3.15 \(\int \cot ^5(c+d x) (a+b \tan (c+d x))^2 (B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\)

Optimal. Leaf size=118 \[ \frac{\left (a^2 B-2 a b C-b^2 B\right ) \cot (c+d x)}{d}+x \left (a^2 B-2 a b C-b^2 B\right )-\frac{a^2 B \cot ^3(c+d x)}{3 d}+\frac{\left (b^2 C-a (a C+2 b B)\right ) \log (\sin (c+d x))}{d}-\frac{a (a C+2 b B) \cot ^2(c+d x)}{2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.31102, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3632, 3604, 3628, 3529, 3531, 3475} \[ \frac{\left (a^2 B-2 a b C-b^2 B\right ) \cot (c+d x)}{d}+x \left (a^2 B-2 a b C-b^2 B\right )-\frac{a^2 B \cot ^3(c+d x)}{3 d}+\frac{\left (b^2 C-a (a C+2 b B)\right ) \log (\sin (c+d x))}{d}-\frac{a (a C+2 b B) \cot ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

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

Rule 3632

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

Rule 3604

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^2*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.)
+ (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[((B*c - A*d)*(b*c - a*d)^2*(c + d*Tan[e + f*x])^(n + 1))/(f*d^2*(n +
1)*(c^2 + d^2)), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f*x])^(n + 1)*Simp[B*(b*c - a*d)^2 + A*d*(a^2
*c - b^2*c + 2*a*b*d) + d*(B*(a^2*c - b^2*c + 2*a*b*d) + A*(2*a*b*c - a^2*d + b^2*d))*Tan[e + f*x] + b^2*B*(c^
2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^
2, 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))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot ^4(c+d x) (a+b \tan (c+d x))^2 (B+C \tan (c+d x)) \, dx\\ &=-\frac{a^2 B \cot ^3(c+d x)}{3 d}+\int \cot ^3(c+d x) \left (a (2 b B+a C)-\left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)+b^2 C \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{a (2 b B+a C) \cot ^2(c+d x)}{2 d}-\frac{a^2 B \cot ^3(c+d x)}{3 d}+\int \cot ^2(c+d x) \left (-a^2 B+b^2 B+2 a b C+\left (b^2 C-a (2 b B+a C)\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{\left (a^2 B-b^2 B-2 a b C\right ) \cot (c+d x)}{d}-\frac{a (2 b B+a C) \cot ^2(c+d x)}{2 d}-\frac{a^2 B \cot ^3(c+d x)}{3 d}+\int \cot (c+d x) \left (b^2 C-a (2 b B+a C)+\left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)\right ) \, dx\\ &=\left (a^2 B-b^2 B-2 a b C\right ) x+\frac{\left (a^2 B-b^2 B-2 a b C\right ) \cot (c+d x)}{d}-\frac{a (2 b B+a C) \cot ^2(c+d x)}{2 d}-\frac{a^2 B \cot ^3(c+d x)}{3 d}+\left (b^2 C-a (2 b B+a C)\right ) \int \cot (c+d x) \, dx\\ &=\left (a^2 B-b^2 B-2 a b C\right ) x+\frac{\left (a^2 B-b^2 B-2 a b C\right ) \cot (c+d x)}{d}-\frac{a (2 b B+a C) \cot ^2(c+d x)}{2 d}-\frac{a^2 B \cot ^3(c+d x)}{3 d}+\frac{\left (b^2 C-a (2 b B+a C)\right ) \log (\sin (c+d x))}{d}\\ \end{align*}

