3.2.64 \(\int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx\) [164]
Optimal. Leaf size=46 \[ -\frac {a^2 (a \sin (e+f x))^{-2+m}}{f (2-m)}-\frac {(a \sin (e+f x))^m}{f m} \]
[Out]
-a^2*(a*sin(f*x+e))^(-2+m)/f/(2-m)-(a*sin(f*x+e))^m/f/m
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Rubi [A]
time = 0.04, 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 = {2672, 14}
\begin {gather*} -\frac {a^2 (a \sin (e+f x))^{m-2}}{f (2-m)}-\frac {(a \sin (e+f x))^m}{f m} \end {gather*}
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 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*} \int \cot ^3(e+f x) (a \sin (e+f x))^m \, dx &=\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*}
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Mathematica [A]
time = 0.07, size = 37, normalized size = 0.80 \begin {gather*} \frac {\left (2-m+m \csc ^2(e+f x)\right ) (a \sin (e+f x))^m}{f (-2+m) m} \end {gather*}
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] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.68, size = 3063, normalized size = 66.59
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\(\text {Expression too large to display}\) |
\(3063\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(f*x+e)^3*(a*sin(f*x+e))^m,x,method=_RETURNVERBOSE)
[Out]
-1/(-2+m)/f/(exp(2*I*(f*x+e))-1)^2/m*(m*(exp(2*I*(f*x+e))-1)^m/(exp(I*(Re(f*x)+Re(e)))^m)*a^m/(2^m)*exp(m*Im(f
*x)+m*Im(e))*exp(1/2*I*m*Pi*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2)*exp(-1/2*I*m*P
i*csgn(I*a)*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(-1/2*I*m*Pi*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)*exp(1/2*I*m*Pi*csgn(
I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2)*exp(-1/2*I*m*Pi*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^3)*e
xp(1/2*I*m*Pi*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)))*exp(1/2
*I*m*Pi*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^3)*exp(-1/2*I*m*Pi*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)))*exp(1/2*I*m*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^3)*exp(1/2*I*m*
Pi*csgn(I*exp(-I*(f*x+e)))*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2)*exp(1/2*I*m*Pi*csgn(I*a)*csgn(a*sin(f*x)*c
os(e)+a*cos(f*x)*sin(e))^2)*exp(1/2*I*m*Pi*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(-I*(f*x+e)))*csgn(sin(f*x)*co
s(e)+cos(f*x)*sin(e)))*exp(-1/2*I*Pi*m)*exp(4*I*f*x)*exp(4*I*e)-2*(exp(2*I*(f*x+e))-1)^m/(exp(I*(Re(f*x)+Re(e)
))^m)*a^m/(2^m)*exp(m*Im(f*x)+m*Im(e))*exp(1/2*I*m*Pi*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x)*cos(e)+cos(f*x)
*sin(e))^2)*exp(-1/2*I*m*Pi*csgn(I*a)*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*csgn(sin(f*x)*cos(e)+cos(f*x)*
sin(e)))*exp(-1/2*I*m*Pi*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)*exp(1/2*I*m*Pi*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2)*exp(-1/2*I*m*Pi*csgn(I*a*sin(f*x)*cos(e)
+I*a*cos(f*x)*sin(e))^3)*exp(1/2*I*m*Pi*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)))*exp(1/2*I*m*Pi*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^3)*exp(-1/2*I*m*Pi*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)))*exp(1/2*I*m*Pi*csgn(sin(f*x)*cos(e)+cos(f*
x)*sin(e))^3)*exp(1/2*I*m*Pi*csgn(I*exp(-I*(f*x+e)))*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2)*exp(1/2*I*m*Pi*c
sgn(I*a)*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2)*exp(1/2*I*m*Pi*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(-I*
(f*x+e)))*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(-1/2*I*Pi*m)*exp(4*I*f*x)*exp(4*I*e)+2*m*(exp(2*I*(f*x+e)
