Integrand size = 25, antiderivative size = 106 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=-\frac {\cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a f}-\frac {\sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \]
-cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a/f-EllipticE(sin(f*x+e),(-b/a)^(1/2) )*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(a+b*sin(f*x+e)^2)^(1/2)/a/f/(1+b*sin(f* x+e)^2/a)^(1/2)
Time = 0.72 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=-\frac {\sqrt {2 a+b-b \cos (2 (e+f x))} \cot (e+f x)}{\sqrt {2} a f}-\frac {\sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
-((Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*Cot[e + f*x])/(Sqrt[2]*a*f)) - (Sqrt [(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)])/(f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])
Time = 0.31 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3675, 377, 25, 330, 327}
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.
| \(\displaystyle \int \frac {\cot ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^2 \sqrt {a+b \sin (e+f x)^2}}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\csc ^2(e+f x) \sqrt {1-\sin ^2(e+f x)}}{\sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 377 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\int -\frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {\int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {\sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {\sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{a \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}\right )}{f}\) |
(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*(-((Csc[e + f*x]*Sqrt[1 - Sin[e + f*x]^ 2]*Sqrt[a + b*Sin[e + f*x]^2])/a) - (EllipticE[ArcSin[Sin[e + f*x]], -(b/a )]*Sqrt[a + b*Sin[e + f*x]^2])/(a*Sqrt[1 + (b*Sin[e + f*x]^2)/a])))/f
3.6.19.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b *c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m , 2, p, q, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 )*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[x^m*((a + b*ff^2*x^2) ^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b , e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 2.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(-\frac {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, a \sin \left (f x +e \right ) E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )}{a \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(120\) |
-(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2+(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e) ^2+(a+b)/a)^(1/2)*a*sin(f*x+e)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2)))/a/sin (f*x+e)/cos(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2)/f
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 723, normalized size of antiderivative = 6.82 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=-\frac {2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} b^{2} \cos \left (f x + e\right ) + {\left (2 i \, \sqrt {-b} b^{2} \sqrt {\frac {a^{2} + a b}{b^{2}}} \sin \left (f x + e\right ) + {\left (2 i \, a b + i \, b^{2}\right )} \sqrt {-b} \sin \left (f x + e\right )\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (-2 i \, \sqrt {-b} b^{2} \sqrt {\frac {a^{2} + a b}{b^{2}}} \sin \left (f x + e\right ) + {\left (-2 i \, a b - i \, b^{2}\right )} \sqrt {-b} \sin \left (f x + e\right )\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + 2 \, {\left (2 \, {\left (-i \, a b - i \, b^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \sin \left (f x + e\right ) + {\left (2 i \, a^{2} + i \, a b\right )} \sqrt {-b} \sin \left (f x + e\right )\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + 2 \, {\left (2 \, {\left (i \, a b + i \, b^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \sin \left (f x + e\right ) + {\left (-2 i \, a^{2} - i \, a b\right )} \sqrt {-b} \sin \left (f x + e\right )\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}})}{2 \, a b^{2} f \sin \left (f x + e\right )} \]
-1/2*(2*sqrt(-b*cos(f*x + e)^2 + a + b)*b^2*cos(f*x + e) + (2*I*sqrt(-b)*b ^2*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + (2*I*a*b + I*b^2)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqr t((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e)) ), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (- 2*I*sqrt(-b)*b^2*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + (-2*I*a*b - I*b^2)*s qrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*ellipt ic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/ b^2))/b^2) + 2*(2*(-I*a*b - I*b^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + (2*I*a^2 + I*a*b)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a*b) /b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2* a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a* b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + 2*(2*(I*a*b + I*b^2)*sqrt(-b)*sqrt( (a^2 + a*b)/b^2)*sin(f*x + e) + (-2*I*a^2 - I*a*b)*sqrt(-b)*sin(f*x + e))* sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b* sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a ^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2))/(a*b^2*f*s in(f*x + e))
\[ \int \frac {\cot ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {\cot ^{2}{\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\cot ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{2}}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \]
\[ \int \frac {\cot ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{2}}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^2}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \]