3.775 \(\int \frac{\sec (c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=37 \[ \frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan ^3(c+d x)}{3 a d} \]

[Out]

Sec[c + d*x]^3/(3*a*d) - Tan[c + d*x]^3/(3*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.093388, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2839, 2606, 30, 2607} \[ \frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan ^3(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[c + d*x]^3/(3*a*d) - Tan[c + d*x]^3/(3*a*d)

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2607

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

Rubi steps

\begin{align*} \int \frac{\sec (c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^3(c+d x) \tan (c+d x) \, dx}{a}-\frac{\int \sec ^2(c+d x) \tan ^2(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan ^3(c+d x)}{3 a d}\\ \end{align*}

Mathematica [B]  time = 0.131107, size = 104, normalized size = 2.81 \[ \frac{-2 \sin (c+d x)+\frac{1}{2} \sin (2 (c+d x))+\cos (c+d x)+\cos (2 (c+d x))-3}{6 a d (\sin (c+d x)+1) \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3 + Cos[c + d*x] + Cos[2*(c + d*x)] - 2*Sin[c + d*x] + Sin[2*(c + d*x)]/2)/(6*a*d*(-Cos[(c + d*x)/2] + Sin[(
c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(1 + Sin[c + d*x]))

________________________________________________________________________________________

Maple [B]  time = 0.056, size = 70, normalized size = 1.9 \begin{align*} 4\,{\frac{1}{da} \left ( -1/8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-1}+1/6\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-3}-1/4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}+1/8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-1} \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]

4/d/a*(-1/8/(tan(1/2*d*x+1/2*c)-1)+1/6/(tan(1/2*d*x+1/2*c)+1)^3-1/4/(tan(1/2*d*x+1/2*c)+1)^2+1/8/(tan(1/2*d*x+
1/2*c)+1))

________________________________________________________________________________________

Maxima [B]  time = 1.02291, size = 149, normalized size = 4.03 \begin{align*} \frac{2 \,{\left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{3 \,{\left (a + \frac{2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{2 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\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]

2/3*(2*sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)/((a + 2*a*sin(d*x + c)/(co
s(d*x + c) + 1) - 2*a*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*d)

________________________________________________________________________________________

Fricas [A]  time = 1.0251, size = 126, normalized size = 3.41 \begin{align*} -\frac{\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right ) - 2}{3 \,{\left (a d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )\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, algorithm="fricas")

[Out]

-1/3*(cos(d*x + c)^2 - sin(d*x + c) - 2)/(a*d*cos(d*x + c)*sin(d*x + c) + a*d*cos(d*x + c))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sin{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \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]

Integral(sin(c + d*x)*sec(c + d*x)**2/(sin(c + d*x) + 1), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.23009, size = 77, normalized size = 2.08 \begin{align*} -\frac{\frac{3}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}} - \frac{3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{3}}}{6 \, 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]

-1/6*(3/(a*(tan(1/2*d*x + 1/2*c) - 1)) - (3*tan(1/2*d*x + 1/2*c)^2 + 1)/(a*(tan(1/2*d*x + 1/2*c) + 1)^3))/d