3.5 \(\int \cot ^3(c+d x) (a+b \tan (c+d x)) (B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\)

Optimal. Leaf size=43 \[ \frac{(a C+b B) \log (\sin (c+d x))}{d}+x (-(a B-b C))-\frac{a B \cot (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.123954, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3632, 3591, 3531, 3475} \[ \frac{(a C+b B) \log (\sin (c+d x))}{d}+x (-(a B-b C))-\frac{a B \cot (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [C]  time = 0.161645, size = 78, normalized size = 1.81 \[ -\frac{a B \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(c+d x)\right )}{d}+\frac{a C (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac{b B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+b C x \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

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

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Maple [A]  time = 0.059, size = 65, normalized size = 1.5 \begin{align*} -aBx+Cbx-{\frac{B\cot \left ( dx+c \right ) a}{d}}+{\frac{Bb\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{Bac}{d}}+{\frac{Ca\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{\frac{Cbc}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(a+b*tan(d*x+c))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x)

[Out]

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

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Maxima [A]  time = 1.73373, size = 92, normalized size = 2.14 \begin{align*} -\frac{2 \,{\left (B a - C b\right )}{\left (d x + c\right )} +{\left (C a + B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \,{\left (C a + B b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac{2 \, B a}{\tan \left (d x + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.33783, size = 178, normalized size = 4.14 \begin{align*} -\frac{2 \,{\left (B a - C b\right )} d x \tan \left (d x + c\right ) -{\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 ) + 2 \, B a}{2 \, 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)^3*(a+b*tan(d*x+c))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 15.0302, size = 116, normalized size = 2.7 \begin{align*} \begin{cases} \text{NaN} & \text{for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan{\left (c \right )}\right ) \left (B \tan{\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{3}{\left (c \right )} & \text{for}\: d = 0 \\\text{NaN} & \text{for}\: c = - d x \\- B a x - \frac{B a}{d \tan{\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{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} + C b x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [B]  time = 1.53822, size = 161, normalized size = 3.74 \begin{align*} \frac{B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \,{\left (B a - C b\right )}{\left (d x + c\right )} - 2 \,{\left (C a + B b\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) + 2 \,{\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{2 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, 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 )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="giac")

[Out]

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