3.304 \(\int \frac{\cot ^3(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\)

Optimal. Leaf size=30 \[ -\frac{B \cot ^2(c+d x)}{2 d}-\frac{B \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0162874, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {21, 3473, 3475} \[ -\frac{B \cot ^2(c+d x)}{2 d}-\frac{B \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^3*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

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

Mathematica [A]  time = 0.0746224, size = 35, normalized size = 1.17 \[ -\frac{B \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/(2*d)

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Maple [A]  time = 0.048, size = 29, normalized size = 1. \begin{align*} -{\frac{B \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{B\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)^3*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.8192, size = 54, normalized size = 1.8 \begin{align*} \frac{B \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, B \log \left (\tan \left (d x + c\right )\right ) - \frac{B}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73243, size = 131, normalized size = 4.37 \begin{align*} -\frac{{\left (B \cos \left (2 \, d x + 2 \, c\right ) - B\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, d x + 2 \, c\right ) + \frac{1}{2}\right ) - 2 \, B}{2 \,{\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*((B*cos(2*d*x + 2*c) - B)*log(-1/2*cos(2*d*x + 2*c) + 1/2) - 2*B)/(d*cos(2*d*x + 2*c) - d)

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Sympy [A]  time = 27.3717, size = 80, normalized size = 2.67 \begin{align*} \begin{cases} \tilde{\infty } B x & \text{for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\\frac{x \left (B a + B b \tan{\left (c \right )}\right ) \cot ^{3}{\left (c \right )}}{a + b \tan{\left (c \right )}} & \text{for}\: d = 0 \\\frac{B \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{B \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{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)**3*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x)

[Out]

Piecewise((zoo*B*x, (Eq(c, 0) | Eq(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x*(B*a + B*b*tan(c))*cot(c)**3/(a +
b*tan(c)), Eq(d, 0)), (B*log(tan(c + d*x)**2 + 1)/(2*d) - B*log(tan(c + d*x))/d - B/(2*d*tan(c + d*x)**2), Tr
ue))

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Giac [B]  time = 1.40382, size = 167, normalized size = 5.57 \begin{align*} -\frac{4 \, B \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 8 \, B \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{{\left (B + \frac{4 \, B{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1} - \frac{B{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*B*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 8*B*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) +
1) + 1)) - (B + 4*B*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))*(cos(d*x + c) + 1)/(cos(d*x + c) - 1) - B*(cos(d*x
+ c) - 1)/(cos(d*x + c) + 1))/d