\(\int (d \csc (e+f x))^m (b (c \tan (e+f x))^n)^p \, dx\) [500]

Optimal result

Integrand size = 25, antiderivative size = 104 \[ \int (d \csc (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {\cos ^2(e+f x)^{\frac {1}{2} (1+n p)} (d \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+n p),\frac {1}{2} (1-m+n p),\frac {1}{2} (3-m+n p),\sin ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-m+n p)} \] Output:

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

Mathematica [C] (warning: unable to verify)

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

Time = 2.28 (sec) , antiderivative size = 319, normalized size of antiderivative = 3.07 \[ \int (d \csc (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=-\frac {d (-3+m-n p) \operatorname {AppellF1}\left (\frac {1}{2} (1-m+n p),n p,1-m,\frac {1}{2} (3-m+n p),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (d \csc (e+f x))^{-1+m} \left (b (c \tan (e+f x))^n\right )^p}{f (-1+m-n p) \left ((-3+m-n p) \operatorname {AppellF1}\left (\frac {1}{2} (1-m+n p),n p,1-m,\frac {1}{2} (3-m+n p),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left ((-1+m) \operatorname {AppellF1}\left (\frac {1}{2} (3-m+n p),n p,2-m,\frac {1}{2} (5-m+n p),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n p \operatorname {AppellF1}\left (\frac {1}{2} (3-m+n p),1+n p,1-m,\frac {1}{2} (5-m+n p),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \] Input:

Integrate[(d*Csc[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]
 

Output:

-((d*(-3 + m - n*p)*AppellF1[(1 - m + n*p)/2, n*p, 1 - m, (3 - m + n*p)/2, 
 Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(d*Csc[e + f*x])^(-1 + m)*(b*(c* 
Tan[e + f*x])^n)^p)/(f*(-1 + m - n*p)*((-3 + m - n*p)*AppellF1[(1 - m + n* 
p)/2, n*p, 1 - m, (3 - m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2 
] - 2*((-1 + m)*AppellF1[(3 - m + n*p)/2, n*p, 2 - m, (5 - m + n*p)/2, Tan 
[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*p*AppellF1[(3 - m + n*p)/2, 1 + 
n*p, 1 - m, (5 - m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan 
[(e + f*x)/2]^2)))
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 4142, 3042, 3098, 3042, 3082, 3042, 3057}

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.

\(\displaystyle \int (d \csc (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (d \csc (e+f x))^m \left (b (c \tan (e+f x))^n\right )^pdx\)

\(\Big \downarrow \) 4142

\(\displaystyle (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p \int (d \csc (e+f x))^m (c \tan (e+f x))^{n p}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p \int (d \csc (e+f x))^m (c \tan (e+f x))^{n p}dx\)

\(\Big \downarrow \) 3098

\(\displaystyle \left (\frac {\sin (e+f x)}{d}\right )^m (d \csc (e+f x))^m (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p \int \left (\frac {\sin (e+f x)}{d}\right )^{-m} (c \tan (e+f x))^{n p}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \left (\frac {\sin (e+f x)}{d}\right )^m (d \csc (e+f x))^m (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p \int \left (\frac {\sin (e+f x)}{d}\right )^{-m} (c \tan (e+f x))^{n p}dx\)

\(\Big \downarrow \) 3082

\(\displaystyle \frac {\sin (e+f x) (d \csc (e+f x))^m \cos ^{n p}(e+f x) \left (b (c \tan (e+f x))^n\right )^p \left (\frac {\sin (e+f x)}{d}\right )^{m-n p-1} \int \cos ^{-n p}(e+f x) \left (\frac {\sin (e+f x)}{d}\right )^{n p-m}dx}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sin (e+f x) (d \csc (e+f x))^m \cos ^{n p}(e+f x) \left (b (c \tan (e+f x))^n\right )^p \left (\frac {\sin (e+f x)}{d}\right )^{m-n p-1} \int \cos (e+f x)^{-n p} \left (\frac {\sin (e+f x)}{d}\right )^{n p-m}dx}{d}\)

\(\Big \downarrow \) 3057

\(\displaystyle \frac {\tan (e+f x) (d \csc (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (n p+1)} \left (b (c \tan (e+f x))^n\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (n p+1),\frac {1}{2} (-m+n p+1),\frac {1}{2} (-m+n p+3),\sin ^2(e+f x)\right )}{f (-m+n p+1)}\)

Input:

Int[(d*Csc[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]
 

Output:

((Cos[e + f*x]^2)^((1 + n*p)/2)*(d*Csc[e + f*x])^m*Hypergeometric2F1[(1 + 
n*p)/2, (1 - m + n*p)/2, (3 - m + n*p)/2, Sin[e + f*x]^2]*Tan[e + f*x]*(b* 
(c*Tan[e + f*x])^n)^p)/(f*(1 - m + n*p))
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3057

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*Frac 
Part[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^2)^Fr 
acPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[ 
e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x]
 

rule 3082

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[a*Cos[e + f*x]^(n + 1)*((b*Tan[e + f*x])^(n + 1)/(b* 
(a*Sin[e + f*x])^(n + 1)))   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]
 

rule 3098

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[(a*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/a)^FracPar 
t[m]   Int[(b*Tan[e + f*x])^n/(Sin[e + f*x]/a)^m, x], x] /; FreeQ[{a, b, e, 
 f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
 

rule 4142

Int[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> S 
imp[b^IntPart[p]*((b*(c*Tan[e + f*x])^n)^FracPart[p]/(c*Tan[e + f*x])^(n*Fr 
acPart[p]))   Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; FreeQ[{ 
b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || Ma 
tchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, 
 cos, tan, cot, sec, csc}, trig]])
 

Maple [F]

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

Input:

int((d*csc(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x)
 

Output:

int((d*csc(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x)
 

Fricas [F]

\[ \int (d \csc (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \] Input:

integrate((d*csc(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm="fricas")
 

Output:

integral(((c*tan(f*x + e))^n*b)^p*(d*csc(f*x + e))^m, x)
 

Sympy [F]

\[ \int (d \csc (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \left (d \csc {\left (e + f x \right )}\right )^{m}\, dx \] Input:

integrate((d*csc(f*x+e))**m*(b*(c*tan(f*x+e))**n)**p,x)
 

Output:

Integral((b*(c*tan(e + f*x))**n)**p*(d*csc(e + f*x))**m, x)
 

Maxima [F]

\[ \int (d \csc (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \] Input:

integrate((d*csc(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm="maxima")
 

Output:

integrate(((c*tan(f*x + e))^n*b)^p*(d*csc(f*x + e))^m, x)
 

Giac [F]

\[ \int (d \csc (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \] Input:

integrate((d*csc(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm="giac")
 

Output:

integrate(((c*tan(f*x + e))^n*b)^p*(d*csc(f*x + e))^m, x)
 

Mupad [F(-1)]

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

int((d/sin(e + f*x))^m*(b*(c*tan(e + f*x))^n)^p,x)
 

Output:

int((d/sin(e + f*x))^m*(b*(c*tan(e + f*x))^n)^p, x)
 

Reduce [F]

\[ \int (d \csc (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=d^{m} c^{n p} b^{p} \left (\int \tan \left (f x +e \right )^{n p} \csc \left (f x +e \right )^{m}d x \right ) \] Input:

int((d*csc(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x)
 

Output:

d**m*c**(n*p)*b**p*int(tan(e + f*x)**(n*p)*csc(e + f*x)**m,x)