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

Optimal. Leaf size=52 \[ -\frac{a^3 \sin ^2(c+d x)}{2 d}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0469152, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 43} \[ -\frac{a^3 \sin ^2(c+d x)}{2 d}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \sec (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{a \operatorname{Subst}\left (\int \frac{(a+x)^2}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (-3 a+\frac{4 a^2}{a-x}-x\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \log (1-\sin (c+d x))}{d}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{a^3 \sin ^2(c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0290341, size = 41, normalized size = 0.79 \[ \frac{a^3 \left (-\frac{1}{2} \sin ^2(c+d x)-3 \sin (c+d x)-4 \log (1-\sin (c+d x))\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.047, size = 69, normalized size = 1.3 \begin{align*} -{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-4\,{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^3*sin(d*x+c)^2/d-4/d*a^3*ln(cos(d*x+c))-3*a^3*sin(d*x+c)/d+4/d*a^3*ln(sec(d*x+c)+tan(d*x+c))

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Maxima [A]  time = 0.96589, size = 58, normalized size = 1.12 \begin{align*} -\frac{a^{3} \sin \left (d x + c\right )^{2} + 8 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, a^{3} \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.65545, size = 108, normalized size = 2.08 \begin{align*} \frac{a^{3} \cos \left (d x + c\right )^{2} - 8 \, a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 6 \, a^{3} \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^3*cos(d*x + c)^2 - 8*a^3*log(-sin(d*x + c) + 1) - 6*a^3*sin(d*x + c))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int 3 \sin{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \sec{\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)*(a+a*sin(d*x+c))**3,x)

[Out]

a**3*(Integral(3*sin(c + d*x)*sec(c + d*x), x) + Integral(3*sin(c + d*x)**2*sec(c + d*x), x) + Integral(sin(c
+ d*x)**3*sec(c + d*x), x) + Integral(sec(c + d*x), x))

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Giac [B]  time = 1.16096, size = 173, normalized size = 3.33 \begin{align*} \frac{2 \,{\left (2 \, a^{3} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 4 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 7 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{3}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*a^3*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 4*a^3*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - (3*a^3*tan(1/2*d*x + 1/2
*c)^4 + 3*a^3*tan(1/2*d*x + 1/2*c)^3 + 7*a^3*tan(1/2*d*x + 1/2*c)^2 + 3*a^3*tan(1/2*d*x + 1/2*c) + 3*a^3)/(tan
(1/2*d*x + 1/2*c)^2 + 1)^2)/d