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

Optimal. Leaf size=87 \[ -\frac{(a C+b B) \cot ^2(c+d x)}{2 d}+\frac{(a B-b C) \cot (c+d x)}{d}-\frac{(a C+b B) \log (\sin (c+d x))}{d}+x (a B-b C)-\frac{a B \cot ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.19293, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{(a C+b B) \cot ^2(c+d x)}{2 d}+\frac{(a B-b C) \cot (c+d x)}{d}-\frac{(a C+b B) \log (\sin (c+d x))}{d}+x (a B-b C)-\frac{a B \cot ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*B - b*C)*x + ((a*B - b*C)*Cot[c + d*x])/d - ((b*B + a*C)*Cot[c + d*x]^2)/(2*d) - (a*B*Cot[c + d*x]^3)/(3*d)
- ((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 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 ^5(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 ^4(c+d x) (a+b \tan (c+d x)) (B+C \tan (c+d x)) \, dx\\ &=-\frac{a B \cot ^3(c+d x)}{3 d}+\int \cot ^3(c+d x) (b B+a C-(a B-b C) \tan (c+d x)) \, dx\\ &=-\frac{(b B+a C) \cot ^2(c+d x)}{2 d}-\frac{a B \cot ^3(c+d x)}{3 d}+\int \cot ^2(c+d x) (-a B+b C-(b B+a C) \tan (c+d x)) \, dx\\ &=\frac{(a B-b C) \cot (c+d x)}{d}-\frac{(b B+a C) \cot ^2(c+d x)}{2 d}-\frac{a B \cot ^3(c+d x)}{3 d}+\int \cot (c+d x) (-b B-a C+(a B-b C) \tan (c+d x)) \, dx\\ &=(a B-b C) x+\frac{(a B-b C) \cot (c+d x)}{d}-\frac{(b B+a C) \cot ^2(c+d x)}{2 d}-\frac{a B \cot ^3(c+d x)}{3 d}+(-b B-a C) \int \cot (c+d x) \, dx\\ &=(a B-b C) x+\frac{(a B-b C) \cot (c+d x)}{d}-\frac{(b B+a C) \cot ^2(c+d x)}{2 d}-\frac{a B \cot ^3(c+d x)}{3 d}-\frac{(b B+a C) \log (\sin (c+d x))}{d}\\ \end{align*}

Mathematica [C]  time = 0.995765, size = 101, normalized size = 1.16 \[ -\frac{2 a B \cot ^3(c+d x) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-\tan ^2(c+d x)\right )+6 b C \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(c+d x)\right )+3 (a C+b B) \left (\cot ^2(c+d x)+2 (\log (\tan (c+d x))+\log (\cos (c+d x)))\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*a*B*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] + 6*b*C*Cot[c + d*x]*Hypergeometric2F
1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 3*(b*B + a*C)*(Cot[c + d*x]^2 + 2*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]])))
/(6*d)

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Maple [A]  time = 0.076, size = 124, normalized size = 1.4 \begin{align*} -{\frac{Bb \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{Bb\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-Cbx-{\frac{C\cot \left ( dx+c \right ) b}{d}}-{\frac{Cbc}{d}}-{\frac{aB \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{B\cot \left ( dx+c \right ) a}{d}}+aBx+{\frac{aBc}{d}}-{\frac{Ca \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{Ca\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))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x)

[Out]

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

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Maxima [A]  time = 1.63866, size = 140, normalized size = 1.61 \begin{align*} \frac{6 \,{\left (B a - C b\right )}{\left (d x + c\right )} + 3 \,{\left (C a + B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \,{\left (C a + B b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac{6 \,{\left (B a - C b\right )} \tan \left (d x + c\right )^{2} - 2 \, B a - 3 \,{\left (C a + B 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))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.2671, size = 292, normalized size = 3.36 \begin{align*} -\frac{3 \,{\left (C a + B 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 )^{3} - 3 \,{\left (2 \,{\left (B a - C b\right )} d x - 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} + 2 \, B a + 3 \,{\left (C a + B 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))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 28.2979, size = 180, normalized size = 2.07 \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 ^{5}{\left (c \right )} & \text{for}\: d = 0 \\B a x + \frac{B a}{d \tan{\left (c + d x \right )}} - \frac{B a}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac{B b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{B b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{B b}{2 d \tan ^{2}{\left (c + d x \right )}} + \frac{C a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{C a \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{C a}{2 d \tan ^{2}{\left (c + d x \right )}} - C b x - \frac{C b}{d \tan{\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)**5*(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)**5, Eq(d, 0)), (B*a*x + B*a/(d*tan(c + d*x)) - B*a/(3*d*tan(c + d*x)**3) + B*b*log(tan(c + d*x)**2 +
1)/(2*d) - B*b*log(tan(c + d*x))/d - B*b/(2*d*tan(c + d*x)**2) + C*a*log(tan(c + d*x)**2 + 1)/(2*d) - C*a*log
(tan(c + d*x))/d - C*a/(2*d*tan(c + d*x)**2) - C*b*x - C*b/(d*tan(c + d*x)), True))

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Giac [B]  time = 1.62126, size = 320, normalized size = 3.68 \begin{align*} \frac{B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \,{\left (B a - C b\right )}{\left (d x + c\right )} + 24 \,{\left (C a + B b\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 24 \,{\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{44 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 44 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - B a}{\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))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="giac")

[Out]

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