Integrand size = 17, antiderivative size = 39 \[ \int (b \csc (e+f x))^m \tan (e+f x) \, dx=\frac {(b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},\csc ^2(e+f x)\right )}{f m} \] Output:
(b*csc(f*x+e))^m*hypergeom([1, 1/2*m],[1+1/2*m],csc(f*x+e)^2)/f/m
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.33 \[ \int (b \csc (e+f x))^m \tan (e+f x) \, dx=-\frac {(b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,1-\frac {m}{2},2-\frac {m}{2},\sin ^2(e+f x)\right ) \sin ^2(e+f x)}{f (-2+m)} \] Input:
Integrate[(b*Csc[e + f*x])^m*Tan[e + f*x],x]
Output:
-(((b*Csc[e + f*x])^m*Hypergeometric2F1[1, 1 - m/2, 2 - m/2, Sin[e + f*x]^ 2]*Sin[e + f*x]^2)/(f*(-2 + m)))
Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3042, 25, 3086, 25, 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.
\(\displaystyle \int \tan (e+f x) (b \csc (e+f x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\left (b \sec \left (e+f x-\frac {\pi }{2}\right )\right )^m}{\tan \left (e+f x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\left (b \sec \left (\frac {1}{2} (2 e-\pi )+f x\right )\right )^m}{\tan \left (\frac {1}{2} (2 e-\pi )+f x\right )}dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {b \int -\frac {(b \csc (e+f x))^{m-1}}{1-\csc ^2(e+f x)}d\csc (e+f x)}{f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b \int \frac {(b \csc (e+f x))^{m-1}}{1-\csc ^2(e+f x)}d\csc (e+f x)}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {(b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},\csc ^2(e+f x)\right )}{f m}\) |
Input:
Int[(b*Csc[e + f*x])^m*Tan[e + f*x],x]
Output:
((b*Csc[e + f*x])^m*Hypergeometric2F1[1, m/2, (2 + m)/2, Csc[e + f*x]^2])/ (f*m)
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[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
\[\int \left (b \csc \left (f x +e \right )\right )^{m} \tan \left (f x +e \right )d x\]
Input:
int((b*csc(f*x+e))^m*tan(f*x+e),x)
Output:
int((b*csc(f*x+e))^m*tan(f*x+e),x)
\[ \int (b \csc (e+f x))^m \tan (e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right ) \,d x } \] Input:
integrate((b*csc(f*x+e))^m*tan(f*x+e),x, algorithm="fricas")
Output:
integral((b*csc(f*x + e))^m*tan(f*x + e), x)
\[ \int (b \csc (e+f x))^m \tan (e+f x) \, dx=\int \left (b \csc {\left (e + f x \right )}\right )^{m} \tan {\left (e + f x \right )}\, dx \] Input:
integrate((b*csc(f*x+e))**m*tan(f*x+e),x)
Output:
Integral((b*csc(e + f*x))**m*tan(e + f*x), x)
\[ \int (b \csc (e+f x))^m \tan (e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right ) \,d x } \] Input:
integrate((b*csc(f*x+e))^m*tan(f*x+e),x, algorithm="maxima")
Output:
integrate((b*csc(f*x + e))^m*tan(f*x + e), x)
\[ \int (b \csc (e+f x))^m \tan (e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right ) \,d x } \] Input:
integrate((b*csc(f*x+e))^m*tan(f*x+e),x, algorithm="giac")
Output:
integrate((b*csc(f*x + e))^m*tan(f*x + e), x)
Timed out. \[ \int (b \csc (e+f x))^m \tan (e+f x) \, dx=\int \mathrm {tan}\left (e+f\,x\right )\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \] Input:
int(tan(e + f*x)*(b/sin(e + f*x))^m,x)
Output:
int(tan(e + f*x)*(b/sin(e + f*x))^m, x)
\[ \int (b \csc (e+f x))^m \tan (e+f x) \, dx=b^{m} \left (\int \csc \left (f x +e \right )^{m} \tan \left (f x +e \right )d x \right ) \] Input:
int((b*csc(f*x+e))^m*tan(f*x+e),x)
Output:
b**m*int(csc(e + f*x)**m*tan(e + f*x),x)