Integrand size = 17, antiderivative size = 78 \[ \int \csc (e+f x) (b \tan (e+f x))^n \, dx=-\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {2-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n}{f (1-n)} \] Output:
-cos(f*x+e)*hypergeom([1-1/2*n, 1/2-1/2*n],[3/2-1/2*n],cos(f*x+e)^2)*(b*ta n(f*x+e))^n/f/(1-n)/((sin(f*x+e)^2)^(1/2*n))
Time = 0.37 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.82 \[ \int \csc (e+f x) (b \tan (e+f x))^n \, dx=\frac {\operatorname {Hypergeometric2F1}\left (\frac {n}{2},n,1+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^n (b \tan (e+f x))^n}{f n} \] Input:
Integrate[Csc[e + f*x]*(b*Tan[e + f*x])^n,x]
Output:
(Hypergeometric2F1[n/2, n, 1 + n/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[ (e + f*x)/2]^2)^n*(b*Tan[e + f*x])^n)/(f*n)
Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3042, 3081, 3042, 3056}
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.
\(\displaystyle \int \csc (e+f x) (b \tan (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(b \tan (e+f x))^n}{\sin (e+f x)}dx\) |
\(\Big \downarrow \) 3081 |
\(\displaystyle \sin ^{-n}(e+f x) \cos ^n(e+f x) (b \tan (e+f x))^n \int \cos ^{-n}(e+f x) \sin ^{n-1}(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sin ^{-n}(e+f x) \cos ^n(e+f x) (b \tan (e+f x))^n \int \cos (e+f x)^{-n} \sin (e+f x)^{n-1}dx\) |
\(\Big \downarrow \) 3056 |
\(\displaystyle -\frac {\cos (e+f x) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {2-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right )}{f (1-n)}\) |
Input:
Int[Csc[e + f*x]*(b*Tan[e + f*x])^n,x]
Output:
-((Cos[e + f*x]*Hypergeometric2F1[(1 - n)/2, (2 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Tan[e + f*x])^n)/(f*(1 - n)*(Sin[e + f*x]^2)^(n/2)))
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-b^(2*IntPart[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*F racPart[(n - 1)/2])*((a*Cos[e + f*x])^(m + 1)/(a*f*(m + 1)*(Sin[e + f*x]^2) ^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, C os[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] && SimplerQ[n, m]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[Cos[e + f*x]^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^ n) Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(- 1)]) || IntegersQ[m - 1/2, n - 1/2])
\[\int \csc \left (f x +e \right ) \left (b \tan \left (f x +e \right )\right )^{n}d x\]
Input:
int(csc(f*x+e)*(b*tan(f*x+e))^n,x)
Output:
int(csc(f*x+e)*(b*tan(f*x+e))^n,x)
\[ \int \csc (e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right ) \,d x } \] Input:
integrate(csc(f*x+e)*(b*tan(f*x+e))^n,x, algorithm="fricas")
Output:
integral((b*tan(f*x + e))^n*csc(f*x + e), x)
\[ \int \csc (e+f x) (b \tan (e+f x))^n \, dx=\int \left (b \tan {\left (e + f x \right )}\right )^{n} \csc {\left (e + f x \right )}\, dx \] Input:
integrate(csc(f*x+e)*(b*tan(f*x+e))**n,x)
Output:
Integral((b*tan(e + f*x))**n*csc(e + f*x), x)
\[ \int \csc (e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right ) \,d x } \] Input:
integrate(csc(f*x+e)*(b*tan(f*x+e))^n,x, algorithm="maxima")
Output:
integrate((b*tan(f*x + e))^n*csc(f*x + e), x)
\[ \int \csc (e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right ) \,d x } \] Input:
integrate(csc(f*x+e)*(b*tan(f*x+e))^n,x, algorithm="giac")
Output:
integrate((b*tan(f*x + e))^n*csc(f*x + e), x)
Timed out. \[ \int \csc (e+f x) (b \tan (e+f x))^n \, dx=\int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{\sin \left (e+f\,x\right )} \,d x \] Input:
int((b*tan(e + f*x))^n/sin(e + f*x),x)
Output:
int((b*tan(e + f*x))^n/sin(e + f*x), x)
\[ \int \csc (e+f x) (b \tan (e+f x))^n \, dx=b^{n} \left (\int \tan \left (f x +e \right )^{n} \csc \left (f x +e \right )d x \right ) \] Input:
int(csc(f*x+e)*(b*tan(f*x+e))^n,x)
Output:
b**n*int(tan(e + f*x)**n*csc(e + f*x),x)