3.60 \(\int \frac{\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=37 \[ \frac{\sin ^2(c+d x)}{2 a d}+\frac{\cos ^3(c+d x)}{3 a d} \]

[Out]

Cos[c + d*x]^3/(3*a*d) + Sin[c + d*x]^2/(2*a*d)

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Rubi [A]  time = 0.126194, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3872, 2835, 2564, 30, 2565} \[ \frac{\sin ^2(c+d x)}{2 a d}+\frac{\cos ^3(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^3/(a + a*Sec[c + d*x]),x]

[Out]

Cos[c + d*x]^3/(3*a*d) + Sin[c + d*x]^2/(2*a*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2835

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rule 2564

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

Rule 30

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

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rubi steps

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

Mathematica [A]  time = 0.110726, size = 32, normalized size = 0.86 \[ \frac{2 \sin ^4\left (\frac{1}{2} (c+d x)\right ) (2 \cos (c+d x)+1)}{3 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^3/(a + a*Sec[c + d*x]),x]

[Out]

(2*(1 + 2*Cos[c + d*x])*Sin[(c + d*x)/2]^4)/(3*a*d)

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Maple [A]  time = 0.056, size = 30, normalized size = 0.8 \begin{align*} -{\frac{1}{da} \left ( -{\frac{1}{3\, \left ( \sec \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{2\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^3/(a+a*sec(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0.991526, size = 39, normalized size = 1.05 \begin{align*} \frac{2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2}}{6 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.66242, size = 66, normalized size = 1.78 \begin{align*} \frac{2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2}}{6 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**3/(a+a*sec(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.22407, size = 43, normalized size = 1.16 \begin{align*} \frac{\frac{2 \, \cos \left (d x + c\right )^{3}}{d} - \frac{3 \, \cos \left (d x + c\right )^{2}}{d}}{6 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

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