3.2.32 \(\int \cot (e+f x) (a+a \sin (e+f x))^m \, dx\) [132]

Optimal. Leaf size=43 \[ -\frac {\, _2F_1(1,1+m;2+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{1+m}}{a f (1+m)} \]

[Out]

-hypergeom([1, 1+m],[2+m],1+sin(f*x+e))*(a+a*sin(f*x+e))^(1+m)/a/f/(1+m)

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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2786, 67} \begin {gather*} -\frac {(a \sin (e+f x)+a)^{m+1} \, _2F_1(1,m+1;m+2;\sin (e+f x)+1)}{a f (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Hypergeometric2F1[1, 1 + m, 2 + m, 1 + Sin[e + f*x]]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 2786

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[x^p*((a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rubi steps

\begin {align*} \int \cot (e+f x) (a+a \sin (e+f x))^m \, dx &=\frac {\text {Subst}\left (\int \frac {(a+x)^m}{x} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=-\frac {\, _2F_1(1,1+m;2+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{1+m}}{a f (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 43, normalized size = 1.00 \begin {gather*} -\frac {\, _2F_1(1,1+m;2+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{1+m}}{a f (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Hypergeometric2F1[1, 1 + m, 2 + m, 1 + Sin[e + f*x]]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)))

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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \cot \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\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+a*sin(f*x+e))**m,x)

[Out]

Integral((a*(sin(e + f*x) + 1))**m*cot(e + f*x), 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)*(a+a*sin(f*x+e))^m,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {cot}\left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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