3.450 \(\int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=97 \[ \frac{b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-x \left (-6 a^2 b^2+a^4+b^4\right )+\frac{4 a^3 b \log (\sin (c+d x))}{d}-\frac{a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}-\frac{4 a b^3 \log (\cos (c+d x))}{d} \]

[Out]

-((a^4 - 6*a^2*b^2 + b^4)*x) - (4*a*b^3*Log[Cos[c + d*x]])/d + (4*a^3*b*Log[Sin[c + d*x]])/d + (b^2*(a^2 + b^2
)*Tan[c + d*x])/d - (a^2*Cot[c + d*x]*(a + b*Tan[c + d*x])^2)/d

________________________________________________________________________________________

Rubi [A]  time = 0.175379, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3565, 3637, 3624, 3475} \[ \frac{b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-x \left (-6 a^2 b^2+a^4+b^4\right )+\frac{4 a^3 b \log (\sin (c+d x))}{d}-\frac{a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}-\frac{4 a b^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]

[Out]

-((a^4 - 6*a^2*b^2 + b^4)*x) - (4*a*b^3*Log[Cos[c + d*x]])/d + (4*a^3*b*Log[Sin[c + d*x]])/d + (b^2*(a^2 + b^2
)*Tan[c + d*x])/d - (a^2*Cot[c + d*x]*(a + b*Tan[c + d*x])^2)/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 3637

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

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))^4 \, dx &=-\frac{a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}+\int \cot (c+d x) (a+b \tan (c+d x)) \left (4 a^2 b-a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (a^2+b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac{a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}-\int \cot (c+d x) \left (-4 a^3 b+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)-4 a b^3 \tan ^2(c+d x)\right ) \, dx\\ &=-\left (a^4-6 a^2 b^2+b^4\right ) x+\frac{b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac{a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}+\left (4 a^3 b\right ) \int \cot (c+d x) \, dx+\left (4 a b^3\right ) \int \tan (c+d x) \, dx\\ &=-\left (a^4-6 a^2 b^2+b^4\right ) x-\frac{4 a b^3 \log (\cos (c+d x))}{d}+\frac{4 a^3 b \log (\sin (c+d x))}{d}+\frac{b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac{a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}\\ \end{align*}

Mathematica [C]  time = 0.263278, size = 94, normalized size = 0.97 \[ -\frac{a^4 \cot (c+d x)-4 a b^3 \log (\tan (c+d x))+\frac{1}{2} i (a-i b)^4 \log (-\cot (c+d x)+i)-\frac{1}{2} i (a+i b)^4 \log (\cot (c+d x)+i)-b^4 \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 112, normalized size = 1.2 \begin{align*} -{a}^{4}x+6\,{a}^{2}{b}^{2}x-{b}^{4}x-{\frac{{a}^{4}\cot \left ( dx+c \right ) }{d}}+{\frac{{b}^{4}\tan \left ( dx+c \right ) }{d}}-4\,{\frac{{b}^{3}a\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{b{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{4}c}{d}}+6\,{\frac{{a}^{2}{b}^{2}c}{d}}-{\frac{{b}^{4}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))^4,x)

[Out]

-a^4*x+6*a^2*b^2*x-b^4*x-a^4*cot(d*x+c)/d+1/d*b^4*tan(d*x+c)-4/d*b^3*a*ln(cos(d*x+c))+4*a^3*b*ln(sin(d*x+c))/d
-1/d*a^4*c+6/d*a^2*b^2*c-1/d*b^4*c

________________________________________________________________________________________

Maxima [A]  time = 1.52433, size = 119, normalized size = 1.23 \begin{align*} \frac{4 \, a^{3} b \log \left (\tan \left (d x + c\right )\right ) + b^{4} \tan \left (d x + c\right ) - \frac{a^{4}}{\tan \left (d x + c\right )} -{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )}{\left (d x + c\right )} - 2 \,{\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{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))^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.21498, size = 274, normalized size = 2.82 \begin{align*} \frac{2 \, a^{3} 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 ) - 2 \, a b^{3} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) + b^{4} \tan \left (d x + c\right )^{2} - a^{4} -{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} d x \tan \left (d x + c\right )}{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))^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 4.90322, size = 133, normalized size = 1.37 \begin{align*} \begin{cases} \tilde{\infty } a^{4} 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 )^{4} \cot ^{2}{\left (c \right )} & \text{for}\: d = 0 \\- a^{4} x - \frac{a^{4}}{d \tan{\left (c + d x \right )}} - \frac{2 a^{3} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{4 a^{3} b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 6 a^{2} b^{2} x + \frac{2 a b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - b^{4} x + \frac{b^{4} \tan{\left (c + d x \right )}}{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))**4,x)

[Out]

Piecewise((zoo*a**4*x, (Eq(c, 0) | Eq(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x*(a + b*tan(c))**4*cot(c)**2, E
q(d, 0)), (-a**4*x - a**4/(d*tan(c + d*x)) - 2*a**3*b*log(tan(c + d*x)**2 + 1)/d + 4*a**3*b*log(tan(c + d*x))/
d + 6*a**2*b**2*x + 2*a*b**3*log(tan(c + d*x)**2 + 1)/d - b**4*x + b**4*tan(c + d*x)/d, True))

________________________________________________________________________________________

Giac [A]  time = 2.80861, size = 138, normalized size = 1.42 \begin{align*} \frac{4 \, a^{3} b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + b^{4} \tan \left (d x + c\right ) -{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )}{\left (d x + c\right )} - 2 \,{\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - \frac{4 \, a^{3} b \tan \left (d x + c\right ) + a^{4}}{\tan \left (d x + c\right )}}{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))^4,x, algorithm="giac")

[Out]

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