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[e + f*x])^(-2 + m))/(f*(2 - m))) - (a*Sin[e + f*x])^m/(f*m)
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Rubi [A] time = 0.0500821, antiderivative size = 46, normalized size of antiderivative = 1.,
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 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]
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]]
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 = 0.0563261, size = 37, normalized size = 0.8 \[ \frac{\left (m \csc ^2(e+f x)-m+2\right ) (a \sin (e+f x))^m}{f (m-2) m} \]
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)
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Maple [C] time = 1.074, size = 3161, normalized size = 68.7 \begin{align*} \text{output too large to display} \end{align*}
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*(1/(exp(I*(Re(f*x)+Re(e)))^m)*(exp(I*(f*x+e))-1)^m*(exp(I*(f*x+e))+1)^m/(
2^m)*a^m*m*exp(m*Im(f*x)+m*Im(e))*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))^2*csgn(I*exp(-I*(f*x+e)))*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*a*sin(f*x+e))^2*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*Pi*csgn(sin(f*x+e))^3)*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(a*sin(f*x+e))^2*Pi*csgn(sin(f*x+e)))*exp(-1/2*I*m*Pi*csgn(I*a*sin(f*x
+e))^3)*exp(1/2*I*m*Pi*csgn(a*sin(f*x+e))^3)*exp(1/2*I*m*csgn(a*sin(f*x+e))^2*csgn(I*a)*Pi)*exp(1/2*I*m*Pi*csg
n(a*sin(f*x+e))*csgn(I*a*sin(f*x+e)))*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(-I*(f*x+e)))*Pi*csgn(s
in(f*x+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))^2*Pi*csgn(a*sin(f*
x+e)))*exp(-1/2*I*Pi*m)*exp(-1/2*I*m*csgn(I*a)*Pi*csgn(sin(f*x+e))*csgn(a*sin(f*x+e)))*exp(4*I*f*x)*exp(4*I*e)
-2/(exp(I*(Re(f*x)+Re(e)))^m)*(exp(I*(f*x+e))-1)^m*(exp(I*(f*x+e))+1)^m/(2^m)*a^m*exp(m*Im(f*x)+m*Im(e))*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))^2*csgn(I*exp(-I*
(f*x+e)))*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)*ex
p(1/2*I*m*csgn(I*a*sin(f*x+e))^2*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
*Pi*csgn(sin(f*x+e))^3)*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*csg
n(a*sin(f*x+e))^2*Pi*csgn(sin(f*x+e)))*exp(-1/2*I*m*Pi*csgn(I*a*sin(f*x+e))^3)*exp(1/2*I*m*Pi*csgn(a*sin(f*x+e
))^3)*exp(1/2*I*m*csgn(a*sin(f*x+e))^2*csgn(I*a)*Pi)*exp(1/2*I*m*Pi*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*x+e)))*e
xp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(-I*(f*x+e)))*Pi*csgn(sin(f*x+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))^2*Pi*csgn(a*sin(f*x+e)))*exp(-1/2*I*Pi*m)*exp(-1/2*I*m*cs
gn(I*a)*Pi*csgn(sin(f*x+e))*csgn(a*sin(f*x+e)))*exp(4*I*f*x)*exp(4*I*e)+2/(exp(I*(Re(f*x)+Re(e)))^m)*(exp(I*(f
*x+e))-1)^m*(exp(I*(f*x+e))+1)^m/(2^m)*a^m*m*exp(m*Im(f*x)+m*Im(e))*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)^2*c
sgn(I*exp(I*(f*x+e))-I)*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(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*a*sin(f*x+e))^2*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*Pi*csgn(sin(f*x+e))^3)*exp(1/2*I*m*c
sgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))+I)*Pi)*exp(-1/2*I*m*csgn(a*sin(f*x+e))^2*Pi*csgn(sin(f*x+e))
)*exp(-1/2*I*m*Pi*csgn(I*a*sin(f*x+e))^3)*exp(1/2*I*m*Pi*csgn(a*sin(f*x+e))^3)*exp(1/2*I*m*csgn(a*sin(f*x+e))^
2*csgn(I*a)*Pi)*exp(1/2*I*m*Pi*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*x+e)))*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)
