\(\int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx\) [164]

Optimal result

Integrand size = 19, antiderivative size = 46 \[ \int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx=-\frac {a^2 (a \sin (e+f x))^{-2+m}}{f (2-m)}-\frac {(a \sin (e+f x))^m}{f m} \]

[Out]

-a^2*(a*sin(f*x+e))^(-2+m)/f/(2-m)-(a*sin(f*x+e))^m/f/m

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2672, 14} \[ \int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx=-\frac {a^2 (a \sin (e+f x))^{m-2}}{f (2-m)}-\frac {(a \sin (e+f x))^m}{f m} \]

[In]

Int[Cot[e + f*x]^3*(a*Sin[e + f*x])^m,x]

[Out]

-((a^2*(a*Sin[e + f*x])^(-2 + m))/(f*(2 - m))) - (a*Sin[e + f*x])^m/(f*m)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2672

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^{-3+m} \left (a^2-x^2\right ) \, dx,x,a \sin (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (a^2 x^{-3+m}-x^{-1+m}\right ) \, dx,x,a \sin (e+f x)\right )}{f} \\ & = -\frac {a^2 (a \sin (e+f x))^{-2+m}}{f (2-m)}-\frac {(a \sin (e+f x))^m}{f m} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx=\frac {\left (2-m+m \csc ^2(e+f x)\right ) (a \sin (e+f x))^m}{f (-2+m) m} \]

[In]

Integrate[Cot[e + f*x]^3*(a*Sin[e + f*x])^m,x]

[Out]

((2 - m + m*Csc[e + f*x]^2)*(a*Sin[e + f*x])^m)/(f*(-2 + m)*m)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.19 (sec) , antiderivative size = 2751, normalized size of antiderivative = 59.80

method	result	size
risch	\(\text {Expression too large to display}\)	\(2751\)

[In]

int(cot(f*x+e)^3*(sin(f*x+e)*a)^m,x,method=_RETURNVERBOSE)

[Out]

-1/(-2+m)/f/(exp(2*I*(f*x+e))-1)^2/m*exp(I*(f*x+e))^(-m)*(exp(2*I*(f*x+e))-1)^m*(1/2)^m*a^m*(m*exp(-1/2*I*m*cs
gn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^3*Pi)*exp(-1/2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2
*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*Pi)*exp(1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^3*Pi)*exp
(1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2*csgn(I*a)*Pi)*exp(-1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f
*x)*sin(e))^2*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(-1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))
*csgn(I*a)*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(1/2*I*m*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^3)*e
xp(1/2*I*Pi*m*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2*csgn(I*exp(-I*(f*x+e))))*exp(1/2*I*Pi*m*csgn(sin(f*x)*co
s(e)+cos(f*x)*sin(e))^2*csgn(I*exp(2*I*(f*x+e))-I))*exp(1/2*I*Pi*m*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))*csgn(
I*exp(-I*(f*x+e)))*csgn(I*exp(2*I*(f*x+e))-I))*exp(1/2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2*Pi)
*exp(1/2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*Pi)*exp(-
1/2*I*Pi*m)*exp(4*I*f*x)*exp(4*I*e)-2*exp(-1/2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^3*Pi)*exp(-1/
2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*Pi)*exp(1/2*I*
m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^3*Pi)*exp(1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2*csgn
(I*a)*Pi)*exp(-1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*e
xp(-1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*csgn(I*a)*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(
1/2*I*m*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^3)*exp(1/2*I*Pi*m*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2*csg
n(I*exp(-I*(f*x+e))))*exp(1/2*I*Pi*m*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2*csgn(I*exp(2*I*(f*x+e))-I))*exp(1
/2*I*Pi*m*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))*csgn(I*exp(-I*(f*x+e)))*csgn(I*exp(2*I*(f*x+e))-I))*exp(1/2*I*
m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2*Pi)*exp(1/2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e)
)*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*Pi)*exp(-1/2*I*Pi*m)*exp(4*I*f*x)*exp(4*I*e)+2*m*exp(-1/2*I*m*csgn
(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^3*Pi)*exp(-1/2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2*c
sgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*Pi)*exp(1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^3*Pi)*exp(1
/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2*csgn(I*a)*Pi)*exp(-1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x
)*sin(e))^2*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(-1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*c
sgn(I*a)*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(1/2*I*m*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^3)*exp
(1/2*I*Pi*m*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2*csgn(I*exp(-I*(f*x+e))))*exp(1/2*I*Pi*m*csgn(sin(f*x)*cos(
e)+cos(f*x)*sin(e))^2*csgn(I*exp(2*I*(f*x+e))-I))*exp(1/2*I*Pi*m*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))*csgn(I*
exp(-I*(f*x+e)))*csgn(I*exp(2*I*(f*x+e))-I))*exp(1/2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2*Pi)*e
xp(1/2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*Pi)*exp(-1/
2*I*Pi*m)*exp(2*I*f*x)*exp(2*I*e)+4*exp(-1/2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^3*Pi)*exp(-1/2*
I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*Pi)*exp(1/2*I*m*
csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^3*Pi)*exp(1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2*csgn(I
*a)*Pi)*exp(-1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp
(-1/2*I*m*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*csgn(I*a)*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(1/
2*I*m*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^3)*exp(1/2*I*Pi*m*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2*csgn(
I*exp(-I*(f*x+e))))*exp(1/2*I*Pi*m*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2*csgn(I*exp(2*I*(f*x+e))-I))*exp(1/2
*I*Pi*m*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))*csgn(I*exp(-I*(f*x+e)))*csgn(I*exp(2*I*(f*x+e))-I))*exp(1/2*I*m*
csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2*Pi)*exp(1/2*I*m*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))*
csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*Pi)*exp(-1/2*I*Pi*m)*exp(2*I*f*x)*exp(2*I*e)+m*exp(-1/2*I*Pi*m*(-csg
n(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^3+csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2*csgn(sin(f*x)*cos(e)+cos(
f*x)*sin(e))+csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))-csgn(si
n(f*x)*cos(e)+cos(f*x)*sin(e))^2*csgn(I*exp(2*I*(f*x+e))-I)+csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^3-cs
gn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^3-csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2*csgn(I*a)+csgn(a*sin(f*x)*co
s(e)+a*cos(f*x)*sin(e))*csgn(I*a)*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))-csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^
2*csgn(I*exp(-I*(f*x+e)))-csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))*csgn(I*exp(-I*(f*x+e)))*csgn(I*exp(2*I*(f*x+e)
)-I)-csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))-csgn(I*a*sin(f*x)
*cos(e)+I*a*cos(f*x)*sin(e))^2+1))-2*exp(-1/2*I*Pi*m*(-csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^3+csgn(a*sin(
f*x)*cos(e)+a*cos(f*x)*sin(e))^2*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))+csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*s
in(e))^2*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))-csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2*csgn(I*exp(2*I*(f*x
+e))-I)+csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^3-csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^3-csgn(a*sin(f*x
)*cos(e)+a*cos(f*x)*sin(e))^2*csgn(I*a)+csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*csgn(I*a)*csgn(sin(f*x)*cos(
e)+cos(f*x)*sin(e))-csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2*csgn(I*exp(-I*(f*x+e)))-csgn(sin(f*x)*cos(e)+cos(f
*x)*sin(e))*csgn(I*exp(-I*(f*x+e)))*csgn(I*exp(2*I*(f*x+e))-I)-csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))*c
sgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))-csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2+1)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.24 \[ \int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx=\frac {{\left ({\left (m - 2\right )} \cos \left (f x + e\right )^{2} + 2\right )} \left (a \sin \left (f x + e\right )\right )^{m}}{f m^{2} - {\left (f m^{2} - 2 \, f m\right )} \cos \left (f x + e\right )^{2} - 2 \, f m} \]

