3.443 \(\int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx\)

Optimal. Leaf size=130 \[ \frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}+b x \left (3 a^2-b^2\right )-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d} \]

[Out]

b*(3*a^2 - b^2)*x + (b*(3*a^2 - b^2)*Cot[c + d*x])/d + (a*(a^2 - 3*b^2)*Cot[c + d*x]^2)/(2*d) - (3*a^2*b*Cot[c
+ d*x]^3)/(4*d) + (a*(a^2 - 3*b^2)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^4*(a + b*Tan[c + d*x]))/(4*d)

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Rubi [A]  time = 0.231662, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3565, 3628, 3529, 3531, 3475} \[ \frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}+b x \left (3 a^2-b^2\right )-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3,x]

[Out]

b*(3*a^2 - b^2)*x + (b*(3*a^2 - b^2)*Cot[c + d*x])/d + (a*(a^2 - 3*b^2)*Cot[c + d*x]^2)/(2*d) - (3*a^2*b*Cot[c
+ d*x]^3)/(4*d) + (a*(a^2 - 3*b^2)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^4*(a + b*Tan[c + d*x]))/(4*d)

Rule 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3628

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]

Rule 3529

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

Rule 3531

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

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx &=-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac{1}{4} \int \cot ^4(c+d x) \left (9 a^2 b-4 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (3 a^2-4 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac{1}{4} \int \cot ^3(c+d x) \left (-4 a \left (a^2-3 b^2\right )-4 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac{1}{4} \int \cot ^2(c+d x) \left (-4 b \left (3 a^2-b^2\right )+4 a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac{1}{4} \int \cot (c+d x) \left (4 a \left (a^2-3 b^2\right )+4 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=b \left (3 a^2-b^2\right ) x+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\left (a \left (a^2-3 b^2\right )\right ) \int \cot (c+d x) \, dx\\ &=b \left (3 a^2-b^2\right ) x+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}+\frac{a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}\\ \end{align*}

Mathematica [C]  time = 1.68483, size = 118, normalized size = 0.91 \[ -\frac{-2 a \left (a^2-3 b^2\right ) \cot ^2(c+d x)+4 b \left (b^2-3 a^2\right ) \cot (c+d x)+4 a^2 b \cot ^3(c+d x)+a^3 \cot ^4(c+d x)+2 (a-i b)^3 \log (-\cot (c+d x)+i)+2 (a+i b)^3 \log (\cot (c+d x)+i)}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3,x]

[Out]

-(4*b*(-3*a^2 + b^2)*Cot[c + d*x] - 2*a*(a^2 - 3*b^2)*Cot[c + d*x]^2 + 4*a^2*b*Cot[c + d*x]^3 + a^3*Cot[c + d*
x]^4 + 2*(a - I*b)^3*Log[I - Cot[c + d*x]] + 2*(a + I*b)^3*Log[I + Cot[c + d*x]])/(4*d)

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Maple [A]  time = 0.056, size = 159, normalized size = 1.2 \begin{align*} -{b}^{3}x-{\frac{\cot \left ( dx+c \right ){b}^{3}}{d}}-{\frac{{b}^{3}c}{d}}-{\frac{3\,a{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-3\,{\frac{a{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{b{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}+3\,x{a}^{2}b+3\,{\frac{b{a}^{2}\cot \left ( dx+c \right ) }{d}}+3\,{\frac{b{a}^{2}c}{d}}-{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^5*(a+b*tan(d*x+c))^3,x)

[Out]

-b^3*x-1/d*cot(d*x+c)*b^3-1/d*b^3*c-3/2/d*a*b^2*cot(d*x+c)^2-3/d*a*b^2*ln(sin(d*x+c))-a^2*b*cot(d*x+c)^3/d+3*x
*a^2*b+3*a^2*b*cot(d*x+c)/d+3/d*a^2*b*c-1/4/d*a^3*cot(d*x+c)^4+1/2*a^3*cot(d*x+c)^2/d+a^3*ln(sin(d*x+c))/d

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Maxima [A]  time = 1.59505, size = 182, normalized size = 1.4 \begin{align*} \frac{4 \,{\left (3 \, a^{2} b - b^{3}\right )}{\left (d x + c\right )} - 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 4 \,{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{4 \, a^{2} b \tan \left (d x + c\right ) - 4 \,{\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} + a^{3} - 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{4}}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/4*(4*(3*a^2*b - b^3)*(d*x + c) - 2*(a^3 - 3*a*b^2)*log(tan(d*x + c)^2 + 1) + 4*(a^3 - 3*a*b^2)*log(tan(d*x +
c)) - (4*a^2*b*tan(d*x + c) - 4*(3*a^2*b - b^3)*tan(d*x + c)^3 + a^3 - 2*(a^3 - 3*a*b^2)*tan(d*x + c)^2)/tan(
d*x + c)^4)/d

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Fricas [A]  time = 1.61506, size = 348, normalized size = 2.68 \begin{align*} \frac{2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} +{\left (3 \, a^{3} - 6 \, a b^{2} + 4 \,{\left (3 \, a^{2} b - b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{4} - 4 \, a^{2} b \tan \left (d x + c\right ) + 4 \,{\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} - a^{3} + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{4 \, d \tan \left (d x + c\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/4*(2*(a^3 - 3*a*b^2)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^4 + (3*a^3 - 6*a*b^2 + 4*(3*a^2*b
- b^3)*d*x)*tan(d*x + c)^4 - 4*a^2*b*tan(d*x + c) + 4*(3*a^2*b - b^3)*tan(d*x + c)^3 - a^3 + 2*(a^3 - 3*a*b^2
)*tan(d*x + c)^2)/(d*tan(d*x + c)^4)

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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)**5*(a+b*tan(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.99658, size = 406, normalized size = 3.12 \begin{align*} -\frac{3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 360 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 192 \,{\left (3 \, a^{2} b - b^{3}\right )}{\left (d x + c\right )} + 192 \,{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 192 \,{\left (a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{400 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1200 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 360 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 96 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{192 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

-1/192*(3*a^3*tan(1/2*d*x + 1/2*c)^4 - 24*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 36*a^3*tan(1/2*d*x + 1/2*c)^2 + 72*a*
b^2*tan(1/2*d*x + 1/2*c)^2 + 360*a^2*b*tan(1/2*d*x + 1/2*c) - 96*b^3*tan(1/2*d*x + 1/2*c) - 192*(3*a^2*b - b^3
)*(d*x + c) + 192*(a^3 - 3*a*b^2)*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 192*(a^3 - 3*a*b^2)*log(abs(tan(1/2*d*x +
1/2*c))) + (400*a^3*tan(1/2*d*x + 1/2*c)^4 - 1200*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 360*a^2*b*tan(1/2*d*x + 1/2*c
)^3 + 96*b^3*tan(1/2*d*x + 1/2*c)^3 - 36*a^3*tan(1/2*d*x + 1/2*c)^2 + 72*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 24*a^2
*b*tan(1/2*d*x + 1/2*c) + 3*a^3)/tan(1/2*d*x + 1/2*c)^4)/d