3.992 \(\int \sec ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx\)

Optimal. Leaf size=43 \[ \frac{a^3 (a A+a B \sin (c+d x))^2}{2 d (A+B) (a-a \sin (c+d x))^2} \]

[Out]

(a^3*(a*A + a*B*Sin[c + d*x])^2)/(2*(A + B)*d*(a - a*Sin[c + d*x])^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0817873, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2836, 37} \[ \frac{a^3 (a A+a B \sin (c+d x))^2}{2 d (A+B) (a-a \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

(a^3*(a*A + a*B*Sin[c + d*x])^2)/(2*(A + B)*d*(a - a*Sin[c + d*x])^2)

Rule 2836

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{A+\frac{B x}{a}}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 (a A+a B \sin (c+d x))^2}{2 (A+B) d (a-a \sin (c+d x))^2}\\ \end{align*}

Mathematica [A]  time = 0.0440062, size = 37, normalized size = 0.86 \[ \frac{a^3 (A+B \sin (c+d x))^2}{2 d (A+B) (\sin (c+d x)-1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.123, size = 312, normalized size = 7.3 \begin{align*}{\frac{{a}^{3}A \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{B{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{B{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{B{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{3\,{a}^{3}A \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{3}A \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{3}A\sin \left ( dx+c \right ) }{8\,d}}+{\frac{3\,B{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{3}A}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,B{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,B{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3}A\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{a}^{3}A\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{B{a}^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x)

[Out]

1/4/d*a^3*A*sin(d*x+c)^4/cos(d*x+c)^4+1/4/d*B*a^3*sin(d*x+c)^5/cos(d*x+c)^4-1/8/d*B*a^3*sin(d*x+c)^5/cos(d*x+c
)^2-1/8/d*B*a^3*sin(d*x+c)^3+3/4/d*a^3*A*sin(d*x+c)^3/cos(d*x+c)^4+3/8/d*a^3*A*sin(d*x+c)^3/cos(d*x+c)^2+3/8/d
*a^3*A*sin(d*x+c)+3/4/d*B*a^3*sin(d*x+c)^4/cos(d*x+c)^4+3/4/d*a^3*A/cos(d*x+c)^4+3/4/d*B*a^3*sin(d*x+c)^3/cos(
d*x+c)^4+3/8/d*B*a^3*sin(d*x+c)^3/cos(d*x+c)^2+1/4/d*a^3*A*tan(d*x+c)*sec(d*x+c)^3+3/8/d*a^3*A*sec(d*x+c)*tan(
d*x+c)+1/4/d*B*a^3/cos(d*x+c)^4

________________________________________________________________________________________

Maxima [A]  time = 1.03375, size = 63, normalized size = 1.47 \begin{align*} \frac{2 \, B a^{3} \sin \left (d x + c\right ) +{\left (A - B\right )} a^{3}}{2 \,{\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.57837, size = 117, normalized size = 2.72 \begin{align*} -\frac{2 \, B a^{3} \sin \left (d x + c\right ) +{\left (A - B\right )} a^{3}}{2 \,{\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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)**5*(a+a*sin(d*x+c))**3*(A+B*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.40641, size = 111, normalized size = 2.58 \begin{align*} \frac{2 \,{\left (A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

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