∫ cot3(e + fx)(a sin(e + fx))m dx

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

Optimal. Leaf size=46 \[ -\frac {a^2 (a \sin (e+f x))^{m-2}}{f (2-m)}-\frac {(a \sin (e+f x))^m}{f m} \]

[Out]

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

________________________________________________________________________________________

Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2592, 14} \[ -\frac {a^2 (a \sin (e+f x))^{m-2}}{f (2-m)}-\frac {(a \sin (e+f x))^m}{f m} \]

Antiderivative was successfully veriﬁed.

[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 2592

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*} \int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx &=\frac {\operatorname {Subst}\left (\int x^{-3+m} \left (a^2-x^2\right ) \, dx,x,a \sin (e+f x)\right )}{f}\\ &=\frac {\operatorname {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] time = 0.06, size = 37, normalized size = 0.80 \[ \frac {\left (m \csc ^2(e+f x)-m+2\right ) (a \sin (e+f x))^m}{f (m-2) m} \]

Antiderivative was successfully veriﬁed.

[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)

________________________________________________________________________________________

fricas [A] time = 0.44, size = 57, normalized size = 1.24 \[ \frac {{\left ({\left (m - 2\right )} \cos \left (f x + e\right )^{2} + 2\right )} \left (a \sin \left (f x + e\right )\right )^{m}}{f m^{2} - {\left (f m^{2} - 2 \, f m\right )} \cos \left (f x + e\right )^{2} - 2 \, f m} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{3}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [C] time = 1.66, size = 3161, normalized size = 68.72 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(-2+m)/f/(exp(2*I*(f*x+e))-1)^2/m*(m/(2^m)*a^m*(exp(I*(f*x+e))+1)^m*(exp(I*(f*x+e))-1)^m/(exp(I*(Re(f*x)+Re

(e)))^m)*exp(m*Im(f*x)+m*Im(e))*exp(-1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^3*Pi)*exp(-1/2*I*m*csgn(I*a*sin(f*x+e)

)^3*Pi)*exp(-1/2*I*m*csgn(sin(f*x+e))*csgn(a*sin(f*x+e))^2*Pi)*exp(-1/2*I*m*csgn(sin(f*x+e))*csgn(a*sin(f*x+e)

)*csgn(I*a)*Pi)*exp(-1/2*I*Pi*m)*exp(1/2*I*m*csgn(I*a*sin(f*x+e))^2*Pi)*exp(-1/2*I*m*csgn(a*sin(f*x+e))*csgn(I

*a*sin(f*x+e))^2*Pi)*exp(-1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(I*(f*x+e))-I)*csgn(I*exp(I*(f*x+e))+I)

*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x+e))^2*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^2*cs

gn(I*exp(I*(f*x+e))-I)*Pi)*exp(1/2*I*m*csgn(sin(f*x+e))^3*Pi)*exp(1/2*I*m*csgn(a*sin(f*x+e))^2*csgn(I*a)*Pi)*e

xp(1/2*I*m*csgn(a*sin(f*x+e))^3*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x+e))*csgn(I*exp(-I*(f*x

+e)))*Pi)*exp(1/2*I*m*csgn(sin(f*x+e))^2*csgn(I*exp(-I*(f*x+e)))*Pi)*exp(1/2*I*m*csgn(a*sin(f*x+e))*csgn(I*a*s

in(f*x+e))*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))+I)*Pi)*exp(4*I*f*x)*exp(4*I*e)-2

/(2^m)*a^m*(exp(I*(f*x+e))+1)^m*(exp(I*(f*x+e))-1)^m/(exp(I*(Re(f*x)+Re(e)))^m)*exp(m*Im(f*x)+m*Im(e))*exp(-1/

2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^3*Pi)*exp(-1/2*I*m*csgn(I*a*sin(f*x+e))^3*Pi)*exp(-1/2*I*m*csgn(sin(f*x+e))*c

sgn(a*sin(f*x+e))^2*Pi)*exp(-1/2*I*m*csgn(sin(f*x+e))*csgn(a*sin(f*x+e))*csgn(I*a)*Pi)*exp(-1/2*I*Pi*m)*exp(1/

2*I*m*csgn(I*a*sin(f*x+e))^2*Pi)*exp(-1/2*I*m*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*x+e))^2*Pi)*exp(-1/2*I*m*csgn(

