Integrand size = 23, antiderivative size = 120 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{3 a f}-\frac {(3 a-b (1-2 p)) \cot (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{3 a f} \]
-1/3*cot(f*x+e)^3*(a+b*tan(f*x+e)^2)^(p+1)/a/f-1/3*(3*a-b*(1-2*p))*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/a/f /((1+b*tan(f*x+e)^2/a)^p)
Time = 1.42 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (-a-b \tan ^2(e+f x)-(3 a+b (-1+2 p)) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right ) \tan ^2(e+f x) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right )}{3 a f} \]
(Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p*(-a - b*Tan[e + f*x]^2 - ((3*a + b*(-1 + 2*p))*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*Ta n[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/a)^p))/(3*a*f)
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4146, 359, 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.
\(\displaystyle \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \tan (e+f x)^2\right )^p}{\sin (e+f x)^4}dx\) |
\(\Big \downarrow \) 4146 |
\(\displaystyle \frac {\int \cot ^4(e+f x) \left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^pd\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {\frac {(3 a-b (1-2 p)) \int \cot ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )^pd\tan (e+f x)}{3 a}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{3 a}}{f}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle \frac {\frac {(3 a-b (1-2 p)) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \int \cot ^2(e+f x) \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^pd\tan (e+f x)}{3 a}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{3 a}}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {-\frac {(3 a-b (1-2 p)) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \tan ^2(e+f x)}{a}\right )}{3 a}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{3 a}}{f}\) |
(-1/3*(Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(1 + p))/a - ((3*a - b*(1 - 2 *p))*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)/(3*a*(1 + (b*Tan[e + f*x]^2)/a)^p))/f
3.2.62.3.1 Defintions of rubi rules used
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
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]
\[\int \csc \left (f x +e \right )^{4} \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
\[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\text {Timed out} \]
\[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{4} \,d x } \]
\[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p}{{\sin \left (e+f\,x\right )}^4} \,d x \]