3.714 \(\int \cot ^3(c+d x) (a+b \tan (c+d x))^n \, dx\)

Optimal. Leaf size=261 \[ \frac{\left (2 a^2+b^2 (1-n) n\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right )}{2 a^3 d (n+1)}+\frac{b (1-n) \cot (c+d x) (a+b \tan (c+d x))^{n+1}}{2 a^2 d}-\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (a-i b)}-\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (a+i b)}-\frac{\cot ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{2 a d} \]

[Out]

(b*(1 - n)*Cot[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(2*a^2*d) - (Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n)
)/(2*a*d) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/
(2*(a - I*b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d
*x])^(1 + n))/(2*(a + I*b)*d*(1 + n)) + ((2*a^2 + b^2*(1 - n)*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan
[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(2*a^3*d*(1 + n))

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Rubi [A]  time = 0.561352, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3569, 3649, 3654, 12, 3539, 3537, 68, 3634, 65} \[ \frac{\left (2 a^2+b^2 (1-n) n\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right )}{2 a^3 d (n+1)}+\frac{b (1-n) \cot (c+d x) (a+b \tan (c+d x))^{n+1}}{2 a^2 d}-\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (a-i b)}-\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (a+i b)}-\frac{\cot ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{2 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]

[Out]

(b*(1 - n)*Cot[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(2*a^2*d) - (Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n)
)/(2*a*d) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/
(2*(a - I*b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d
*x])^(1 + n))/(2*(a + I*b)*d*(1 + n)) + ((2*a^2 + b^2*(1 - n)*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan
[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(2*a^3*d*(1 + n))

Rule 3569

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3654

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*ta
n[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*
C)*Tan[e + f*x], x], x], x] + Dist[(A*b^2 + a^2*C)/(a^2 + b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^
2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^
2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3539

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*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 3537

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

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

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 65

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

Rubi steps

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

Mathematica [A]  time = 1.85643, size = 212, normalized size = 0.81 \[ -\frac{\tan (c+d x) (a \cot (c+d x)+b) (a+b \tan (c+d x))^n \left ((a-i b) \left ((a+i b) \left (\left (b^2 (n-1) n-2 a^2\right ) \, _2F_1\left (1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right )+a (n+1) \cot (c+d x) (a \cot (c+d x)+b (n-1))\right )+a^3 \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right )\right )+a^3 (a+i b) \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right )\right )}{2 a^3 d (n+1) (a-i b) (a+i b)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]

[Out]

-((b + a*Cot[c + d*x])*(a^3*(a + I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] + (a
- I*b)*(a^3*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)] + (a + I*b)*(a*(1 + n)*Cot[c +
d*x]*(b*(-1 + n) + a*Cot[c + d*x]) + (-2*a^2 + b^2*(-1 + n)*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c
+ d*x])/a])))*Tan[c + d*x]*(a + b*Tan[c + d*x])^n)/(2*a^3*(a - I*b)*(a + I*b)*d*(1 + n))

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Maple [F]  time = 0.217, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^n,x)

[Out]

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^n*cot(d*x + c)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*tan(d*x + c) + a)^n*cot(d*x + c)^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3*(a+b*tan(d*x+c))**n,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*tan(d*x + c) + a)^n*cot(d*x + c)^3, x)