Integrand size = 25, antiderivative size = 68 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot (e+f x)}{(a+b) f \sqrt {a+b+b \tan ^2(e+f x)}}-\frac {2 b \tan (e+f x)}{(a+b)^2 f \sqrt {a+b+b \tan ^2(e+f x)}} \]
-cot(f*x+e)/(a+b)/f/(a+b+b*tan(f*x+e)^2)^(1/2)-2*b*tan(f*x+e)/(a+b)^2/f/(a +b+b*tan(f*x+e)^2)^(1/2)
Time = 1.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {(a+2 b+a \cos (2 (e+f x))) (a+3 b+(a-b) \cos (2 (e+f x))) \csc (e+f x) \sec ^3(e+f x)}{4 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
-1/4*((a + 2*b + a*Cos[2*(e + f*x)])*(a + 3*b + (a - b)*Cos[2*(e + f*x)])* Csc[e + f*x]*Sec[e + f*x]^3)/((a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2))
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 4620, 245, 208}
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.
\(\displaystyle \int \frac {\csc ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (e+f x)^2 \left (a+b \sec (e+f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4620 |
\(\displaystyle \frac {\int \frac {\cot ^2(e+f x)}{\left (b \tan ^2(e+f x)+a+b\right )^{3/2}}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \frac {-\frac {2 b \int \frac {1}{\left (b \tan ^2(e+f x)+a+b\right )^{3/2}}d\tan (e+f x)}{a+b}-\frac {\cot (e+f x)}{(a+b) \sqrt {a+b \tan ^2(e+f x)+b}}}{f}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {-\frac {2 b \tan (e+f x)}{(a+b)^2 \sqrt {a+b \tan ^2(e+f x)+b}}-\frac {\cot (e+f x)}{(a+b) \sqrt {a+b \tan ^2(e+f x)+b}}}{f}\) |
(-(Cot[e + f*x]/((a + b)*Sqrt[a + b + b*Tan[e + f*x]^2])) - (2*b*Tan[e + f *x])/((a + b)^2*Sqrt[a + b + b*Tan[e + f*x]^2]))/f
3.2.16.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1)/f Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1 + f f^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
Time = 3.99 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.24
method | result | size |
default | \(-\frac {\left (b +a \cos \left (f x +e \right )^{2}\right ) \left (a \cos \left (f x +e \right )^{2}-\cos \left (f x +e \right )^{2} b +2 b \right ) \sec \left (f x +e \right )^{3} \csc \left (f x +e \right )}{f \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \sec \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}\) | \(84\) |
-1/f/(a^2+2*a*b+b^2)*(b+a*cos(f*x+e)^2)*(a*cos(f*x+e)^2-cos(f*x+e)^2*b+2*b )/(a+b*sec(f*x+e)^2)^(3/2)*sec(f*x+e)^3*csc(f*x+e)
Time = 0.34 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.50 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{{\left ({\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} f\right )} \sin \left (f x + e\right )} \]
-((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/c os(f*x + e)^2)/(((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 + (a^2*b + 2*a*b ^2 + b^3)*f)*sin(f*x + e))
\[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\csc ^{2}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\frac {2 \, b \tan \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a + b} {\left (a + b\right )}^{2}} + \frac {1}{\sqrt {b \tan \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} \tan \left (f x + e\right )}}{f} \]
-(2*b*tan(f*x + e)/(sqrt(b*tan(f*x + e)^2 + a + b)*(a + b)^2) + 1/(sqrt(b* tan(f*x + e)^2 + a + b)*(a + b)*tan(f*x + e)))/f
\[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{2}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Time = 26.10 (sec) , antiderivative size = 2151, normalized size of antiderivative = 31.63 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
-((a + b/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2)^2)^(1/2)*(2*exp(e *2i + f*x*2i) + exp(e*4i + f*x*4i) + 1)*(exp(e*4i + f*x*4i)*(((a + 4*b)*(( (a + 4*b)*((((a^2*(a + 3*b)*(a*(a - b) - (a + 3*b)^2))/(a*b + a^2) + (a*(a + 3*b)^2*(a*(a + 3*b) - a*(a + 4*b)))/(a*b + a^2))*1i)/(8*f*(a*b^2 + a^2* b)*(a + 3*b)) - (a^3*(a + 3*b)*3i)/(8*f*(a*b + a^2)*(a*b^2 + a^2*b)) + (a^ 2*(a + 3*b)*(a + 4*b)*1i)/(8*f*(a*b + a^2)*(a*b^2 + a^2*b))))/a - (((a + 3 *b)^3 - ((a + 3*b)*(a*(a - b) - (a + 3*b)^2)*(a*(a + 3*b) - a*(a + 4*b)))/ (a*b + a^2))*1i)/(8*f*(a*b^2 + a^2*b)*(a + 3*b)) - (((a^2*(a + 3*b)*(a*(a - b) - (a + 3*b)^2))/(a*b + a^2) + (a*(a + 3*b)^2*(a*(a + 3*b) - a*(a + 4* b)))/(a*b + a^2))*3i)/(8*f*(a*b^2 + a^2*b)*(a + 3*b)) + (a^3*(a + 3*b)*3i) /(8*f*(a*b + a^2)*(a*b^2 + a^2*b)) + (a^2*(a + 3*b)*(a + 4*b)*1i)/(8*f*(a* b + a^2)*(a*b^2 + a^2*b))))/a + ((a + 4*b)*((((a^2*(a + 3*b)*(a*(a - b) - (a + 3*b)^2))/(a*b + a^2) + (a*(a + 3*b)^2*(a*(a + 3*b) - a*(a + 4*b)))/(a *b + a^2))*1i)/(8*f*(a*b^2 + a^2*b)*(a + 3*b)) - (a^3*(a + 3*b)*3i)/(8*f*( a*b + a^2)*(a*b^2 + a^2*b)) + (a^2*(a + 3*b)*(a + 4*b)*1i)/(8*f*(a*b + a^2 )*(a*b^2 + a^2*b))))/a + (((a + 3*b)^3 - ((a + 3*b)*(a*(a - b) - (a + 3*b) ^2)*(a*(a + 3*b) - a*(a + 4*b)))/(a*b + a^2))*3i)/(8*f*(a*b^2 + a^2*b)*(a + 3*b)) + (((a^2*(a + 3*b)*(a*(a - b) - (a + 3*b)^2))/(a*b + a^2) + (a*(a + 3*b)^2*(a*(a + 3*b) - a*(a + 4*b)))/(a*b + a^2))*3i)/(8*f*(a*b^2 + a^2*b )*(a + 3*b)) - (a^3*(a + 3*b)*1i)/(4*f*(a*b + a^2)*(a*b^2 + a^2*b))) + ...