3.8 \(\int \cot ^6(c+d x) (a+b \tan (c+d x)) (B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\)

Optimal. Leaf size=108 \[ -\frac{(a C+b B) \cot ^3(c+d x)}{3 d}+\frac{(a B-b C) \cot ^2(c+d x)}{2 d}+\frac{(a C+b B) \cot (c+d x)}{d}+\frac{(a B-b C) \log (\sin (c+d x))}{d}+x (a C+b B)-\frac{a B \cot ^4(c+d x)}{4 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.226194, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {3632, 3591, 3529, 3531, 3475} \[ -\frac{(a C+b B) \cot ^3(c+d x)}{3 d}+\frac{(a B-b C) \cot ^2(c+d x)}{2 d}+\frac{(a C+b B) \cot (c+d x)}{d}+\frac{(a B-b C) \log (\sin (c+d x))}{d}+x (a C+b B)-\frac{a B \cot ^4(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

(b*B + a*C)*x + ((b*B + a*C)*Cot[c + d*x])/d + ((a*B - b*C)*Cot[c + d*x]^2)/(2*d) - ((b*B + a*C)*Cot[c + d*x]^
3)/(3*d) - (a*B*Cot[c + d*x]^4)/(4*d) + ((a*B - b*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 3591

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((b*c - a*d)*(A*b - a*B)*(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[a*A*c + b*B*c + A*b*d - a*B*d - (A*b*
c - a*B*c - a*A*d - b*B*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 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 ^6(c+d x) (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot ^5(c+d x) (a+b \tan (c+d x)) (B+C \tan (c+d x)) \, dx\\ &=-\frac{a B \cot ^4(c+d x)}{4 d}+\int \cot ^4(c+d x) (b B+a C-(a B-b C) \tan (c+d x)) \, dx\\ &=-\frac{(b B+a C) \cot ^3(c+d x)}{3 d}-\frac{a B \cot ^4(c+d x)}{4 d}+\int \cot ^3(c+d x) (-a B+b C-(b B+a C) \tan (c+d x)) \, dx\\ &=\frac{(a B-b C) \cot ^2(c+d x)}{2 d}-\frac{(b B+a C) \cot ^3(c+d x)}{3 d}-\frac{a B \cot ^4(c+d x)}{4 d}+\int \cot ^2(c+d x) (-b B-a C+(a B-b C) \tan (c+d x)) \, dx\\ &=\frac{(b B+a C) \cot (c+d x)}{d}+\frac{(a B-b C) \cot ^2(c+d x)}{2 d}-\frac{(b B+a C) \cot ^3(c+d x)}{3 d}-\frac{a B \cot ^4(c+d x)}{4 d}+\int \cot (c+d x) (a B-b C+(b B+a C) \tan (c+d x)) \, dx\\ &=(b B+a C) x+\frac{(b B+a C) \cot (c+d x)}{d}+\frac{(a B-b C) \cot ^2(c+d x)}{2 d}-\frac{(b B+a C) \cot ^3(c+d x)}{3 d}-\frac{a B \cot ^4(c+d x)}{4 d}+(a B-b C) \int \cot (c+d x) \, dx\\ &=(b B+a C) x+\frac{(b B+a C) \cot (c+d x)}{d}+\frac{(a B-b C) \cot ^2(c+d x)}{2 d}-\frac{(b B+a C) \cot ^3(c+d x)}{3 d}-\frac{a B \cot ^4(c+d x)}{4 d}+\frac{(a B-b C) \log (\sin (c+d x))}{d}\\ \end{align*}

Mathematica [C]  time = 1.14117, size = 100, normalized size = 0.93 \[ -\frac{4 (a C+b B) \cot ^3(c+d x) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-\tan ^2(c+d x)\right )+3 \left ((2 b C-2 a B) \cot ^2(c+d x)-4 (a B-b C) (\log (\tan (c+d x))+\log (\cos (c+d x)))+a B \cot ^4(c+d x)\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

-(4*(b*B + a*C)*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] + 3*((-2*a*B + 2*b*C)*Cot[c +
d*x]^2 + a*B*Cot[c + d*x]^4 - 4*(a*B - b*C)*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]])))/(12*d)