)-1)^m/(exp(I*(Re(f*x)+Re(e)))^m)*a^m/(2^m)*exp(m*Im(f*x)+m*Im(e))*exp(1/2*I*m*Pi*csgn(I*exp(2*I*(f*x+e))-I)*c
sgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2)*exp(-1/2*I*m*Pi*csgn(I*a)*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*cs
gn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(-1/2*I*m*Pi*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)*exp(1/2*I*m*Pi*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^2)*exp(-1/2*I*m
*Pi*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^3)*exp(1/2*I*m*Pi*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)))*exp(1/2*I*m*Pi*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^3)*exp
(-1/2*I*m*Pi*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)))*exp(1/2*I*m*Pi
*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^3)*exp(1/2*I*m*Pi*csgn(I*exp(-I*(f*x+e)))*csgn(sin(f*x)*cos(e)+cos(f*x)
*sin(e))^2)*exp(1/2*I*m*Pi*csgn(I*a)*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2)*exp(1/2*I*m*Pi*csgn(I*exp(2*
I*(f*x+e))-I)*csgn(I*exp(-I*(f*x+e)))*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(-1/2*I*Pi*m)*exp(2*I*f*x)*exp
(2*I*e)+4*(exp(2*I*(f*x+e))-1)^m/(exp(I*(Re(f*x)+Re(e)))^m)*a^m/(2^m)*exp(m*Im(f*x)+m*Im(e))*exp(1/2*I*m*Pi*cs
gn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2)*exp(-1/2*I*m*Pi*csgn(I*a)*csgn(a*sin(f*x)*co
s(e)+a*cos(f*x)*sin(e))*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(-1/2*I*m*Pi*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)*exp(1/2*I*m*Pi*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*
x)*sin(e))^2)*exp(-1/2*I*m*Pi*csgn(I*a*sin(f*x)*cos(e)+I*a*cos(f*x)*sin(e))^3)*exp(1/2*I*m*Pi*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)))*exp(1/2*I*m*Pi*csgn(a*sin(f*x)*cos(e)
+a*cos(f*x)*sin(e))^3)*exp(-1/2*I*m*Pi*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)))*exp(1/2*I*m*Pi*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^3)*exp(1/2*I*m*Pi*csgn(I*exp(-I*(f*x+e)))*csg
n(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2)*exp(1/2*I*m*Pi*csgn(I*a)*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))^2)*ex
p(1/2*I*m*Pi*csgn(I*exp(2*I*(f*x+e))-I)*csgn(I*exp(-I*(f*x+e)))*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)))*exp(-1/
2*I*Pi*m)*exp(2*I*f*x)*exp(2*I*e)+m*(exp(2*I*(f*x+e))-1)^m/(exp(I*(Re(f*x)+Re(e)))^m)*a^m/(2^m)*exp(1/2*m*(I*P
i*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^3+I*Pi*csgn(I*exp(2*I*(f*x+e))-I)*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e)
)^2+I*Pi*csgn(I*exp(-I*(f*x+e)))*csgn(sin(f*x)*cos(e)+cos(f*x)*sin(e))^2-I*Pi*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))-I*Pi*csgn(I*a)*csgn(a*sin(f*x)*cos(e)+a*cos(f*x)*sin(e))*csg
n(sin(f*x)*cos(e)+cos(f*x)*sin(e))+I*Pi*csgn(I*...
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Maxima [A]
time = 0.28, size = 50, normalized size = 1.09 \begin {gather*} -\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 {gather*}
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 = 0.35, size = 60, normalized size = 1.30 \begin {gather*} \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 {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \sin {\left (e + f x \right )}\right )^{m} \cot ^{3}{\left (e + f x \right )}\, dx \end {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
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Mupad [B]
time = 3.31, size = 91, normalized size = 1.98 \begin {gather*} -\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 )} \end {gather*}
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))
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