3.442 \(\int \cot ^4(c+d x) (a+b \tan (c+d x))^3 \, dx\)

Optimal. Leaf size=104 \[ \frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}-\frac{b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}+a x \left (a^2-3 b^2\right )-\frac{7 a^2 b \cot ^2(c+d x)}{6 d}-\frac{a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d} \]

[Out]

a*(a^2 - 3*b^2)*x + (a*(a^2 - 3*b^2)*Cot[c + d*x])/d - (7*a^2*b*Cot[c + d*x]^2)/(6*d) - (b*(3*a^2 - b^2)*Log[S
in[c + d*x]])/d - (a^2*Cot[c + d*x]^3*(a + b*Tan[c + d*x]))/(3*d)

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Rubi [A]  time = 0.190841, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}-\frac{b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}+a x \left (a^2-3 b^2\right )-\frac{7 a^2 b \cot ^2(c+d x)}{6 d}-\frac{a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]

[Out]

a*(a^2 - 3*b^2)*x + (a*(a^2 - 3*b^2)*Cot[c + d*x])/d - (7*a^2*b*Cot[c + d*x]^2)/(6*d) - (b*(3*a^2 - b^2)*Log[S
in[c + d*x]])/d - (a^2*Cot[c + d*x]^3*(a + b*Tan[c + d*x]))/(3*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 ^4(c+d x) (a+b \tan (c+d x))^3 \, dx &=-\frac{a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}+\frac{1}{3} \int \cot ^3(c+d x) \left (7 a^2 b-3 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (2 a^2-3 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{7 a^2 b \cot ^2(c+d x)}{6 d}-\frac{a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}+\frac{1}{3} \int \cot ^2(c+d x) \left (-3 a \left (a^2-3 b^2\right )-3 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}-\frac{7 a^2 b \cot ^2(c+d x)}{6 d}-\frac{a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}+\frac{1}{3} \int \cot (c+d x) \left (-3 b \left (3 a^2-b^2\right )+3 a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx\\ &=a \left (a^2-3 b^2\right ) x+\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}-\frac{7 a^2 b \cot ^2(c+d x)}{6 d}-\frac{a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}-\left (b \left (3 a^2-b^2\right )\right ) \int \cot (c+d x) \, dx\\ &=a \left (a^2-3 b^2\right ) x+\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}-\frac{7 a^2 b \cot ^2(c+d x)}{6 d}-\frac{b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}\\ \end{align*}

Mathematica [C]  time = 1.13975, size = 120, normalized size = 1.15 \[ \frac{6 a \left (a^2-3 b^2\right ) \cot (c+d x)+6 b \left (b^2-3 a^2\right ) \log (\tan (c+d x))-9 a^2 b \cot ^2(c+d x)-2 a^3 \cot ^3(c+d x)+3 (-b+i a)^3 \log (-\tan (c+d x)+i)-3 (b+i a)^3 \log (\tan (c+d x)+i)}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]

[Out]

(6*a*(a^2 - 3*b^2)*Cot[c + d*x] - 9*a^2*b*Cot[c + d*x]^2 - 2*a^3*Cot[c + d*x]^3 + 3*(I*a - b)^3*Log[I - Tan[c
+ d*x]] + 6*b*(-3*a^2 + b^2)*Log[Tan[c + d*x]] - 3*(I*a + b)^3*Log[I + Tan[c + d*x]])/(6*d)

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Maple [A]  time = 0.051, size = 123, normalized size = 1.2 \begin{align*}{\frac{{b}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-3\,a{b}^{2}x-3\,{\frac{a{b}^{2}\cot \left ( dx+c \right ) }{d}}-3\,{\frac{a{b}^{2}c}{d}}-{\frac{3\,b{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-3\,{\frac{b{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}+{a}^{3}x+{\frac{{a}^{3}c}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.50766, size = 158, normalized size = 1.52 \begin{align*} \frac{6 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )} + 3 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{9 \, a^{2} b \tan \left (d x + c\right ) + 2 \, a^{3} - 6 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/6*(6*(a^3 - 3*a*b^2)*(d*x + c) + 3*(3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1) - 6*(3*a^2*b - b^3)*log(tan(d*x +
c)) - (9*a^2*b*tan(d*x + c) + 2*a^3 - 6*(a^3 - 3*a*b^2)*tan(d*x + c)^2)/tan(d*x + c)^3)/d

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Fricas [A]  time = 1.82226, size = 297, normalized size = 2.86 \begin{align*} -\frac{3 \,{\left (3 \, a^{2} b - b^{3}\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 )^{3} + 9 \, a^{2} b \tan \left (d x + c\right ) + 3 \,{\left (3 \, a^{2} b - 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, a^{3} - 6 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{6 \, d \tan \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/6*(3*(3*a^2*b - b^3)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^3 + 9*a^2*b*tan(d*x + c) + 3*(3*
a^2*b - 2*(a^3 - 3*a*b^2)*d*x)*tan(d*x + c)^3 + 2*a^3 - 6*(a^3 - 3*a*b^2)*tan(d*x + c)^2)/(d*tan(d*x + c)^3)

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Sympy [A]  time = 9.29194, size = 177, normalized size = 1.7 \begin{align*} \begin{cases} \tilde{\infty } a^{3} x & \text{for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan{\left (c \right )}\right )^{3} \cot ^{4}{\left (c \right )} & \text{for}\: d = 0 \\a^{3} x + \frac{a^{3}}{d \tan{\left (c + d x \right )}} - \frac{a^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac{3 a^{2} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{3 a^{2} b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{3 a^{2} b}{2 d \tan ^{2}{\left (c + d x \right )}} - 3 a b^{2} x - \frac{3 a b^{2}}{d \tan{\left (c + d x \right )}} - \frac{b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b^{3} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a+b*tan(d*x+c))**3,x)

[Out]

Piecewise((zoo*a**3*x, (Eq(c, 0) | Eq(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x*(a + b*tan(c))**3*cot(c)**4, E
q(d, 0)), (a**3*x + a**3/(d*tan(c + d*x)) - a**3/(3*d*tan(c + d*x)**3) + 3*a**2*b*log(tan(c + d*x)**2 + 1)/(2*
d) - 3*a**2*b*log(tan(c + d*x))/d - 3*a**2*b/(2*d*tan(c + d*x)**2) - 3*a*b**2*x - 3*a*b**2/(d*tan(c + d*x)) -
b**3*log(tan(c + d*x)**2 + 1)/(2*d) + b**3*log(tan(c + d*x))/d, True))

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Giac [B]  time = 2.09889, size = 319, normalized size = 3.07 \begin{align*} \frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 36 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )} + 24 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 24 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{132 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 44 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 36 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/24*(a^3*tan(1/2*d*x + 1/2*c)^3 - 9*a^2*b*tan(1/2*d*x + 1/2*c)^2 - 15*a^3*tan(1/2*d*x + 1/2*c) + 36*a*b^2*tan
(1/2*d*x + 1/2*c) + 24*(a^3 - 3*a*b^2)*(d*x + c) + 24*(3*a^2*b - b^3)*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 24*(3*
a^2*b - b^3)*log(abs(tan(1/2*d*x + 1/2*c))) + (132*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 44*b^3*tan(1/2*d*x + 1/2*c)^
3 + 15*a^3*tan(1/2*d*x + 1/2*c)^2 - 36*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 9*a^2*b*tan(1/2*d*x + 1/2*c) - a^3)/tan(
1/2*d*x + 1/2*c)^3)/d