3.754 \(\int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx\)

Optimal. Leaf size=27 \[ \frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-a x \]

[Out]

-(a*x) + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d

________________________________________________________________________________________

Rubi [A]  time = 0.0465878, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2838, 2606, 8, 3473} \[ \frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-a x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*x) + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d

Rule 2838

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 8

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

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

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

Mathematica [A]  time = 0.0197155, size = 36, normalized size = 1.33 \[ -\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*ArcTan[Tan[c + d*x]])/d) + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d

________________________________________________________________________________________

Maple [A]  time = 0.037, size = 32, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( a \left ( \tan \left ( dx+c \right ) -dx-c \right ) +{\frac{a}{\cos \left ( dx+c \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2*sin(d*x+c)*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(tan(d*x+c)-d*x-c)+a/cos(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 1.65113, size = 43, normalized size = 1.59 \begin{align*} -\frac{{\left (d x + c - \tan \left (d x + c\right )\right )} a - \frac{a}{\cos \left (d x + c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-((d*x + c - tan(d*x + c))*a - a/cos(d*x + c))/d

________________________________________________________________________________________

Fricas [B]  time = 1.06085, size = 143, normalized size = 5.3 \begin{align*} -\frac{a d x +{\left (a d x - a\right )} \cos \left (d x + c\right ) -{\left (a d x + a\right )} \sin \left (d x + c\right ) - a}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-(a*d*x + (a*d*x - a)*cos(d*x + c) - (a*d*x + a)*sin(d*x + c) - a)/(d*cos(d*x + c) - d*sin(d*x + c) + d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2*sin(d*x+c)*(a+a*sin(d*x+c)),x)

[Out]

a*(Integral(sin(c + d*x)*sec(c + d*x)**2, x) + Integral(sin(c + d*x)**2*sec(c + d*x)**2, x))

________________________________________________________________________________________

Giac [A]  time = 1.28639, size = 39, normalized size = 1.44 \begin{align*} -\frac{{\left (d x + c\right )} a + \frac{2 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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