3.800 \(\int \sec ^3(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx\)

Optimal. Leaf size=33 \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0838112, antiderivative size = 33, 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 = {2838, 2606, 30, 2607} \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0212681, size = 33, normalized size = 1. \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.047, size = 36, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{a}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*sin(d*x+c)*(a+a*sin(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.04008, size = 35, normalized size = 1.06 \begin{align*} \frac{a \tan \left (d x + c\right )^{3} + \frac{a}{\cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/3*(a*tan(d*x + c)^3 + a/cos(d*x + c)^3)/d

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Fricas [A]  time = 1.56659, size = 127, normalized size = 3.85 \begin{align*} \frac{a \cos \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) - 2 \, a}{3 \,{\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 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)^4*sin(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

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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(sec(d*x+c)**4*sin(d*x+c)*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.24004, size = 72, normalized size = 2.18 \begin{align*} \frac{\frac{3 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1} - \frac{3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 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)^4*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*a*tan(1/2*d*x + 1/2*c)^2 + a)/(tan(1/2*d*x + 1/2*c) - 1)^3)/d