3.883 \(\int \frac{(d \cot (e+f x))^n}{a+b \tan (e+f x)} \, dx\)

Optimal. Leaf size=182 \[ -\frac{b (d \cot (e+f x))^{n+2} \, _2F_1\left (1,\frac{n+2}{2};\frac{n+4}{2};-\cot ^2(e+f x)\right )}{d^2 f (n+2) \left (a^2+b^2\right )}+\frac{a (d \cot (e+f x))^{n+3} \, _2F_1\left (1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right )}{d^3 f (n+3) \left (a^2+b^2\right )}-\frac{a^2 (d \cot (e+f x))^{n+2} \, _2F_1\left (1,n+2;n+3;-\frac{a \cot (e+f x)}{b}\right )}{b d^2 f (n+2) \left (a^2+b^2\right )} \]

[Out]

-((b*(d*Cot[e + f*x])^(2 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Cot[e + f*x]^2])/((a^2 + b^2)*d^2*f*
(2 + n))) - (a^2*(d*Cot[e + f*x])^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + n, -((a*Cot[e + f*x])/b)])/(b*(a^2 +
b^2)*d^2*f*(2 + n)) + (a*(d*Cot[e + f*x])^(3 + n)*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2]
)/((a^2 + b^2)*d^3*f*(3 + n))

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Rubi [A]  time = 0.389639, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3673, 3574, 3538, 3476, 364, 3634, 64} \[ -\frac{b (d \cot (e+f x))^{n+2} \, _2F_1\left (1,\frac{n+2}{2};\frac{n+4}{2};-\cot ^2(e+f x)\right )}{d^2 f (n+2) \left (a^2+b^2\right )}+\frac{a (d \cot (e+f x))^{n+3} \, _2F_1\left (1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right )}{d^3 f (n+3) \left (a^2+b^2\right )}-\frac{a^2 (d \cot (e+f x))^{n+2} \, _2F_1\left (1,n+2;n+3;-\frac{a \cot (e+f x)}{b}\right )}{b d^2 f (n+2) \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(d*Cot[e + f*x])^(2 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Cot[e + f*x]^2])/((a^2 + b^2)*d^2*f*
(2 + n))) - (a^2*(d*Cot[e + f*x])^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + n, -((a*Cot[e + f*x])/b)])/(b*(a^2 +
b^2)*d^2*f*(2 + n)) + (a*(d*Cot[e + f*x])^(3 + n)*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2]
)/((a^2 + b^2)*d^3*f*(3 + n))

Rule 3673

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

Rule 3574

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/
(c^2 + d^2), Int[(a + b*Tan[e + f*x])^m*(c - d*Tan[e + f*x]), x], x] + Dist[d^2/(c^2 + d^2), Int[((a + b*Tan[e
+ f*x])^m*(1 + Tan[e + f*x]^2))/(c + d*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3538

Int[((b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*T
an[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Tan[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x] && NeQ
[c^2 + d^2, 0] &&  !IntegerQ[2*m]

Rule 3476

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]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 64

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

Rubi steps

\begin{align*} \int \frac{(d \cot (e+f x))^n}{a+b \tan (e+f x)} \, dx &=\frac{\int \frac{(d \cot (e+f x))^{1+n}}{b+a \cot (e+f x)} \, dx}{d}\\ &=\frac{\int (d \cot (e+f x))^{1+n} (b-a \cot (e+f x)) \, dx}{\left (a^2+b^2\right ) d}+\frac{a^2 \int \frac{(d \cot (e+f x))^{1+n} \left (1+\cot ^2(e+f x)\right )}{b+a \cot (e+f x)} \, dx}{\left (a^2+b^2\right ) d}\\ &=-\frac{a \int (d \cot (e+f x))^{2+n} \, dx}{\left (a^2+b^2\right ) d^2}+\frac{b \int (d \cot (e+f x))^{1+n} \, dx}{\left (a^2+b^2\right ) d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{(-d x)^{1+n}}{b-a x} \, dx,x,-\cot (e+f x)\right )}{\left (a^2+b^2\right ) d f}\\ &=-\frac{a^2 (d \cot (e+f x))^{2+n} \, _2F_1\left (1,2+n;3+n;-\frac{a \cot (e+f x)}{b}\right )}{b \left (a^2+b^2\right ) d^2 f (2+n)}-\frac{b \operatorname{Subst}\left (\int \frac{x^{1+n}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{\left (a^2+b^2\right ) f}+\frac{a \operatorname{Subst}\left (\int \frac{x^{2+n}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{\left (a^2+b^2\right ) d f}\\ &=-\frac{b (d \cot (e+f x))^{2+n} \, _2F_1\left (1,\frac{2+n}{2};\frac{4+n}{2};-\cot ^2(e+f x)\right )}{\left (a^2+b^2\right ) d^2 f (2+n)}-\frac{a^2 (d \cot (e+f x))^{2+n} \, _2F_1\left (1,2+n;3+n;-\frac{a \cot (e+f x)}{b}\right )}{b \left (a^2+b^2\right ) d^2 f (2+n)}+\frac{a (d \cot (e+f x))^{3+n} \, _2F_1\left (1,\frac{3+n}{2};\frac{5+n}{2};-\cot ^2(e+f x)\right )}{\left (a^2+b^2\right ) d^3 f (3+n)}\\ \end{align*}

Mathematica [A]  time = 0.537, size = 145, normalized size = 0.8 \[ -\frac{\cot ^2(e+f x) (d \cot (e+f x))^n \left (a \left (a (n+3) \, _2F_1\left (1,n+2;n+3;-\frac{a \cot (e+f x)}{b}\right )-b (n+2) \cot (e+f x) \, _2F_1\left (1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right )\right )+b^2 (n+3) \, _2F_1\left (1,\frac{n+2}{2};\frac{n+4}{2};-\cot ^2(e+f x)\right )\right )}{b f (n+2) (n+3) \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Cot[e + f*x]^2*(d*Cot[e + f*x])^n*(b^2*(3 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Cot[e + f*x]^2]
+ a*(a*(3 + n)*Hypergeometric2F1[1, 2 + n, 3 + n, -((a*Cot[e + f*x])/b)] - b*(2 + n)*Cot[e + f*x]*Hypergeometr
ic2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2])))/(b*(a^2 + b^2)*f*(2 + n)*(3 + n)))

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Maple [F]  time = 0.224, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\cot \left ( fx+e \right ) \right ) ^{n}}{a+b\tan \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cot(f*x+e))^n/(a+b*tan(f*x+e)),x)

[Out]

int((d*cot(f*x+e))^n/(a+b*tan(f*x+e)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \cot \left (f x + e\right )\right )^{n}}{b \tan \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n/(a+b*tan(f*x+e)),x, algorithm="maxima")

[Out]

integrate((d*cot(f*x + e))^n/(b*tan(f*x + e) + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (d \cot \left (f x + e\right )\right )^{n}}{b \tan \left (f x + e\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*cot(f*x + e))^n/(b*tan(f*x + e) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \cot{\left (e + f x \right )}\right )^{n}}{a + b \tan{\left (e + f x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))**n/(a+b*tan(f*x+e)),x)

[Out]

Integral((d*cot(e + f*x))**n/(a + b*tan(e + f*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \cot \left (f x + e\right )\right )^{n}}{b \tan \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n/(a+b*tan(f*x+e)),x, algorithm="giac")

[Out]

integrate((d*cot(f*x + e))^n/(b*tan(f*x + e) + a), x)