3.453 \(\int \cot ^5(c+d x) (a+b \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=141 \[ \frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}+\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\sin (c+d x))}{d}+4 a b x \left (a^2-b^2\right )-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d} \]

[Out]

4*a*b*(a^2 - b^2)*x + (4*a*b*(a^2 - b^2)*Cot[c + d*x])/d + (a^2*(2*a^2 - 11*b^2)*Cot[c + d*x]^2)/(4*d) - (5*a^
3*b*Cot[c + d*x]^3)/(6*d) + ((a^4 - 6*a^2*b^2 + b^4)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^4*(a + b*Tan[c +
d*x])^2)/(4*d)

________________________________________________________________________________________

Rubi [A]  time = 0.341677, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3565, 3635, 3628, 3529, 3531, 3475} \[ \frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}+\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\sin (c+d x))}{d}+4 a b x \left (a^2-b^2\right )-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^4,x]

[Out]

4*a*b*(a^2 - b^2)*x + (4*a*b*(a^2 - b^2)*Cot[c + d*x])/d + (a^2*(2*a^2 - 11*b^2)*Cot[c + d*x]^2)/(4*d) - (5*a^
3*b*Cot[c + d*x]^3)/(6*d) + ((a^4 - 6*a^2*b^2 + b^4)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^4*(a + b*Tan[c +
d*x])^2)/(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 3635

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 - a*d)*(c^2*C - B*c*d + A*d^2)*
(c + d*Tan[e + f*x])^(n + 1))/(d^2*f*(n + 1)*(c^2 + d^2)), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f*x
])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d +
a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] &&
NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]

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))^4 \, dx &=-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot ^4(c+d x) (a+b \tan (c+d x)) \left (10 a^2 b-4 a \left (a^2-3 b^2\right ) \tan (c+d x)-2 b \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot ^3(c+d x) \left (-2 a^2 \left (2 a^2-11 b^2\right )-16 a b \left (a^2-b^2\right ) \tan (c+d x)-2 b^2 \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot ^2(c+d x) \left (-16 a b \left (a^2-b^2\right )+4 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot (c+d x) \left (4 \left (a^4-6 a^2 b^2+b^4\right )+16 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=4 a b \left (a^2-b^2\right ) x+\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\left (a^4-6 a^2 b^2+b^4\right ) \int \cot (c+d x) \, dx\\ &=4 a b \left (a^2-b^2\right ) x+\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}+\frac{\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\\ \end{align*}

Mathematica [C]  time = 2.25636, size = 147, normalized size = 1.04 \[ \frac{6 a^2 \left (a^2-6 b^2\right ) \cot ^2(c+d x)+48 a b \left (a^2-b^2\right ) \cot (c+d x)-6 \left (-2 \left (-6 a^2 b^2+a^4+b^4\right ) \log (\tan (c+d x))+(a-i b)^4 \log (\tan (c+d x)+i)+(a+i b)^4 \log (-\tan (c+d x)+i)\right )-16 a^3 b \cot ^3(c+d x)-3 a^4 \cot ^4(c+d x)}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^4,x]

[Out]

(48*a*b*(a^2 - b^2)*Cot[c + d*x] + 6*a^2*(a^2 - 6*b^2)*Cot[c + d*x]^2 - 16*a^3*b*Cot[c + d*x]^3 - 3*a^4*Cot[c
+ d*x]^4 - 6*((a + I*b)^4*Log[I - Tan[c + d*x]] - 2*(a^4 - 6*a^2*b^2 + b^4)*Log[Tan[c + d*x]] + (a - I*b)^4*Lo
g[I + Tan[c + d*x]]))/(12*d)

________________________________________________________________________________________

Maple [A]  time = 0.063, size = 180, normalized size = 1.3 \begin{align*}{\frac{{b}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-4\,{b}^{3}ax-4\,{\frac{\cot \left ( dx+c \right ) a{b}^{3}}{d}}-4\,{\frac{a{b}^{3}c}{d}}-3\,{\frac{{a}^{2}{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-6\,{\frac{{a}^{2}{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{4\,b{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+4\,{\frac{b{a}^{3}\cot \left ( dx+c \right ) }{d}}+4\,x{a}^{3}b+4\,{\frac{b{a}^{3}c}{d}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{a}^{4}\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))^4,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.53747, size = 201, normalized size = 1.43 \begin{align*} \frac{48 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )} - 6 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{16 \, a^{3} b \tan \left (d x + c\right ) + 3 \, a^{4} - 48 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 6 \,{\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{4}}}{12 \, 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))^4,x, algorithm="maxima")

[Out]

1/12*(48*(a^3*b - a*b^3)*(d*x + c) - 6*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1) + 12*(a^4 - 6*a^2*b^2 +
b^4)*log(tan(d*x + c)) - (16*a^3*b*tan(d*x + c) + 3*a^4 - 48*(a^3*b - a*b^3)*tan(d*x + c)^3 - 6*(a^4 - 6*a^2*
b^2)*tan(d*x + c)^2)/tan(d*x + c)^4)/d

________________________________________________________________________________________

Fricas [A]  time = 2.05052, size = 377, normalized size = 2.67 \begin{align*} \frac{6 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\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} - 16 \, a^{3} b \tan \left (d x + c\right ) + 3 \,{\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 16 \,{\left (a^{3} b - a b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{4} - 3 \, a^{4} + 48 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} + 6 \,{\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{12 \, 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))^4,x, algorithm="fricas")

[Out]

1/12*(6*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^4 - 16*a^3*b*tan(d*x + c
) + 3*(3*a^4 - 12*a^2*b^2 + 16*(a^3*b - a*b^3)*d*x)*tan(d*x + c)^4 - 3*a^4 + 48*(a^3*b - a*b^3)*tan(d*x + c)^3
+ 6*(a^4 - 6*a^2*b^2)*tan(d*x + c)^2)/(d*tan(d*x + c)^4)

________________________________________________________________________________________

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))**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.84577, size = 452, normalized size = 3.21 \begin{align*} -\frac{3 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 32 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 144 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 480 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 384 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 768 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )} + 192 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 192 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{400 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 2400 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 400 \, b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 480 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 384 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 144 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 32 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{4}}{\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))^4,x, algorithm="giac")

[Out]

-1/192*(3*a^4*tan(1/2*d*x + 1/2*c)^4 - 32*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 36*a^4*tan(1/2*d*x + 1/2*c)^2 + 144*a
^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 480*a^3*b*tan(1/2*d*x + 1/2*c) - 384*a*b^3*tan(1/2*d*x + 1/2*c) - 768*(a^3*b -
a*b^3)*(d*x + c) + 192*(a^4 - 6*a^2*b^2 + b^4)*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 192*(a^4 - 6*a^2*b^2 + b^4)*
log(abs(tan(1/2*d*x + 1/2*c))) + (400*a^4*tan(1/2*d*x + 1/2*c)^4 - 2400*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 400*b
^4*tan(1/2*d*x + 1/2*c)^4 - 480*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 384*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 36*a^4*tan(1
/2*d*x + 1/2*c)^2 + 144*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 32*a^3*b*tan(1/2*d*x + 1/2*c) + 3*a^4)/tan(1/2*d*x +
1/2*c)^4)/d