {\left (\cos \left (3 \, f x + 3 \, e\right ) \sin \left (6 \, f x + 6 \, e\right ) - 3 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (4 \, f x + 4 \, e\right ) - {\left (3 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \sin \left (3 \, f x + 3 \, e\right ) - \cos \left (6 \, f x + 6 \, e\right ) \sin \left (3 \, f x + 3 \, e\right ) + 3 \, \cos \left (4 \, f x + 4 \, e\right ) \sin \left (3 \, f x + 3 \, e\right ) + 3 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right )\right )} \sqrt {a}}{3 \, {\left (a^{2} \cos \left (6 \, f x + 6 \, e\right )^{2} + 9 \, a^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 9 \, a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + a^{2} \sin \left (6 \, f x + 6 \, e\right )^{2} + 9 \, a^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} - 18 \, a^{2} \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 9 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} - 6 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2} - 2 \, {\left (3 \, a^{2} \cos \left (4 \, f x + 4 \, e\right ) - 3 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \cos \left (6 \, f x + 6 \, e\right ) - 6 \, {\left (3 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) - a^{2}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 6 \, {\left (a^{2} \sin \left (4 \, f x + 4 \, e\right ) - a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \sin \left (6 \, f x + 6 \, e\right )\right )} f} \] Input:
integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")
Output:
8/3*(cos(3*f*x + 3*e)*sin(6*f*x + 6*e) - 3*cos(3*f*x + 3*e)*sin(4*f*x + 4* e) - (3*cos(2*f*x + 2*e) - 1)*sin(3*f*x + 3*e) - cos(6*f*x + 6*e)*sin(3*f* x + 3*e) + 3*cos(4*f*x + 4*e)*sin(3*f*x + 3*e) + 3*cos(3*f*x + 3*e)*sin(2* f*x + 2*e))*sqrt(a)/((a^2*cos(6*f*x + 6*e)^2 + 9*a^2*cos(4*f*x + 4*e)^2 + 9*a^2*cos(2*f*x + 2*e)^2 + a^2*sin(6*f*x + 6*e)^2 + 9*a^2*sin(4*f*x + 4*e) ^2 - 18*a^2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 9*a^2*sin(2*f*x + 2*e)^2 - 6*a^2*cos(2*f*x + 2*e) + a^2 - 2*(3*a^2*cos(4*f*x + 4*e) - 3*a^2*cos(2*f* x + 2*e) + a^2)*cos(6*f*x + 6*e) - 6*(3*a^2*cos(2*f*x + 2*e) - a^2)*cos(4* f*x + 4*e) - 6*(a^2*sin(4*f*x + 4*e) - a^2*sin(2*f*x + 2*e))*sin(6*f*x + 6 *e))*f)
Exception generated. \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(3/2),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 = 39.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.32 \[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,16{}\mathrm {i}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )} \] Input:
int(cot(e + f*x)^4/(a - a*sin(e + f*x)^2)^(3/2),x)
Output:
(exp(e*4i + f*x*4i)*(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)^3*(exp(e*2i + f *x*2i) + 1))
\[ \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\sin \left (f x +e \right )^{2}+1}\, \cot \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{4}-2 \sin \left (f x +e \right )^{2}+1}d x \right )}{a^{2}} \] Input:
int(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(3/2),x)
Output:
(sqrt(a)*int((sqrt( - sin(e + f*x)**2 + 1)*cot(e + f*x)**4)/(sin(e + f*x)* *4 - 2*sin(e + f*x)**2 + 1),x))/a**2