3.429 \(\int \cot (c+d x) (a+b \tan (c+d x))^2 \, dx\)

Optimal. Leaf size=35 \[ \frac{a^2 \log (\sin (c+d x))}{d}+2 a b x-\frac{b^2 \log (\cos (c+d x))}{d} \]

[Out]

2*a*b*x - (b^2*Log[Cos[c + d*x]])/d + (a^2*Log[Sin[c + d*x]])/d

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Rubi [A]  time = 0.0367422, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3541, 3475} \[ \frac{a^2 \log (\sin (c+d x))}{d}+2 a b x-\frac{b^2 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*b*x - (b^2*Log[Cos[c + d*x]])/d + (a^2*Log[Sin[c + d*x]])/d

Rule 3541

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d*(2
*b*c - a*d)*x)/b^2, x] + (Dist[d^2/b, Int[Tan[e + f*x], x], x] + Dist[(b*c - a*d)^2/b^2, Int[1/(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]

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

Mathematica [A]  time = 0.0600533, size = 43, normalized size = 1.23 \[ \frac{a^2 (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+2 a b x-\frac{b^2 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 44, normalized size = 1.3 \begin{align*} 2\,abx-{\frac{{b}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{abc}{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))^2,x)

[Out]

2*a*b*x-b^2*ln(cos(d*x+c))/d+a^2*ln(sin(d*x+c))/d+2/d*a*b*c

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Maxima [A]  time = 1.54114, size = 66, normalized size = 1.89 \begin{align*} \frac{4 \,{\left (d x + c\right )} a b + 2 \, a^{2} \log \left (\tan \left (d x + c\right )\right ) -{\left (a^{2} - b^{2}\right )} \log \left (\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))^2,x, algorithm="maxima")

[Out]

1/2*(4*(d*x + c)*a*b + 2*a^2*log(tan(d*x + c)) - (a^2 - b^2)*log(tan(d*x + c)^2 + 1))/d

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Fricas [A]  time = 1.71908, size = 135, normalized size = 3.86 \begin{align*} \frac{4 \, a b d x + a^{2} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - b^{2} \log \left (\frac{1}{\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))^2,x, algorithm="fricas")

[Out]

1/2*(4*a*b*d*x + a^2*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - b^2*log(1/(tan(d*x + c)^2 + 1)))/d

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Sympy [A]  time = 0.690321, size = 70, normalized size = 2. \begin{align*} \begin{cases} - \frac{a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{2} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 2 a b x + \frac{b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right )^{2} \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))**2,x)

[Out]

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

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Giac [A]  time = 1.47952, size = 68, normalized size = 1.94 \begin{align*} \frac{4 \,{\left (d x + c\right )} a b + 2 \, a^{2} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) -{\left (a^{2} - b^{2}\right )} \log \left (\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))^2,x, algorithm="giac")

[Out]

1/2*(4*(d*x + c)*a*b + 2*a^2*log(abs(tan(d*x + c))) - (a^2 - b^2)*log(tan(d*x + c)^2 + 1))/d