\(\int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx\) [184]

Optimal result

Integrand size = 19, antiderivative size = 78 \[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=-\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {6-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n}{f (1-n)} \]

[Out]

-cos(f*x+e)*hypergeom([3-1/2*n, 1/2-1/2*n],[3/2-1/2*n],cos(f*x+e)^2)*(b*tan(f*x+e))^n/f/(1-n)/((sin(f*x+e)^2)^
(1/2*n))

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2681, 2656} \[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=-\frac {\cos (e+f x) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {6-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right )}{f (1-n)} \]

[In]

Int[Csc[e + f*x]^5*(b*Tan[e + f*x])^n,x]

[Out]

-((Cos[e + f*x]*Hypergeometric2F1[(1 - n)/2, (6 - 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)))

Rule 2656

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^(2*IntPar
t[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*FracPart[(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, Cos[e + f*x]^2], x] /; FreeQ[{a
, b, e, f, m, n}, x] && SimplerQ[n, m]

Rule 2681

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[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])

Rubi steps \begin{align*} \text {integral}& = \left (\cos ^n(e+f x) \sin ^{-n}(e+f x) (b \tan (e+f x))^n\right ) \int \cos ^{-n}(e+f x) \sin ^{-5+n}(e+f x) \, dx \\ & = -\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {6-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)} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 14.17 (sec) , antiderivative size = 1017, normalized size of antiderivative = 13.04 \[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\frac {3 \cot ^2\left (\frac {1}{2} (e+f x)\right ) \operatorname {Hypergeometric2F1}\left (-1+\frac {n}{2},n,\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}{16 f (-2+n)}+\frac {\cot ^2\left (\frac {1}{2} (e+f x)\right ) \left ((-2+n) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \operatorname {Hypergeometric2F1}\left (-2+\frac {n}{2},n,-1+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(-4+n) \operatorname {Hypergeometric2F1}\left (-1+\frac {n}{2},n,\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\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}{16 f (-4+n) (-2+n)}+\frac {3 (4+n) \operatorname {AppellF1}\left (1+\frac {n}{2},n,1,2+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^2(e+f x) (b \tan (e+f x))^n}{16 f (2+n) \left (2 \left (\operatorname {AppellF1}\left (2+\frac {n}{2},n,2,3+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n \operatorname {AppellF1}\left (2+\frac {n}{2},1+n,1,3+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (-1+\cos (e+f x))+(4+n) \operatorname {AppellF1}\left (1+\frac {n}{2},n,1,2+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (1+\cos (e+f x))\right )}+\frac {3 \operatorname {Hypergeometric2F1}\left (1+\frac {n}{2},n,2+\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 \tan ^2\left (\frac {1}{2} (e+f x)\right ) (b \tan (e+f x))^n}{16 f (2+n)}+\frac {\left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^n \tan ^2\left (\frac {1}{2} (e+f x)\right ) \left ((4+n) \operatorname {Hypergeometric2F1}\left (1+\frac {n}{2},n,2+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(2+n) \operatorname {Hypergeometric2F1}\left (2+\frac {n}{2},n,3+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (b \tan (e+f x))^n}{16 f (2+n) (4+n)}+\frac {3 \cos ^2\left (\frac {1}{2} (e+f x)\right ) \cot \left (\frac {1}{2} (e+f x)\right ) \left ((2+n) \operatorname {Hypergeometric2F1}\left (\frac {n}{2},n,1+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n \operatorname {AppellF1}\left (1+\frac {n}{2},n,1,2+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (b \tan (e+f x))^n}{f n (2+n) \left (-8 \operatorname {AppellF1}\left (1+\frac {n}{2},n,1,2+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )+\frac {8 \left (2 \operatorname {AppellF1}\left (2+\frac {n}{2},n,2,3+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 n \operatorname {AppellF1}\left (2+\frac {n}{2},1+n,1,3+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(4+n) \cot ^4\left (\frac {1}{2} (e+f x)\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n}\right ) \tan ^3\left (\frac {1}{2} (e+f x)\right )}{4+n}\right )} \]

[In]

