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

Optimal. Leaf size=44 \[ \frac{a \tan (c+d x)}{d}-a x+\frac{b \tan ^2(c+d x)}{2 d}+\frac{b \log (\cos (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0379417, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3528, 3525, 3475} \[ \frac{a \tan (c+d x)}{d}-a x+\frac{b \tan ^2(c+d x)}{2 d}+\frac{b \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3528

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*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] && GtQ[m, 0]

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

Mathematica [A]  time = 0.10126, size = 51, normalized size = 1.16 \[ -\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{b \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 57, normalized size = 1.3 \begin{align*}{\frac{b \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\tan \left ( dx+c \right ) }{d}}-{\frac{b\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}-{\frac{a\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)^2*(a+b*tan(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.69948, size = 63, normalized size = 1.43 \begin{align*} \frac{b \tan \left (d x + c\right )^{2} - 2 \,{\left (d x + c\right )} a - b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, a \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)^2*(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.73999, size = 119, normalized size = 2.7 \begin{align*} -\frac{2 \, a d x - b \tan \left (d x + c\right )^{2} - b \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, a \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)^2*(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.25122, size = 56, normalized size = 1.27 \begin{align*} \begin{cases} - a x + \frac{a \tan{\left (c + d x \right )}}{d} - \frac{b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right ) \tan ^{2}{\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)**2*(a+b*tan(d*x+c)),x)

[Out]

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

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Giac [B]  time = 1.66851, size = 441, normalized size = 10.02 \begin{align*} -\frac{2 \, a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - b \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 )^{2} \tan \left (c\right )^{2} - 4 \, a d x \tan \left (d x\right ) \tan \left (c\right ) - b \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 2 \, b \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 \, a \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, a \tan \left (d x\right ) \tan \left (c\right )^{2} + 2 \, a d x - b \tan \left (d x\right )^{2} - b \tan \left (c\right )^{2} - b \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 \, a \tan \left (d x\right ) - 2 \, a \tan \left (c\right ) - b}{2 \,{\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, 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)^2*(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

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