3.3.24 \(\int (a \cot (e+f x))^m \tan ^n(e+f x) \, dx\) [224]

Optimal. Leaf size=64 \[ \frac {(a \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (1-m+n);\frac {1}{2} (3-m+n);-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)} \]

[Out]

(a*cot(f*x+e))^m*hypergeom([1, 1/2-1/2*m+1/2*n],[3/2-1/2*m+1/2*n],-tan(f*x+e)^2)*tan(f*x+e)^(1+n)/f/(1-m+n)

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Rubi [A]
time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2684, 3557, 371} \begin {gather*} \frac {\tan ^{n+1}(e+f x) (a \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (-m+n+1);\frac {1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{f (-m+n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*Cot[e + f*x])^m*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n))/
(f*(1 - m + n))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2684

Int[(cot[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cot[e + f*
x])^m*(b*Tan[e + f*x])^m, Int[(b*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[m
] &&  !IntegerQ[n]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int (a \cot (e+f x))^m \tan ^n(e+f x) \, dx &=\left ((a \cot (e+f x))^m \tan ^m(e+f x)\right ) \int \tan ^{-m+n}(e+f x) \, dx\\ &=\frac {\left ((a \cot (e+f x))^m \tan ^m(e+f x)\right ) \text {Subst}\left (\int \frac {x^{-m+n}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (1-m+n);\frac {1}{2} (3-m+n);-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 64, normalized size = 1.00 \begin {gather*} \frac {(a \cot (e+f x))^m \, _2F_1\left (1,\frac {1}{2} (1-m+n);\frac {1}{2} (3-m+n);-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*Cot[e + f*x])^m*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n))/
(f*(1 - m + n))

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Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \left (a \cot \left (f x +e \right )\right )^{m} \left (\tan ^{n}\left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*cot(f*x+e))^m*tan(f*x+e)^n,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((a*cot(f*x+e))^m*tan(f*x+e)^n,x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \cot {\left (e + f x \right )}\right )^{m} \tan ^{n}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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