________________________________________________________________________________________

Maple [A]  time = 0.073, size = 150, normalized size = 1.4 \begin{align*} -{\frac{Bb \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{B\cot \left ( dx+c \right ) b}{d}}+Bxb+{\frac{Bbc}{d}}-{\frac{Cb \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{Cb\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{aB \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{aB \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{aB\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{Ca \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{C\cot \left ( dx+c \right ) a}{d}}+Cxa+{\frac{Cac}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6*(a+b*tan(d*x+c))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.67937, size = 165, normalized size = 1.53 \begin{align*} \frac{12 \,{\left (C a + B b\right )}{\left (d x + c\right )} - 6 \,{\left (B a - C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \,{\left (B a - C b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac{12 \,{\left (C a + B b\right )} \tan \left (d x + c\right )^{3} + 6 \,{\left (B a - C b\right )} \tan \left (d x + c\right )^{2} - 3 \, B a - 4 \,{\left (C a + B b\right )} \tan \left (d x + c\right )}{\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)^6*(a+b*tan(d*x+c))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.38618, size = 340, normalized size = 3.15 \begin{align*} \frac{6 \,{\left (B a - C b\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} + 3 \,{\left (4 \,{\left (C a + B b\right )} d x + 3 \, B a - 2 \, C b\right )} \tan \left (d x + c\right )^{4} + 12 \,{\left (C a + B b\right )} \tan \left (d x + c\right )^{3} + 6 \,{\left (B a - C b\right )} \tan \left (d x + c\right )^{2} - 3 \, B a - 4 \,{\left (C a + B b\right )} \tan \left (d x + c\right )}{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)^6*(a+b*tan(d*x+c))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

1/12*(6*(B*a - C*b)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^4 + 3*(4*(C*a + B*b)*d*x + 3*B*a - 2
*C*b)*tan(d*x + c)^4 + 12*(C*a + B*b)*tan(d*x + c)^3 + 6*(B*a - C*b)*tan(d*x + c)^2 - 3*B*a - 4*(C*a + B*b)*ta
n(d*x + c))/(d*tan(d*x + c)^4)

________________________________________________________________________________________

Sympy [A]  time = 88.9161, size = 211, normalized size = 1.95 \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 ) \left (B \tan{\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{6}{\left (c \right )} & \text{for}\: d = 0 \\- \frac{B a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B a \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + \frac{B a}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac{B a}{4 d \tan ^{4}{\left (c + d x \right )}} + B b x + \frac{B b}{d \tan{\left (c + d x \right )}} - \frac{B b}{3 d \tan ^{3}{\left (c + d x \right )}} + C a x + \frac{C a}{d \tan{\left (c + d x \right )}} - \frac{C a}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac{C b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{C b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{C b}{2 d \tan ^{2}{\left (c + d x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6*(a+b*tan(d*x+c))*(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))*(B*tan(c) + C*tan(c)**
2)*cot(c)**6, Eq(d, 0)), (-B*a*log(tan(c + d*x)**2 + 1)/(2*d) + B*a*log(tan(c + d*x))/d + B*a/(2*d*tan(c + d*x
)**2) - B*a/(4*d*tan(c + d*x)**4) + B*b*x + B*b/(d*tan(c + d*x)) - B*b/(3*d*tan(c + d*x)**3) + C*a*x + C*a/(d*
tan(c + d*x)) - C*a/(3*d*tan(c + d*x)**3) + C*b*log(tan(c + d*x)**2 + 1)/(2*d) - C*b*log(tan(c + d*x))/d - C*b
/(2*d*tan(c + d*x)**2), True))

________________________________________________________________________________________

Giac [B]  time = 1.58677, size = 404, normalized size = 3.74 \begin{align*} -\frac{3 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 8 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 8 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 120 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 120 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 192 \,{\left (C a + B b\right )}{\left (d x + c\right )} + 192 \,{\left (B a - C b\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 192 \,{\left (B a - C b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{400 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 400 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 120 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 120 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 8 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, B a}{\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)^6*(a+b*tan(d*x+c))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="giac")

[Out]

-1/192*(3*B*a*tan(1/2*d*x + 1/2*c)^4 - 8*C*a*tan(1/2*d*x + 1/2*c)^3 - 8*B*b*tan(1/2*d*x + 1/2*c)^3 - 36*B*a*ta
n(1/2*d*x + 1/2*c)^2 + 24*C*b*tan(1/2*d*x + 1/2*c)^2 + 120*C*a*tan(1/2*d*x + 1/2*c) + 120*B*b*tan(1/2*d*x + 1/
2*c) - 192*(C*a + B*b)*(d*x + c) + 192*(B*a - C*b)*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 192*(B*a - C*b)*log(abs(t
an(1/2*d*x + 1/2*c))) + (400*B*a*tan(1/2*d*x + 1/2*c)^4 - 400*C*b*tan(1/2*d*x + 1/2*c)^4 - 120*C*a*tan(1/2*d*x
+ 1/2*c)^3 - 120*B*b*tan(1/2*d*x + 1/2*c)^3 - 36*B*a*tan(1/2*d*x + 1/2*c)^2 + 24*C*b*tan(1/2*d*x + 1/2*c)^2 +
8*C*a*tan(1/2*d*x + 1/2*c) + 8*B*b*tan(1/2*d*x + 1/2*c) + 3*B*a)/tan(1/2*d*x + 1/2*c)^4)/d