I*exp(2*I*(f*x+e))-I)*csgn(I*exp(I*(f*x+e))-I)*csgn(I*exp(I*(f*x+e))+I)*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e)

)-I)*csgn(sin(f*x+e))^2*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))-I)*Pi)*exp(1/2*I*m*

csgn(sin(f*x+e))^3*Pi)*exp(1/2*I*m*csgn(a*sin(f*x+e))^2*csgn(I*a)*Pi)*exp(1/2*I*m*csgn(a*sin(f*x+e))^3*Pi)*exp

(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x+e))*csgn(I*exp(-I*(f*x+e)))*Pi)*exp(1/2*I*m*csgn(sin(f*x+e))^

2*csgn(I*exp(-I*(f*x+e)))*Pi)*exp(1/2*I*m*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*x+e))*Pi)*exp(1/2*I*m*csgn(I*exp(2

*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))+I)*Pi)*exp(4*I*f*x)*exp(4*I*e)+2*m/(2^m)*a^m*(exp(I*(f*x+e))+1)^m*(exp(

I*(f*x+e))-1)^m/(exp(I*(Re(f*x)+Re(e)))^m)*exp(m*Im(f*x)+m*Im(e))*exp(-1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^3*Pi

)*exp(-1/2*I*m*csgn(I*a*sin(f*x+e))^3*Pi)*exp(-1/2*I*m*csgn(sin(f*x+e))*csgn(a*sin(f*x+e))^2*Pi)*exp(-1/2*I*m*

csgn(sin(f*x+e))*csgn(a*sin(f*x+e))*csgn(I*a)*Pi)*exp(-1/2*I*Pi*m)*exp(1/2*I*m*csgn(I*a*sin(f*x+e))^2*Pi)*exp(

-1/2*I*m*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*x+e))^2*Pi)*exp(-1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(I*(f

*x+e))-I)*csgn(I*exp(I*(f*x+e))+I)*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x+e))^2*Pi)*exp(1/2*I

*m*csgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))-I)*Pi)*exp(1/2*I*m*csgn(sin(f*x+e))^3*Pi)*exp(1/2*I*m*cs

gn(a*sin(f*x+e))^2*csgn(I*a)*Pi)*exp(1/2*I*m*csgn(a*sin(f*x+e))^3*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*c

sgn(sin(f*x+e))*csgn(I*exp(-I*(f*x+e)))*Pi)*exp(1/2*I*m*csgn(sin(f*x+e))^2*csgn(I*exp(-I*(f*x+e)))*Pi)*exp(1/2

*I*m*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*x+e))*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e)

)+I)*Pi)*exp(2*I*f*x)*exp(2*I*e)+4/(2^m)*a^m*(exp(I*(f*x+e))+1)^m*(exp(I*(f*x+e))-1)^m/(exp(I*(Re(f*x)+Re(e)))

^m)*exp(m*Im(f*x)+m*Im(e))*exp(-1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^3*Pi)*exp(-1/2*I*m*csgn(I*a*sin(f*x+e))^3*P

i)*exp(-1/2*I*m*csgn(sin(f*x+e))*csgn(a*sin(f*x+e))^2*Pi)*exp(-1/2*I*m*csgn(sin(f*x+e))*csgn(a*sin(f*x+e))*csg

n(I*a)*Pi)*exp(-1/2*I*Pi*m)*exp(1/2*I*m*csgn(I*a*sin(f*x+e))^2*Pi)*exp(-1/2*I*m*csgn(a*sin(f*x+e))*csgn(I*a*si

n(f*x+e))^2*Pi)*exp(-1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(I*(f*x+e))-I)*csgn(I*exp(I*(f*x+e))+I)*Pi)*

exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x+e))^2*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*

exp(I*(f*x+e))-I)*Pi)*exp(1/2*I*m*csgn(sin(f*x+e))^3*Pi)*exp(1/2*I*m*csgn(a*sin(f*x+e))^2*csgn(I*a)*Pi)*exp(1/

2*I*m*csgn(a*sin(f*x+e))^3*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x+e))*csgn(I*exp(-I*(f*x+e)))

