3.2.63 \(\int \cot (e+f x) (a \sin (e+f x))^m \, dx\) [163]

Optimal. Leaf size=17 \[ \frac {(a \sin (e+f x))^m}{f m} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2672, 30} \begin {gather*} \frac {(a \sin (e+f x))^m}{f m} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sin[e + f*x])^m/(f*m)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {(a \sin (e+f x))^m}{f m} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sin[e + f*x])^m/(f*m)

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Maple [A]
time = 0.34, size = 18, normalized size = 1.06

method	result	size
derivativedivides	\(\frac {\left (a \sin \left (f x +e \right )\right )^{m}}{f m}\)	\(18\)
default	\(\frac {\left (a \sin \left (f x +e \right )\right )^{m}}{f m}\)	\(18\)
risch	\(\frac {{\mathrm e}^{-\frac {m \left (-i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )\right ) \mathrm {csgn}\left (\sin \left (f x +e \right )\right )^{2}-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i \left (f x +e \right )}\right ) \mathrm {csgn}\left (\sin \left (f x +e \right )\right )^{2}+i \pi \,\mathrm {csgn}\left (a \sin \left (f x +e \right )\right ) \mathrm {csgn}\left (i a \sin \left (f x +e \right )\right )^{2}-i \pi \mathrm {csgn}\left (i a \sin \left (f x +e \right )\right )^{2}+i \pi \mathrm {csgn}\left (i a \sin \left (f x +e \right )\right )^{3}-i \pi \,\mathrm {csgn}\left (a \sin \left (f x +e \right )\right ) \mathrm {csgn}\left (i a \sin \left (f x +e \right )\right )-i \pi \mathrm {csgn}\left (a \sin \left (f x +e \right )\right )^{3}+i \pi \mathrm {csgn}\left (a \sin \left (f x +e \right )\right )^{2} \mathrm {csgn}\left (\sin \left (f x +e \right )\right )-i \pi \mathrm {csgn}\left (\sin \left (f x +e \right )\right )^{3}+i \pi \,\mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (a \sin \left (f x +e \right )\right ) \mathrm {csgn}\left (\sin \left (f x +e \right )\right )-i \pi \,\mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (a \sin \left (f x +e \right )\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i \left (f x +e \right )}\right ) \mathrm {csgn}\left (\sin \left (f x +e \right )\right )+i \pi +2 \ln \left (2\right )-2 \ln \left (a \right )+2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )-2 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )\right )}{2}}}{f m}\)	\(323\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 19, normalized size = 1.12 \begin {gather*} \frac {a^{m} \sin \left (f x + e\right )^{m}}{f m} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^m*sin(f*x + e)^m/(f*m)

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Fricas [A]
time = 0.41, size = 18, normalized size = 1.06 \begin {gather*} \frac {\left (a \sin \left (f x + e\right )\right )^{m}}{f m} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*sin(f*x + e))^m/(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 {\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.44, size = 18, normalized size = 1.06 \begin {gather*} \frac {\left (a \sin \left (f x + e\right )\right )^{m}}{f m} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*sin(f*x + e))^m/(f*m)

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Mupad [B]
time = 2.45, size = 17, normalized size = 1.00 \begin {gather*} \frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^m}{f\,m} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*sin(e + f*x))^m/(f*m)

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