*csgn(I*exp(-I*(f*x+e)))*Pi*csgn(sin(f*x+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))^2*Pi*csgn(a*sin(f*x+e)))*exp(-1/2*I*Pi*m)*exp(-1/2*I*m*csgn(I*a)*Pi*csgn(sin(f*x+e))*csgn(a*si
n(f*x+e)))*exp(2*I*f*x)*exp(2*I*e)+4/(exp(I*(Re(f*x)+Re(e)))^m)*(exp(I*(f*x+e))-1)^m*(exp(I*(f*x+e))+1)^m/(2^m
)*a^m*exp(m*Im(f*x)+m*Im(e))*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))^2*csgn(I*exp(-I*(f*x+e)))*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*a*sin(f*x+e))^2*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*Pi*csgn(sin(f*x+e))^3)*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(a*sin(f*x+e))^2*Pi*csgn(sin(f*x+e)))*exp(-1/2*I*m*Pi*csgn(I*a*sin(f*x+e))^
3)*exp(1/2*I*m*Pi*csgn(a*sin(f*x+e))^3)*exp(1/2*I*m*csgn(a*sin(f*x+e))^2*csgn(I*a)*Pi)*exp(1/2*I*m*Pi*csgn(a*s
in(f*x+e))*csgn(I*a*sin(f*x+e)))*exp(1/2*I*m*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(-I*(f*x+e)))*Pi*csgn(sin(f*
x+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))^2*Pi*csgn(a*sin(f*x+e))
)*exp(-1/2*I*Pi*m)*exp(-1/2*I*m*csgn(I*a)*Pi*csgn(sin(f*x+e))*csgn(a*sin(f*x+e)))*exp(2*I*f*x)*exp(2*I*e)+1/(e
xp(I*(Re(f*x)+Re(e)))^m)*(exp(I*(f*x+e))-1)^m*(exp(I*(f*x+e))+1)^m/(2^m)*a^m*m*exp(1/2*m*(I*Pi*csgn(a*sin(f*x+
e))*csgn(I*a*sin(f*x+e))+I*csgn(I*a*sin(f*x+e))^2*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*a*sin(f*x+e))^2*Pi*csgn(a*sin(f*x+e))+I*csgn(I*exp(
2*I*(f*x+e))-I)*csgn(sin(f*x+e))^2*Pi-I*csgn(a*sin(f*x+e))^2*Pi*csgn(sin(f*x+e))+I*csgn(I*exp(2*I*(f*x+e))-I)*
csgn(I*exp(-I*(f*x+e)))*Pi*csgn(sin(f*x+e))+I*Pi*csgn(sin(f*x+e))^3-I*csgn(I*exp(2*I*(f*x+e))-I)^3*Pi-I*Pi*csg
n(I*a*sin(f*x+e))^3+I*csgn(I*exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))+I)*Pi-I*csgn(I*a)*Pi*csgn(sin(f*x+e))
*csgn(a*sin(f*x+e))+I*csgn(a*sin(f*x+e))^2*csgn(I*a)*Pi+I*Pi*csgn(a*sin(f*x+e))^3-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*Pi+2*Im(f*x)+2*Im(e)))-2/(exp(I*(Re(f*x)+Re(e)))^m)*(e
xp(I*(f*x+e))-1)^m*(exp(I*(f*x+e))+1)^m/(2^m)*a^m*exp(1/2*m*(I*Pi*csgn(a*sin(f*x+e))*csgn(I*a*sin(f*x+e))+I*cs
gn(I*a*sin(f*x+e))^2*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*a*sin(f*x+e))^2*Pi*csgn(a*sin(f*x+e))+I*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x+
e))^2*Pi-I*csgn(a*sin(f*x+e))^2*Pi*csgn(sin(f*x+e))+I*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(-I*(f*x+e)))*Pi*cs
gn(sin(f*x+e))+I*Pi*csgn(sin(f*x+e))^3-I*csgn(I*exp(2*I*(f*x+e))-I)^3*Pi-I*Pi*csgn(I*a*sin(f*x+e))^3+I*csgn(I*
exp(2*I*(f*x+e))-I)^2*csgn(I*exp(I*(f*x+e))+I)*Pi-I*csgn(I*a)*Pi*csgn(sin(f*x+e))*csgn(a*sin(f*x+e))+I*csgn(a*
sin(f*x+e))^2*csgn(I*a)*Pi+I*Pi*csgn(a*sin(f*x+e))^3-I*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(I*(f*x+e))-I)*csg
n(I*exp(I*(f*x+e))+I)*Pi-I*Pi+2*Im(f*x)+2*Im(e))))
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Maxima [A] time = 0.991215, size = 63, normalized size = 1.37 \begin{align*} -\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} \end{align*}
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
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Fricas [A] time = 1.59485, size = 132, normalized size = 2.87 \begin{align*} \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} \end{align*}
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)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin{\left (e + f x \right )}\right )^{m} \cot ^{3}{\left (e + f x \right )}\, dx \end{align*}
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)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{3}\,{d x} \end{align*}
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)