Integrate[Csc[e + f*x]^5*(b*Tan[e + f*x])^n,x]

[Out]

(3*Cot[(e + f*x)/2]^2*Hypergeometric2F1[-1 + n/2, n, 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)/(16*f*(-2 + n)) + (Cot[(e + f*x)/2]^2*((-2 + n)*Cot[(e + f*x)/2]^2*Hypergeometric2F1[-
2 + n/2, n, -1 + n/2, Tan[(e + f*x)/2]^2] + (-4 + n)*Hypergeometric2F1[-1 + n/2, n, 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)/(16*f*(-4 + n)*(-2 + n)) + (3*(4 + n)*AppellF1[1 + n/2
, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[e + f*x]^2*(b*Tan[e + f*x])^n)/(16*f*(2 + n)*(2*
(AppellF1[2 + n/2, n, 2, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[2 + n/2, 1 + n, 1, 3 +
n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x]) + (4 + n)*AppellF1[1 + n/2, n, 1, 2 + n/2,
Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x]))) + (3*Hypergeometric2F1[1 + n/2, n, 2 + n/2, Tan
[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2*(b*Tan[e + f*x])^n)/(16*f*(2 + n)) + (
(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2*((4 + n)*Hypergeometric2F1[1 + n/2, n, 2 + n/2, Tan[(e
+ f*x)/2]^2] + (2 + n)*Hypergeometric2F1[2 + n/2, n, 3 + n/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)*(b*Tan[e
+ f*x])^n)/(16*f*(2 + n)*(4 + n)) + (3*Cos[(e + f*x)/2]^2*Cot[(e + f*x)/2]*((2 + n)*Hypergeometric2F1[n/2, n,
1 + n/2, Tan[(e + f*x)/2]^2] - n*AppellF1[1 + n/2, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Ta
n[(e + f*x)/2]^2)*(b*Tan[e + f*x])^n)/(f*n*(2 + n)*(-8*AppellF1[1 + n/2, n, 1, 2 + n/2, Tan[(e + f*x)/2]^2, -T
an[(e + f*x)/2]^2]*Tan[(e + f*x)/2] + (8*(2*AppellF1[2 + n/2, n, 2, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x
)/2]^2] - 2*n*AppellF1[2 + n/2, 1 + n, 1, 3 + n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + ((4 + n)*Cot[(e
+ f*x)/2]^4)/(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n)*Tan[(e + f*x)/2]^3)/(4 + n)))

Maple [F]

\[\int \left (\csc ^{5}\left (f x +e \right )\right ) \left (b \tan \left (f x +e \right )\right )^{n}d x\]

[In]

int(csc(f*x+e)^5*(b*tan(f*x+e))^n,x)

[Out]

int(csc(f*x+e)^5*(b*tan(f*x+e))^n,x)

Fricas [F]

\[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{5} \,d x } \]

[In]

integrate(csc(f*x+e)^5*(b*tan(f*x+e))^n,x, algorithm="fricas")

[Out]

integral((b*tan(f*x + e))^n*csc(f*x + e)^5, x)

Sympy [F]

\[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\int \left (b \tan {\left (e + f x \right )}\right )^{n} \csc ^{5}{\left (e + f x \right )}\, dx \]

[In]

integrate(csc(f*x+e)**5*(b*tan(f*x+e))**n,x)

[Out]

Integral((b*tan(e + f*x))**n*csc(e + f*x)**5, x)

Maxima [F]

\[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{5} \,d x } \]

[In]

integrate(csc(f*x+e)^5*(b*tan(f*x+e))^n,x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e))^n*csc(f*x + e)^5, x)

Giac [F]

\[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{5} \,d x } \]

[In]

integrate(csc(f*x+e)^5*(b*tan(f*x+e))^n,x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e))^n*csc(f*x + e)^5, x)

Mupad [F(-1)]

Timed out. \[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\sin \left (e+f\,x\right )}^5} \,d x \]

[In]

int((b*tan(e + f*x))^n/sin(e + f*x)^5,x)

[Out]

int((b*tan(e + f*x))^n/sin(e + f*x)^5, x)