3.989 \(\int \cos (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx\)

Optimal. Leaf size=51 \[ \frac{B (a \sin (c+d x)+a)^5}{5 a^2 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{4 a d} \]

[Out]

((A - B)*(a + a*Sin[c + d*x])^4)/(4*a*d) + (B*(a + a*Sin[c + d*x])^5)/(5*a^2*d)

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Rubi [A]  time = 0.062236, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2833, 43} \[ \frac{B (a \sin (c+d x)+a)^5}{5 a^2 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{4 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A - B)*(a + a*Sin[c + d*x])^4)/(4*a*d) + (B*(a + a*Sin[c + d*x])^5)/(5*a^2*d)

Rule 2833

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

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 \cos (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^3 \left (A+\frac{B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left ((A-B) (a+x)^3+\frac{B (a+x)^4}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{(A-B) (a+a \sin (c+d x))^4}{4 a d}+\frac{B (a+a \sin (c+d x))^5}{5 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0882942, size = 36, normalized size = 0.71 \[ \frac{a^3 (\sin (c+d x)+1)^4 (5 A+4 B \sin (c+d x)-B)}{20 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + Sin[c + d*x])^4*(5*A - B + 4*B*Sin[c + d*x]))/(20*d)

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Maple [B]  time = 0.033, size = 98, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({\frac{B{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ({a}^{3}A+3\,B{a}^{3} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{ \left ( 3\,{a}^{3}A+3\,B{a}^{3} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{ \left ( 3\,{a}^{3}A+B{a}^{3} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2}}+{a}^{3}A\sin \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/5*B*a^3*sin(d*x+c)^5+1/4*(A*a^3+3*B*a^3)*sin(d*x+c)^4+1/3*(3*A*a^3+3*B*a^3)*sin(d*x+c)^3+1/2*(3*A*a^3+B
*a^3)*sin(d*x+c)^2+a^3*A*sin(d*x+c))

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Maxima [A]  time = 1.05471, size = 113, normalized size = 2.22 \begin{align*} \frac{4 \, B a^{3} \sin \left (d x + c\right )^{5} + 5 \,{\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{4} + 20 \,{\left (A + B\right )} a^{3} \sin \left (d x + c\right )^{3} + 10 \,{\left (3 \, A + B\right )} a^{3} \sin \left (d x + c\right )^{2} + 20 \, A a^{3} \sin \left (d x + c\right )}{20 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*B*a^3*sin(d*x + c)^5 + 5*(A + 3*B)*a^3*sin(d*x + c)^4 + 20*(A + B)*a^3*sin(d*x + c)^3 + 10*(3*A + B)*a
^3*sin(d*x + c)^2 + 20*A*a^3*sin(d*x + c))/d

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Fricas [B]  time = 1.68586, size = 224, normalized size = 4.39 \begin{align*} \frac{5 \,{\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} - 40 \,{\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + 4 \,{\left (B a^{3} \cos \left (d x + c\right )^{4} -{\left (5 \, A + 7 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 2 \,{\left (5 \, A + 3 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{20 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(5*(A + 3*B)*a^3*cos(d*x + c)^4 - 40*(A + B)*a^3*cos(d*x + c)^2 + 4*(B*a^3*cos(d*x + c)^4 - (5*A + 7*B)*a
^3*cos(d*x + c)^2 + 2*(5*A + 3*B)*a^3)*sin(d*x + c))/d

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Sympy [A]  time = 2.70239, size = 204, normalized size = 4. \begin{align*} \begin{cases} \frac{A a^{3} \sin ^{3}{\left (c + d x \right )}}{d} - \frac{A a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac{A a^{3} \sin{\left (c + d x \right )}}{d} - \frac{A a^{3} \cos ^{4}{\left (c + d x \right )}}{4 d} - \frac{3 A a^{3} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac{B a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac{B a^{3} \sin ^{3}{\left (c + d x \right )}}{d} - \frac{3 B a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} - \frac{3 B a^{3} \cos ^{4}{\left (c + d x \right )}}{4 d} - \frac{B a^{3} \cos ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (A + B \sin{\left (c \right )}\right ) \left (a \sin{\left (c \right )} + a\right )^{3} \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((A*a**3*sin(c + d*x)**3/d - A*a**3*sin(c + d*x)**2*cos(c + d*x)**2/(2*d) + A*a**3*sin(c + d*x)/d - A
*a**3*cos(c + d*x)**4/(4*d) - 3*A*a**3*cos(c + d*x)**2/(2*d) + B*a**3*sin(c + d*x)**5/(5*d) + B*a**3*sin(c + d
*x)**3/d - 3*B*a**3*sin(c + d*x)**2*cos(c + d*x)**2/(2*d) - 3*B*a**3*cos(c + d*x)**4/(4*d) - B*a**3*cos(c + d*
x)**2/(2*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c) + a)**3*cos(c), True))

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Giac [B]  time = 1.32333, size = 157, normalized size = 3.08 \begin{align*} \frac{4 \, B a^{3} \sin \left (d x + c\right )^{5} + 5 \, A a^{3} \sin \left (d x + c\right )^{4} + 15 \, B a^{3} \sin \left (d x + c\right )^{4} + 20 \, A a^{3} \sin \left (d x + c\right )^{3} + 20 \, B a^{3} \sin \left (d x + c\right )^{3} + 30 \, A a^{3} \sin \left (d x + c\right )^{2} + 10 \, B a^{3} \sin \left (d x + c\right )^{2} + 20 \, A a^{3} \sin \left (d x + c\right )}{20 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*B*a^3*sin(d*x + c)^5 + 5*A*a^3*sin(d*x + c)^4 + 15*B*a^3*sin(d*x + c)^4 + 20*A*a^3*sin(d*x + c)^3 + 20
*B*a^3*sin(d*x + c)^3 + 30*A*a^3*sin(d*x + c)^2 + 10*B*a^3*sin(d*x + c)^2 + 20*A*a^3*sin(d*x + c))/d