*Pi)*exp(1/2*I*m*csgn(sin(f*x+e))^2*csgn(I*exp(-I*(f*x+e)))*Pi)*exp(1/2*I*m*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*

x+e))*Pi)*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))+I)*Pi)*exp(2*I*f*x)*exp(2*I*e)+m/(2^m

)*a^m*(exp(I*(f*x+e))+1)^m*(exp(I*(f*x+e))-1)^m/(exp(I*(Re(f*x)+Re(e)))^m)*exp(1/2*m*(-I*csgn(I*exp(2*I*(f*x+e

))-I)*csgn(I*exp(I*(f*x+e))-I)*csgn(I*exp(I*(f*x+e))+I)*Pi+I*csgn(a*sin(f*x+e))^2*csgn(I*a)*Pi-I*Pi-I*csgn(sin

(f*x+e))*csgn(a*sin(f*x+e))*csgn(I*a)*Pi-I*csgn(sin(f*x+e))*csgn(a*sin(f*x+e))^2*Pi+I*csgn(I*exp(2*I*(f*x+e))-

I)*csgn(sin(f*x+e))*csgn(I*exp(-I*(f*x+e)))*Pi+I*csgn(sin(f*x+e))^2*csgn(I*exp(-I*(f*x+e)))*Pi+I*csgn(I*exp(2*

I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))-I)*Pi+I*csgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))+I)*Pi-I*csgn(

I*a*sin(f*x+e))^3*Pi+I*csgn(I*a*sin(f*x+e))^2*Pi+I*csgn(sin(f*x+e))^3*Pi+I*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*x

+e))*Pi+I*csgn(a*sin(f*x+e))^3*Pi+I*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x+e))^2*Pi-I*csgn(I*exp(2*I*(f*x+e))

-I)^3*Pi-I*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*x+e))^2*Pi+2*Im(f*x)+2*Im(e)))-2/(2^m)*a^m*(exp(I*(f*x+e))+1)^m*(

exp(I*(f*x+e))-1)^m/(exp(I*(Re(f*x)+Re(e)))^m)*exp(1/2*m*(-I*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(I*(f*x+e))-

I)*csgn(I*exp(I*(f*x+e))+I)*Pi+I*csgn(a*sin(f*x+e))^2*csgn(I*a)*Pi-I*Pi-I*csgn(sin(f*x+e))*csgn(a*sin(f*x+e))*

csgn(I*a)*Pi-I*csgn(sin(f*x+e))*csgn(a*sin(f*x+e))^2*Pi+I*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x+e))*csgn(I*e

xp(-I*(f*x+e)))*Pi+I*csgn(sin(f*x+e))^2*csgn(I*exp(-I*(f*x+e)))*Pi+I*csgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I

*(f*x+e))-I)*Pi+I*csgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))+I)*Pi-I*csgn(I*a*sin(f*x+e))^3*Pi+I*csgn(

I*a*sin(f*x+e))^2*Pi+I*csgn(sin(f*x+e))^3*Pi+I*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*x+e))*Pi+I*csgn(a*sin(f*x+e))

^3*Pi+I*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x+e))^2*Pi-I*csgn(I*exp(2*I*(f*x+e))-I)^3*Pi-I*csgn(a*sin(f*x+e)

)*csgn(I*a*sin(f*x+e))^2*Pi+2*Im(f*x)+2*Im(e))))

________________________________________________________________________________________

maxima [A] time = 0.56, size = 47, normalized size = 1.02 \[ -\frac {\frac {a^{m} \sin \left (f x + e\right )^{m}}{m} - \frac {a^{m} \sin \left (f x + e\right )^{m}}{{\left (m - 2\right )} \sin \left (f x + e\right )^{2}}}{f} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [B] time = 3.31, size = 91, normalized size = 1.98 \[ -\frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^m\,\left (m-4\,{\sin \left (2\,e+2\,f\,x\right )}^2+m\,\left (2\,{\sin \left (2\,e+2\,f\,x\right )}^2-1\right )+16\,{\sin \left (e+f\,x\right )}^2\right )}{f\,m\,\left (2\,{\sin \left (2\,e+2\,f\,x\right )}^2-8\,{\sin \left (e+f\,x\right )}^2\right )\,\left (m-2\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin {\left (e + f x \right )}\right )^{m} \cot ^{3}{\left (e + f x \right )}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]