Integrand size = 24, antiderivative size = 53 \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {1}{a f \sqrt {a \cos ^2(e+f x)}} \] Output:
-arctanh((a*cos(f*x+e)^2)^(1/2)/a^(1/2))/a^(3/2)/f+1/a/f/(a*cos(f*x+e)^2)^ (1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\cos ^2(e+f x)\right )}{a f \sqrt {a \cos ^2(e+f x)}} \] Input:
Integrate[Cot[e + f*x]/(a - a*Sin[e + f*x]^2)^(3/2),x]
Output:
Hypergeometric2F1[-1/2, 1, 1/2, Cos[e + f*x]^2]/(a*f*Sqrt[a*Cos[e + f*x]^2 ])
Time = 0.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3655, 3042, 25, 3684, 61, 73, 219}
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.
\(\displaystyle \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (e+f x) \left (a-a \sin (e+f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle \int \frac {\cot (e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\tan \left (e+f x+\frac {\pi }{2}\right )}{\left (a \sin \left (e+f x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\tan \left (\frac {1}{2} (2 e+\pi )+f x\right )}{\left (a \sin \left (\frac {1}{2} (2 e+\pi )+f x\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3684 |
\(\displaystyle -\frac {\int \frac {1}{\left (a \cos ^2(e+f x)\right )^{3/2} \left (1-\cos ^2(e+f x)\right )}d\cos ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {\frac {\int \frac {1}{\sqrt {a \cos ^2(e+f x)} \left (1-\cos ^2(e+f x)\right )}d\cos ^2(e+f x)}{a}-\frac {2}{a \sqrt {a \cos ^2(e+f x)}}}{2 f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {\frac {2 \int \frac {1}{1-\frac {\cos ^4(e+f x)}{a}}d\sqrt {a \cos ^2(e+f x)}}{a^2}-\frac {2}{a \sqrt {a \cos ^2(e+f x)}}}{2 f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {a \cos ^2(e+f x)}}}{2 f}\) |
Input:
Int[Cot[e + f*x]/(a - a*Sin[e + f*x]^2)^(3/2),x]
Output:
-1/2*((2*ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]])/a^(3/2) - 2/(a*Sqrt[a*Co s[e + f*x]^2]))/f
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_. ), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Simp[ff^((m + 1 )/2)/(2*f) Subst[Int[x^((m - 1)/2)*((b*ff^(n/2)*x^(n/2))^p/(1 - ff*x)^((m + 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{b, e, f, p}, x] && Inte gerQ[(m - 1)/2] && IntegerQ[n/2]
Time = 0.62 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.42
method | result | size |
default | \(\frac {-\ln \left (\frac {2 \sqrt {a}\, \sqrt {a \cos \left (f x +e \right )^{2}}+2 a}{\sin \left (f x +e \right )}\right ) a^{2} \cos \left (f x +e \right )^{2}+\sqrt {a \cos \left (f x +e \right )^{2}}\, a^{\frac {3}{2}}}{a^{\frac {7}{2}} \cos \left (f x +e \right )^{2} f}\) | \(75\) |
risch | \(\frac {2}{a \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, f}+\frac {2 \ln \left ({\mathrm e}^{i f x}-{\mathrm e}^{-i e}\right ) \cos \left (f x +e \right )}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}-\frac {2 \ln \left ({\mathrm e}^{i f x}+{\mathrm e}^{-i e}\right ) \cos \left (f x +e \right )}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}\) | \(144\) |
Input:
int(cot(f*x+e)/(a-a*sin(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/a^(7/2)/cos(f*x+e)^2*(-ln(2/sin(f*x+e)*(a^(1/2)*(a*cos(f*x+e)^2)^(1/2)+a ))*a^2*cos(f*x+e)^2+(a*cos(f*x+e)^2)^(1/2)*a^(3/2))/f
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09 \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (\cos \left (f x + e\right ) \log \left (-\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1}\right ) - 2\right )}}{2 \, a^{2} f \cos \left (f x + e\right )^{2}} \] Input:
integrate(cot(f*x+e)/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="fricas")
Output:
-1/2*sqrt(a*cos(f*x + e)^2)*(cos(f*x + e)*log(-(cos(f*x + e) + 1)/(cos(f*x + e) - 1)) - 2)/(a^2*f*cos(f*x + e)^2)
\[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot {\left (e + f x \right )}}{\left (- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(f*x+e)/(a-a*sin(f*x+e)**2)**(3/2),x)
Output:
Integral(cot(e + f*x)/(-a*(sin(e + f*x) - 1)*(sin(e + f*x) + 1))**(3/2), x )
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.38 \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\frac {\log \left (\frac {2 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (f x + e\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {3}{2}}} - \frac {1}{\sqrt {-a \sin \left (f x + e\right )^{2} + a} a}}{f} \] Input:
integrate(cot(f*x+e)/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")
Output:
-(log(2*sqrt(-a*sin(f*x + e)^2 + a)*sqrt(a)/abs(sin(f*x + e)) + 2*a/abs(si n(f*x + e)))/a^(3/2) - 1/(sqrt(-a*sin(f*x + e)^2 + a)*a))/f
Exception generated. \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(f*x+e)/(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 i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\mathrm {cot}\left (e+f\,x\right )}{{\left (a-a\,{\sin \left (e+f\,x\right )}^2\right )}^{3/2}} \,d x \] Input:
int(cot(e + f*x)/(a - a*sin(e + f*x)^2)^(3/2),x)
Output:
int(cot(e + f*x)/(a - a*sin(e + f*x)^2)^(3/2), x)
\[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\sin \left (f x +e \right )^{2}+1}\, \cot \left (f x +e \right )}{\sin \left (f x +e \right )^{4}-2 \sin \left (f x +e \right )^{2}+1}d x \right )}{a^{2}} \] Input:
int(cot(f*x+e)/(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))/(sin(e + f*x)**4 - 2*sin(e + f*x)**2 + 1),x))/a**2