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*ln(sin(d*x+c))/d

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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]

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*}

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Mathematica [A]  time = 0.03, size = 24, normalized size = 1.50 \[ \frac {a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+b x \]

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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fricas [B]  time = 0.48, size = 35, normalized size = 2.19 \[ \frac {2 \, b d x + a \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \]

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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giac [B]  time = 0.59, size = 42, normalized size = 2.62 \[ \frac {{\left (d x + c\right )} b - a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{d} \]

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

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maple [A]  time = 0.29, size = 23, normalized size = 1.44 \[ b x +\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {b c}{d} \]

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+b*c/d

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maxima [B]  time = 0.87, size = 38, normalized size = 2.38 \[ \frac {2 \, {\left (d x + c\right )} b - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, a \log \left (\tan \left (d x + c\right )\right )}{2 \, d} \]

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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mupad [B]  time = 4.15, size = 79, normalized size = 4.94 \[ \frac {a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d}-\frac {a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d}-\frac {b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d}+\frac {b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.29, size = 42, normalized size = 2.62 \[ \begin {cases} - \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + b x & \text {for}\: d \neq 0 \\x \left (a + b \tan {\relax (c )}\right ) \cot {\relax (c )} & \text {otherwise} \end {cases} \]

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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