Integrand size = 23, antiderivative size = 69 \[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {a+b \tan ^2(e+f x)}{a}\right )^{-p}}{f} \] Output:
-cot(f*x+e)*hypergeom([-1/2, -p],[1/2],-b*tan(f*x+e)^2/a)*(a+b*tan(f*x+e)^ 2)^p/f/(((a+b*tan(f*x+e)^2)/a)^p)
Time = 0.49 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f} \] Input:
Integrate[Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]
Output:
-((Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]* (a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p))
Time = 0.41 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4146, 279, 278}
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.
\(\displaystyle \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \tan (e+f x)^2\right )^p}{\sin (e+f x)^2}dx\) |
\(\Big \downarrow \) 4146 |
\(\displaystyle \frac {\int \cot ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )^pd\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle \frac {\left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \int \cot ^2(e+f x) \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^pd\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right )}{f}\) |
Input:
Int[Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]
Output:
-((Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]* (a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
\[\int \csc \left (f x +e \right )^{2} \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
Input:
int(csc(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x)
Output:
int(csc(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x)
\[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{2} \,d x } \] Input:
integrate(csc(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x, algorithm="fricas")
Output:
integral((b*tan(f*x + e)^2 + a)^p*csc(f*x + e)^2, x)
Timed out. \[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\text {Timed out} \] Input:
integrate(csc(f*x+e)**2*(a+b*tan(f*x+e)**2)**p,x)
Output:
Timed out
\[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{2} \,d x } \] Input:
integrate(csc(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x, algorithm="maxima")
Output:
integrate((b*tan(f*x + e)^2 + a)^p*csc(f*x + e)^2, x)
\[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{2} \,d x } \] Input:
integrate(csc(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x, algorithm="giac")
Output:
integrate((b*tan(f*x + e)^2 + a)^p*csc(f*x + e)^2, x)
Timed out. \[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p}{{\sin \left (e+f\,x\right )}^2} \,d x \] Input:
int((a + b*tan(e + f*x)^2)^p/sin(e + f*x)^2,x)
Output:
int((a + b*tan(e + f*x)^2)^p/sin(e + f*x)^2, x)
\[ \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \csc \left (f x +e \right )^{2}d x \] Input:
int(csc(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x)
Output:
int((tan(e + f*x)**2*b + a)**p*csc(e + f*x)**2,x)