\(\int (b \csc (e+f x))^m \tan ^2(e+f x) \, dx\) [380]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int (b \csc (e+f x))^m \tan ^2(e+f x) \, dx=\frac {(b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-1+m),\frac {1}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1}{2} (-1+m)} \tan (e+f x)}{f} \]

[Out]

(b*csc(f*x+e))^m*hypergeom([-1/2, -1/2+1/2*m],[1/2],cos(f*x+e)^2)*(sin(f*x+e)^2)^(-1/2+1/2*m)*tan(f*x+e)/f

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2697} \[ \int (b \csc (e+f x))^m \tan ^2(e+f x) \, dx=\frac {\tan (e+f x) \sin ^2(e+f x)^{\frac {m-1}{2}} (b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {m-1}{2},\frac {1}{2},\cos ^2(e+f x)\right )}{f} \]

[In]

Int[(b*Csc[e + f*x])^m*Tan[e + f*x]^2,x]

[Out]

((b*Csc[e + f*x])^m*Hypergeometric2F1[-1/2, (-1 + m)/2, 1/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((-1 + m)/2)*Tan
[e + f*x])/f

Rule 2697

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*Sec[e + f
*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {(b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-1+m),\frac {1}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1}{2} (-1+m)} \tan (e+f x)}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.36 \[ \int (b \csc (e+f x))^m \tan ^2(e+f x) \, dx=\frac {(b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (1-\frac {m}{2},\frac {3}{2}-\frac {m}{2},\frac {5}{2}-\frac {m}{2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{-m/2} \tan ^3(e+f x)}{f (3-m)} \]

[In]

Integrate[(b*Csc[e + f*x])^m*Tan[e + f*x]^2,x]

[Out]

((b*Csc[e + f*x])^m*Hypergeometric2F1[1 - m/2, 3/2 - m/2, 5/2 - m/2, -Tan[e + f*x]^2]*Tan[e + f*x]^3)/(f*(3 -
m)*(Sec[e + f*x]^2)^(m/2))

Maple [F]

\[\int \left (b \csc \left (f x +e \right )\right )^{m} \left (\tan ^{2}\left (f x +e \right )\right )d x\]

[In]

int((b*csc(f*x+e))^m*tan(f*x+e)^2,x)

[Out]

int((b*csc(f*x+e))^m*tan(f*x+e)^2,x)

Fricas [F]

\[ \int (b \csc (e+f x))^m \tan ^2(e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{2} \,d x } \]

[In]

integrate((b*csc(f*x+e))^m*tan(f*x+e)^2,x, algorithm="fricas")

[Out]

integral((b*csc(f*x + e))^m*tan(f*x + e)^2, x)

Sympy [F]

\[ \int (b \csc (e+f x))^m \tan ^2(e+f x) \, dx=\int \left (b \csc {\left (e + f x \right )}\right )^{m} \tan ^{2}{\left (e + f x \right )}\, dx \]

[In]

integrate((b*csc(f*x+e))**m*tan(f*x+e)**2,x)

[Out]

Integral((b*csc(e + f*x))**m*tan(e + f*x)**2, x)

Maxima [F]

\[ \int (b \csc (e+f x))^m \tan ^2(e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{2} \,d x } \]

[In]

integrate((b*csc(f*x+e))^m*tan(f*x+e)^2,x, algorithm="maxima")

[Out]

integrate((b*csc(f*x + e))^m*tan(f*x + e)^2, x)

Giac [F]

\[ \int (b \csc (e+f x))^m \tan ^2(e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{2} \,d x } \]

[In]

integrate((b*csc(f*x+e))^m*tan(f*x+e)^2,x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e))^m*tan(f*x + e)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (b \csc (e+f x))^m \tan ^2(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]

[In]

int(tan(e + f*x)^2*(b/sin(e + f*x))^m,x)

[Out]

int(tan(e + f*x)^2*(b/sin(e + f*x))^m, x)