[In]

integrate(cot(f*x+e)^3*(a*sin(f*x+e))^m,x, algorithm="fricas")

[Out]

((m - 2)*cos(f*x + e)^2 + 2)*(a*sin(f*x + e))^m/(f*m^2 - (f*m^2 - 2*f*m)*cos(f*x + e)^2 - 2*f*m)

Sympy [F]

\[ \int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx=\int \left (a \sin {\left (e + f x \right )}\right )^{m} \cot ^{3}{\left (e + f x \right )}\, dx \]

[In]

integrate(cot(f*x+e)**3*(a*sin(f*x+e))**m,x)

[Out]

Integral((a*sin(e + f*x))**m*cot(e + f*x)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.02 \[ \int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx=-\frac {\frac {a^{m} \sin \left (f x + e\right )^{m}}{m} - \frac {a^{m} \sin \left (f x + e\right )^{m}}{{\left (m - 2\right )} \sin \left (f x + e\right )^{2}}}{f} \]

[In]

integrate(cot(f*x+e)^3*(a*sin(f*x+e))^m,x, algorithm="maxima")

[Out]

-(a^m*sin(f*x + e)^m/m - a^m*sin(f*x + e)^m/((m - 2)*sin(f*x + e)^2))/f

Giac [F]

\[ \int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{3} \,d x } \]

[In]

integrate(cot(f*x+e)^3*(a*sin(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e))^m*cot(f*x + e)^3, x)

Mupad [B] (verification not implemented)

Time = 4.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.98 \[ \int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx=-\frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^m\,\left (m-4\,{\sin \left (2\,e+2\,f\,x\right )}^2+m\,\left (2\,{\sin \left (2\,e+2\,f\,x\right )}^2-1\right )+16\,{\sin \left (e+f\,x\right )}^2\right )}{f\,m\,\left (2\,{\sin \left (2\,e+2\,f\,x\right )}^2-8\,{\sin \left (e+f\,x\right )}^2\right )\,\left (m-2\right )} \]

[In]

int(cot(e + f*x)^3*(a*sin(e + f*x))^m,x)

[Out]

-((a*sin(e + f*x))^m*(m - 4*sin(2*e + 2*f*x)^2 + m*(2*sin(2*e + 2*f*x)^2 - 1) + 16*sin(e + f*x)^2))/(f*m*(2*si
n(2*e + 2*f*x)^2 - 8*sin(e + f*x)^2)*(m - 2))