Mathematica [C]  time = 1.1463, size = 152, normalized size = 1.29 \[ \frac{6 \left (a^2 B-2 a b C-b^2 B\right ) \cot (c+d x)-6 \left (a^2 C+2 a b B-b^2 C\right ) \log (\tan (c+d x))-2 a^2 B \cot ^3(c+d x)-3 a (a C+2 b B) \cot ^2(c+d x)+3 (a+i b)^2 (C-i B) \log (-\tan (c+d x)+i)+3 (a-i b)^2 (C+i B) \log (\tan (c+d x)+i)}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.093, size = 188, normalized size = 1.6 \begin{align*} -{b}^{2}Bx-{\frac{B\cot \left ( dx+c \right ){b}^{2}}{d}}-{\frac{B{b}^{2}c}{d}}+{\frac{{b}^{2}C\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{Bab \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-2\,{\frac{Bab\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-2\,Cabx-2\,{\frac{C\cot \left ( dx+c \right ) ab}{d}}-2\,{\frac{Cabc}{d}}-{\frac{{a}^{2}B \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{B\cot \left ( dx+c \right ){a}^{2}}{d}}+{a}^{2}Bx+{\frac{{a}^{2}Bc}{d}}-{\frac{C{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{C{a}^{2}\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))^2*(B*tan(d*x+c)+C*tan(d*x+c)^2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.68782, size = 201, normalized size = 1.7 \begin{align*} \frac{6 \,{\left (B a^{2} - 2 \, C a b - B b^{2}\right )}{\left (d x + c\right )} + 3 \,{\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \,{\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{2 \, B a^{2} - 6 \,{\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (C a^{2} + 2 \, B a b\right )} \tan \left (d x + c\right )}{\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)^5*(a+b*tan(d*x+c))^2*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.50593, size = 367, normalized size = 3.11 \begin{align*} -\frac{3 \,{\left (C a^{2} + 2 \, B a b - C 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 )^{3} + 3 \,{\left (C a^{2} + 2 \, B a b - 2 \,{\left (B a^{2} - 2 \, C a b - B b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, B a^{2} - 6 \,{\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (C a^{2} + 2 \, B a b\right )} \tan \left (d x + c\right )}{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)^5*(a+b*tan(d*x+c))^2*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 47.6416, size = 258, normalized size = 2.19 \begin{align*} \begin{cases} \text{NaN} & \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 )^{2} \left (B \tan{\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{5}{\left (c \right )} & \text{for}\: d = 0 \\B a^{2} x + \frac{B a^{2}}{d \tan{\left (c + d x \right )}} - \frac{B a^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac{B a b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac{2 B a b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{B a b}{d \tan ^{2}{\left (c + d x \right )}} - B b^{2} x - \frac{B b^{2}}{d \tan{\left (c + d x \right )}} + \frac{C a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{C a^{2} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{C a^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - 2 C a b x - \frac{2 C a b}{d \tan{\left (c + d x \right )}} - \frac{C b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{C b^{2} \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)**5*(a+b*tan(d*x+c))**2*(B*tan(d*x+c)+C*tan(d*x+c)**2),x)

[Out]

Piecewise((nan, (Eq(c, 0) | Eq(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x*(a + b*tan(c))**2*(B*tan(c) + C*tan(c
)**2)*cot(c)**5, Eq(d, 0)), (B*a**2*x + B*a**2/(d*tan(c + d*x)) - B*a**2/(3*d*tan(c + d*x)**3) + B*a*b*log(tan
(c + d*x)**2 + 1)/d - 2*B*a*b*log(tan(c + d*x))/d - B*a*b/(d*tan(c + d*x)**2) - B*b**2*x - B*b**2/(d*tan(c + d
*x)) + C*a**2*log(tan(c + d*x)**2 + 1)/(2*d) - C*a**2*log(tan(c + d*x))/d - C*a**2/(2*d*tan(c + d*x)**2) - 2*C
*a*b*x - 2*C*a*b/(d*tan(c + d*x)) - C*b**2*log(tan(c + d*x)**2 + 1)/(2*d) + C*b**2*log(tan(c + d*x))/d, True))

________________________________________________________________________________________

Giac [B]  time = 2.0847, size = 451, normalized size = 3.82 \begin{align*} \frac{B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 6 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, B b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \,{\left (B a^{2} - 2 \, C a b - B b^{2}\right )}{\left (d x + c\right )} + 24 \,{\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 24 \,{\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{44 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 88 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 44 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, B b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - B a^{2}}{\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)^5*(a+b*tan(d*x+c))^2*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="giac")

[Out]

1/24*(B*a^2*tan(1/2*d*x + 1/2*c)^3 - 3*C*a^2*tan(1/2*d*x + 1/2*c)^2 - 6*B*a*b*tan(1/2*d*x + 1/2*c)^2 - 15*B*a^
2*tan(1/2*d*x + 1/2*c) + 24*C*a*b*tan(1/2*d*x + 1/2*c) + 12*B*b^2*tan(1/2*d*x + 1/2*c) + 24*(B*a^2 - 2*C*a*b -
B*b^2)*(d*x + c) + 24*(C*a^2 + 2*B*a*b - C*b^2)*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 24*(C*a^2 + 2*B*a*b - C*b^2
)*log(abs(tan(1/2*d*x + 1/2*c))) + (44*C*a^2*tan(1/2*d*x + 1/2*c)^3 + 88*B*a*b*tan(1/2*d*x + 1/2*c)^3 - 44*C*b
^2*tan(1/2*d*x + 1/2*c)^3 + 15*B*a^2*tan(1/2*d*x + 1/2*c)^2 - 24*C*a*b*tan(1/2*d*x + 1/2*c)^2 - 12*B*b^2*tan(1
/2*d*x + 1/2*c)^2 - 3*C*a^2*tan(1/2*d*x + 1/2*c) - 6*B*a*b*tan(1/2*d*x + 1/2*c) - B*a^2)/tan(1/2*d*x + 1/2*c)^
3)/d