3.418 \(\int \cot (c+d x) (a+b \tan (c+d x)) \, dx\)

Optimal. Leaf size=16 \[ \frac{a \log (\sin (c+d x))}{d}+b x \]

[Out]

b*x + (a*Log[Sin[c + d*x]])/d

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Rubi [A]  time = 0.019953, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3531, 3475} \[ \frac{a \log (\sin (c+d x))}{d}+b x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*(a + b*Tan[c + d*x]),x]

[Out]

b*x + (a*Log[Sin[c + d*x]])/d

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

Mathematica [A]  time = 0.0254985, size = 24, normalized size = 1.5 \[ \frac{a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+b x \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x]),x]

[Out]

b*x + (a*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d

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Maple [A]  time = 0.03, size = 23, normalized size = 1.4 \begin{align*} bx+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{\frac{bc}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)*(a+b*tan(d*x+c)),x)

[Out]

b*x+a*ln(sin(d*x+c))/d+1/d*b*c

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Maxima [B]  time = 1.73076, size = 51, normalized size = 3.19 \begin{align*} \frac{2 \,{\left (d x + c\right )} b - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, a \log \left (\tan \left (d x + c\right )\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.68923, size = 84, normalized size = 5.25 \begin{align*} \frac{2 \, b d x + a \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.378081, size = 42, normalized size = 2.62 \begin{align*} \begin{cases} - \frac{a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + b x & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right ) \cot{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+b*tan(d*x+c)),x)

[Out]

Piecewise((-a*log(tan(c + d*x)**2 + 1)/(2*d) + a*log(tan(c + d*x))/d + b*x, Ne(d, 0)), (x*(a + b*tan(c))*cot(c
), True))

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Giac [B]  time = 1.33313, size = 57, normalized size = 3.56 \begin{align*} \frac{{\left (d x + c\right )} b - a \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) + a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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