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

Optimal. Leaf size=29 \[ -\frac{a \log (\cos (c+d x))}{d}+\frac{b \tan (c+d x)}{d}-b x \]

[Out]

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

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Rubi [A]  time = 0.0161197, antiderivative size = 29, 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 = {3525, 3475} \[ -\frac{a \log (\cos (c+d x))}{d}+\frac{b \tan (c+d x)}{d}-b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3525

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[(b*d*Tan[e + f*x])/f, x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*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 \tan (c+d x) (a+b \tan (c+d x)) \, dx &=-b x+\frac{b \tan (c+d x)}{d}+a \int \tan (c+d x) \, dx\\ &=-b x-\frac{a \log (\cos (c+d x))}{d}+\frac{b \tan (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0190083, size = 38, normalized size = 1.31 \[ -\frac{a \log (\cos (c+d x))}{d}-\frac{b \tan ^{-1}(\tan (c+d x))}{d}+\frac{b \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 43, normalized size = 1.5 \begin{align*}{\frac{b\tan \left ( dx+c \right ) }{d}}+{\frac{a\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}-{\frac{b\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.71512, size = 50, normalized size = 1.72 \begin{align*} -\frac{2 \,{\left (d x + c\right )} b - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, b \tan \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(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*b*tan(d*x + c))/d

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Fricas [A]  time = 1.67867, size = 93, normalized size = 3.21 \begin{align*} -\frac{2 \, b d x + a \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, b \tan \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.196688, size = 41, normalized size = 1.41 \begin{align*} \begin{cases} \frac{a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - b x + \frac{b \tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right ) \tan{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.20795, size = 235, normalized size = 8.1 \begin{align*} -\frac{2 \, b d x \tan \left (d x\right ) \tan \left (c\right ) + a \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) - 2 \, b d x - a \log \left (\frac{4 \,{\left (\tan \left (c\right )^{2} + 1\right )}}{\tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1}\right ) + 2 \, b \tan \left (d x\right ) + 2 \, b \tan \left (c\right )}{2 \,{\left (d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*d*x*tan(d*x)*tan(c) + a*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2
*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c) - 2*b*d*x - a*log(4*(tan(c)^2 + 1)/(tan(d*x)^
4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) + 2*b*tan(d*x) +
2*b*tan(c))/(d*tan(d*x)*tan(c) - d)