3.440 \(\int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx\)

Optimal. Leaf size=69 \[ -a x \left (a^2-3 b^2\right )+\frac{3 a^2 b \log (\sin (c+d x))}{d}-\frac{a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}-\frac{b^3 \log (\cos (c+d x))}{d} \]

[Out]

-(a*(a^2 - 3*b^2)*x) - (b^3*Log[Cos[c + d*x]])/d + (3*a^2*b*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]*(a + b*Ta
n[c + d*x]))/d

________________________________________________________________________________________

Rubi [A]  time = 0.0874544, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3565, 3624, 3475} \[ -a x \left (a^2-3 b^2\right )+\frac{3 a^2 b \log (\sin (c+d x))}{d}-\frac{a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}-\frac{b^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]

[Out]

-(a*(a^2 - 3*b^2)*x) - (b^3*Log[Cos[c + d*x]])/d + (3*a^2*b*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]*(a + b*Ta
n[c + d*x]))/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 3624

Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/tan[(e_.) + (f_.)*(x_)], x_Symbol
] :> Simp[B*x, x] + (Dist[A, Int[1/Tan[e + f*x], x], x] + Dist[C, Int[Tan[e + f*x], x], x]) /; FreeQ[{e, f, A,
B, C}, x] && NeQ[A, C]

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

Mathematica [C]  time = 0.136075, size = 80, normalized size = 1.16 \[ -\frac{a^3 \cot (c+d x)-\frac{1}{2} (b+i a)^3 \log (-\cot (c+d x)+i)+\frac{1}{2} (-b+i a)^3 \log (\cot (c+d x)+i)-b^3 \log (\tan (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 79, normalized size = 1.1 \begin{align*} -{a}^{3}x+3\,a{b}^{2}x-{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}+3\,{\frac{b{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{b}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3}c}{d}}+3\,{\frac{a{b}^{2}c}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2*(a+b*tan(d*x+c))^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.49508, size = 100, normalized size = 1.45 \begin{align*} \frac{6 \, a^{2} b \log \left (\tan \left (d x + c\right )\right ) - 2 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )} -{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - \frac{2 \, a^{3}}{\tan \left (d x + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.67587, size = 240, normalized size = 3.48 \begin{align*} \frac{3 \, a^{2} b \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 ) - b^{3} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) - 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} d x \tan \left (d x + c\right ) - 2 \, a^{3}}{2 \, d \tan \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 2.89713, size = 114, normalized size = 1.65 \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 ^{2}{\left (c \right )} & \text{for}\: d = 0 \\- a^{3} x - \frac{a^{3}}{d \tan{\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} + 3 a b^{2} x + \frac{b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2*(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)**2, E
q(d, 0)), (-a**3*x - a**3/(d*tan(c + d*x)) - 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*b**2*x + b**3*log(tan(c + d*x)**2 + 1)/(2*d), True))

________________________________________________________________________________________

Giac [A]  time = 2.03087, size = 119, normalized size = 1.72 \begin{align*} \frac{6 \, a^{2} b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - 2 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )} -{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - \frac{2 \,{\left (3 \, a^{2} b \tan \left (d x + c\right ) + a^{3}\right )}}{\tan \left (d x + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

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