Integrand size = 21, antiderivative size = 83 \[ \int \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \sec ^2(e+f x)}{a-b}\right ) \left (a-b+b \sec ^2(e+f x)\right )^p \left (\frac {a-b+b \sec ^2(e+f x)}{a-b}\right )^{-p}}{f} \] Output:
-cos(f*x+e)*hypergeom([-1/2, -p],[1/2],-b*sec(f*x+e)^2/(a-b))*(a-b+b*sec(f *x+e)^2)^p/f/(((a-b+b*sec(f*x+e)^2)/(a-b))^p)
Time = 0.79 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \sec ^2(e+f x)}{a-b}\right ) \left (\frac {a-b+b \sec ^2(e+f x)}{a-b}\right )^{-p} \left (a+b \tan ^2(e+f x)\right )^p}{f} \] Input:
Integrate[Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^p,x]
Output:
-((Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))]*(a + b*Tan[e + f*x]^2)^p)/(f*((a - b + b*Sec[e + f*x]^2)/(a - b))^p) )
Time = 0.39 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 4147, 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.
\(\displaystyle \int \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (e+f x) \left (a+b \tan (e+f x)^2\right )^pdx\) |
\(\Big \downarrow \) 4147 |
\(\displaystyle \frac {\int \cos ^2(e+f x) \left (b \sec ^2(e+f x)+a-b\right )^pd\sec (e+f x)}{f}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle \frac {\left (a+b \sec ^2(e+f x)-b\right )^p \left (\frac {b \sec ^2(e+f x)}{a-b}+1\right )^{-p} \int \cos ^2(e+f x) \left (\frac {b \sec ^2(e+f x)}{a-b}+1\right )^pd\sec (e+f x)}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {\cos (e+f x) \left (a+b \sec ^2(e+f x)-b\right )^p \left (\frac {b \sec ^2(e+f x)}{a-b}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \sec ^2(e+f x)}{a-b}\right )}{f}\) |
Input:
Int[Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^p,x]
Output:
-((Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))]*(a - b + b*Sec[e + f*x]^2)^p)/(f*(1 + (b*Sec[e + f*x]^2)/(a - b))^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_.)*tan[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[1/(f*ff^ m) Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m + 1 )), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[( m - 1)/2]
\[\int \sin \left (f x +e \right ) \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
Input:
int(sin(f*x+e)*(a+b*tan(f*x+e)^2)^p,x)
Output:
int(sin(f*x+e)*(a+b*tan(f*x+e)^2)^p,x)
\[ \int \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right ) \,d x } \] Input:
integrate(sin(f*x+e)*(a+b*tan(f*x+e)^2)^p,x, algorithm="fricas")
Output:
integral((b*tan(f*x + e)^2 + a)^p*sin(f*x + e), x)
Timed out. \[ \int \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\text {Timed out} \] Input:
integrate(sin(f*x+e)*(a+b*tan(f*x+e)**2)**p,x)
Output:
Timed out
\[ \int \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right ) \,d x } \] Input:
integrate(sin(f*x+e)*(a+b*tan(f*x+e)^2)^p,x, algorithm="maxima")
Output:
integrate((b*tan(f*x + e)^2 + a)^p*sin(f*x + e), x)
\[ \int \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right ) \,d x } \] Input:
integrate(sin(f*x+e)*(a+b*tan(f*x+e)^2)^p,x, algorithm="giac")
Output:
integrate((b*tan(f*x + e)^2 + a)^p*sin(f*x + e), x)
Timed out. \[ \int \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \sin \left (e+f\,x\right )\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \] Input:
int(sin(e + f*x)*(a + b*tan(e + f*x)^2)^p,x)
Output:
int(sin(e + f*x)*(a + b*tan(e + f*x)^2)^p, x)
\[ \int \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \sin \left (f x +e \right )d x \] Input:
int(sin(f*x+e)*(a+b*tan(f*x+e)^2)^p,x)
Output:
int((tan(e + f*x)**2*b + a)**p*sin(e + f*x),x)