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

Optimal. Leaf size=43 \[ \frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \cos ^2(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0281819, antiderivative size = 43, 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 = {3486, 2635, 8} \[ \frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \cos ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac{b \cos ^2(c+d x)}{2 d}+a \int \cos ^2(c+d x) \, dx\\ &=-\frac{b \cos ^2(c+d x)}{2 d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} a \int 1 \, dx\\ &=\frac{a x}{2}-\frac{b \cos ^2(c+d x)}{2 d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0538012, size = 46, normalized size = 1.07 \[ \frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{b \cos ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(c + d*x))/(2*d) - (b*Cos[c + d*x]^2)/(2*d) + (a*Sin[2*(c + d*x)])/(4*d)

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Maple [A]  time = 0.042, size = 41, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2}}+a \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(a+b*tan(d*x+c)),x)

[Out]

1/d*(-1/2*b*cos(d*x+c)^2+a*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))

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Maxima [A]  time = 2.34818, size = 51, normalized size = 1.19 \begin{align*} \frac{{\left (d x + c\right )} a + \frac{a \tan \left (d x + c\right ) - b}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.84076, size = 86, normalized size = 2. \begin{align*} \frac{a d x - b \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/2*(a*d*x - b*cos(d*x + c)^2 + a*cos(d*x + c)*sin(d*x + c))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*(a+b*tan(d*x+c)),x)

[Out]

Integral((a + b*tan(c + d*x))*cos(c + d*x)**2, x)

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Giac [B]  time = 1.35676, size = 197, normalized size = 4.58 \begin{align*} \frac{2 \, a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 2 \, a d x \tan \left (d x\right )^{2} + 2 \, a d x \tan \left (c\right )^{2} - b \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 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} + 4 \, b \tan \left (d x\right ) \tan \left (c\right ) + b \tan \left (c\right )^{2} + 2 \, a \tan \left (d x\right ) + 2 \, a \tan \left (c\right ) - b}{4 \,{\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + d \tan \left (d x\right )^{2} + d \tan \left (c\right )^{2} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

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