3.881 \(\int (d \cot (e+f x))^n (a+b \tan (e+f x))^2 \, dx\)

Optimal. Leaf size=132 \[ -\frac{d \left (a^2-b^2\right ) (d \cot (e+f x))^{n-1} \, _2F_1\left (1,\frac{n-1}{2};\frac{n+1}{2};-\cot ^2(e+f x)\right )}{f (1-n)}+\frac{a^2 d (d \cot (e+f x))^{n-1}}{f (1-n)}-\frac{2 a b (d \cot (e+f x))^n \, _2F_1\left (1,\frac{n}{2};\frac{n+2}{2};-\cot ^2(e+f x)\right )}{f n} \]

[Out]

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

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Rubi [A]  time = 0.217563, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3673, 3543, 3538, 3476, 364} \[ -\frac{d \left (a^2-b^2\right ) (d \cot (e+f x))^{n-1} \, _2F_1\left (1,\frac{n-1}{2};\frac{n+1}{2};-\cot ^2(e+f x)\right )}{f (1-n)}+\frac{a^2 d (d \cot (e+f x))^{n-1}}{f (1-n)}-\frac{2 a b (d \cot (e+f x))^n \, _2F_1\left (1,\frac{n}{2};\frac{n+2}{2};-\cot ^2(e+f x)\right )}{f n} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cot[e + f*x])^n*(a + b*Tan[e + f*x])^2,x]

[Out]

(a^2*d*(d*Cot[e + f*x])^(-1 + n))/(f*(1 - n)) - ((a^2 - b^2)*d*(d*Cot[e + f*x])^(-1 + n)*Hypergeometric2F1[1,
(-1 + n)/2, (1 + n)/2, -Cot[e + f*x]^2])/(f*(1 - n)) - (2*a*b*(d*Cot[e + f*x])^n*Hypergeometric2F1[1, n/2, (2
+ n)/2, -Cot[e + f*x]^2])/(f*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 3543

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

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])

Rubi steps

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

Mathematica [A]  time = 0.447957, size = 107, normalized size = 0.81 \[ -\frac{d (d \cot (e+f x))^{n-1} \left (a \left (a n+2 b (n-1) \cot (e+f x) \, _2F_1\left (1,\frac{n}{2};\frac{n+2}{2};-\cot ^2(e+f x)\right )\right )-n \left (a^2-b^2\right ) \, _2F_1\left (1,\frac{n-1}{2};\frac{n+1}{2};-\cot ^2(e+f x)\right )\right )}{f (n-1) n} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cot[e + f*x])^n*(a + b*Tan[e + f*x])^2,x]

[Out]

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

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Maple [F]  time = 0.322, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n} \left ( a+b\tan \left ( fx+e \right ) \right ) ^{2}\, 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))^2,x)

[Out]

int((d*cot(f*x+e))^n*(a+b*tan(f*x+e))^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (f x + e\right ) + a\right )}^{2} \left (d \cot \left (f x + e\right )\right )^{n}\,{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))^2,x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e) + a)^2*(d*cot(f*x + e))^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}\right )} \left (d \cot \left (f x + e\right )\right )^{n}, 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))^2,x, algorithm="fricas")

[Out]

integral((b^2*tan(f*x + e)^2 + 2*a*b*tan(f*x + e) + a^2)*(d*cot(f*x + e))^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cot{\left (e + f x \right )}\right )^{n} \left (a + b \tan{\left (e + f x \right )}\right )^{2}\, 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))**2,x)

[Out]

Integral((d*cot(e + f*x))**n*(a + b*tan(e + f*x))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (f x + e\right ) + a\right )}^{2} \left (d \cot \left (f x + e\right )\right )^{n}\,{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))^2,x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e) + a)^2*(d*cot(f*x + e))^n, x)