3.8.14 \(\int \cot ^3(c+d x) (a+b \tan (c+d x))^n \, dx\) [714]

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

[Out]

1/2*b*(1-n)*cot(d*x+c)*(a+b*tan(d*x+c))^(1+n)/a^2/d-1/2*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1+n)/a/d-1/2*hypergeom(
[1, 1+n],[2+n],(a+b*tan(d*x+c))/(a-I*b))*(a+b*tan(d*x+c))^(1+n)/(a-I*b)/d/(1+n)-1/2*hypergeom([1, 1+n],[2+n],(
a+b*tan(d*x+c))/(a+I*b))*(a+b*tan(d*x+c))^(1+n)/(a+I*b)/d/(1+n)+1/2*(2*a^2+b^2*(1-n)*n)*hypergeom([1, 1+n],[2+
n],1+b*tan(d*x+c)/a)*(a+b*tan(d*x+c))^(1+n)/a^3/d/(1+n)

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Rubi [A]
time = 0.39, antiderivative size = 261, normalized size of antiderivative = 1.00, 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 = {3650, 3730, 3735, 12, 3620, 3618, 70, 3715, 67} \begin {gather*} \frac {b (1-n) \cot (c+d x) (a+b \tan (c+d x))^{n+1}}{2 a^2 d}+\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 {(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} \end {gather*}

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 12

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

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 70

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

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^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 3620

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 3650

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 3715

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 3730

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*Ta
n[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 3735

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]

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 ) \text {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 {\text {Subst}\left (\int \frac {(a-i b x)^n}{-1+x} \, dx,x,i \tan (c+d x)\right )}{2 d}+\frac {\text {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*}

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Mathematica [A]
time = 2.04, size = 212, normalized size = 0.81 \begin {gather*} -\frac {(b+a \cot (c+d x)) \left (a^3 (a+i b) \, _2F_1\left (1,1+n;2+n;\frac {a+b \tan (c+d x)}{a-i b}\right )+(a-i b) \left (a^3 \, _2F_1\left (1,1+n;2+n;\frac {a+b \tan (c+d x)}{a+i b}\right )+(a+i b) \left (a (1+n) \cot (c+d x) (b (-1+n)+a \cot (c+d x))+\left (-2 a^2+b^2 (-1+n) n\right ) \, _2F_1\left (1,1+n;2+n;1+\frac {b \tan (c+d x)}{a}\right )\right )\right )\right ) \tan (c+d x) (a+b \tan (c+d x))^n}{2 a^3 (a-i b) (a+i b) d (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.35, size = 0, normalized size = 0.00 \[\int \left (\cot ^{3}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx\]

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 = 0.00, 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(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 = 0.00, 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(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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (c + d x \right )}\right )^{n} \cot ^{3}{\left (c + d x \right )}\, dx \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {cot}\left (c+d\,x\right )}^3\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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