3.1545 \(\int \frac{\cos ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=202 \[ -\frac{\left (a^2 (-B)+a A b+2 b^2 B\right ) \sin ^3(c+d x)}{3 b^3 d}+\frac{\left (a^2-2 b^2\right ) (A b-a B) \sin ^2(c+d x)}{2 b^4 d}-\frac{\left (a^3 A b+2 a^2 b^2 B+a^4 (-B)-2 a A b^3-b^4 B\right ) \sin (c+d x)}{b^5 d}+\frac{\left (a^2-b^2\right )^2 (A b-a B) \log (a+b \sin (c+d x))}{b^6 d}+\frac{(A b-a B) \sin ^4(c+d x)}{4 b^2 d}+\frac{B \sin ^5(c+d x)}{5 b d} \]

[Out]

((a^2 - b^2)^2*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^6*d) - ((a^3*A*b - 2*a*A*b^3 - a^4*B + 2*a^2*b^2*B - b^
4*B)*Sin[c + d*x])/(b^5*d) + ((a^2 - 2*b^2)*(A*b - a*B)*Sin[c + d*x]^2)/(2*b^4*d) - ((a*A*b - a^2*B + 2*b^2*B)
*Sin[c + d*x]^3)/(3*b^3*d) + ((A*b - a*B)*Sin[c + d*x]^4)/(4*b^2*d) + (B*Sin[c + d*x]^5)/(5*b*d)

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Rubi [A]  time = 0.247878, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2837, 772} \[ -\frac{\left (a^2 (-B)+a A b+2 b^2 B\right ) \sin ^3(c+d x)}{3 b^3 d}+\frac{\left (a^2-2 b^2\right ) (A b-a B) \sin ^2(c+d x)}{2 b^4 d}-\frac{\left (a^3 A b+2 a^2 b^2 B+a^4 (-B)-2 a A b^3-b^4 B\right ) \sin (c+d x)}{b^5 d}+\frac{\left (a^2-b^2\right )^2 (A b-a B) \log (a+b \sin (c+d x))}{b^6 d}+\frac{(A b-a B) \sin ^4(c+d x)}{4 b^2 d}+\frac{B \sin ^5(c+d x)}{5 b d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]

[Out]

((a^2 - b^2)^2*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^6*d) - ((a^3*A*b - 2*a*A*b^3 - a^4*B + 2*a^2*b^2*B - b^
4*B)*Sin[c + d*x])/(b^5*d) + ((a^2 - 2*b^2)*(A*b - a*B)*Sin[c + d*x]^2)/(2*b^4*d) - ((a*A*b - a^2*B + 2*b^2*B)
*Sin[c + d*x]^3)/(3*b^3*d) + ((A*b - a*B)*Sin[c + d*x]^4)/(4*b^2*d) + (B*Sin[c + d*x]^5)/(5*b*d)

Rule 2837

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*(c + (d*x)/b)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (A+\frac{B x}{b}\right ) \left (b^2-x^2\right )^2}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{-a^3 A b+2 a A b^3+a^4 B-2 a^2 b^2 B+b^4 B}{b}-\frac{\left (-a^2+2 b^2\right ) (A b-a B) x}{b}-\frac{\left (a A b-a^2 B+2 b^2 B\right ) x^2}{b}+\frac{(A b-a B) x^3}{b}+\frac{B x^4}{b}+\frac{\left (-a^2+b^2\right )^2 (A b-a B)}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\left (a^2-b^2\right )^2 (A b-a B) \log (a+b \sin (c+d x))}{b^6 d}-\frac{\left (a^3 A b-2 a A b^3-a^4 B+2 a^2 b^2 B-b^4 B\right ) \sin (c+d x)}{b^5 d}+\frac{\left (a^2-2 b^2\right ) (A b-a B) \sin ^2(c+d x)}{2 b^4 d}-\frac{\left (a A b-a^2 B+2 b^2 B\right ) \sin ^3(c+d x)}{3 b^3 d}+\frac{(A b-a B) \sin ^4(c+d x)}{4 b^2 d}+\frac{B \sin ^5(c+d x)}{5 b d}\\ \end{align*}

Mathematica [A]  time = 0.450887, size = 148, normalized size = 0.73 \[ \frac{20 (A b-a B) \left (6 b^2 \left (a^2-b^2\right ) \sin ^2(c+d x)-12 a b \left (a^2-2 b^2\right ) \sin (c+d x)+12 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))-4 a b^3 \sin ^3(c+d x)+3 b^4 \cos ^4(c+d x)\right )+b^5 B (150 \sin (c+d x)+25 \sin (3 (c+d x))+3 \sin (5 (c+d x)))}{240 b^6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]

[Out]

(20*(A*b - a*B)*(3*b^4*Cos[c + d*x]^4 + 12*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]] - 12*a*b*(a^2 - 2*b^2)*Sin[c
+ d*x] + 6*b^2*(a^2 - b^2)*Sin[c + d*x]^2 - 4*a*b^3*Sin[c + d*x]^3) + b^5*B*(150*Sin[c + d*x] + 25*Sin[3*(c +
d*x)] + 3*Sin[5*(c + d*x)]))/(240*b^6*d)

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Maple [B]  time = 0.072, size = 397, normalized size = 2. \begin{align*}{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,bd}}+{\frac{A \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,bd}}-{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{4}a}{4\,{b}^{2}d}}-{\frac{A \left ( \sin \left ( dx+c \right ) \right ) ^{3}a}{3\,{b}^{2}d}}+{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{3}{a}^{2}}{3\,d{b}^{3}}}-{\frac{2\,B \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,bd}}+{\frac{A \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{2\,d{b}^{3}}}-{\frac{A \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{bd}}-{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{3}}{2\,d{b}^{4}}}+{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{{b}^{2}d}}-{\frac{{a}^{3}A\sin \left ( dx+c \right ) }{d{b}^{4}}}+2\,{\frac{A\sin \left ( dx+c \right ) a}{{b}^{2}d}}+{\frac{B{a}^{4}\sin \left ( dx+c \right ) }{d{b}^{5}}}-2\,{\frac{B{a}^{2}\sin \left ( dx+c \right ) }{d{b}^{3}}}+{\frac{B\sin \left ( dx+c \right ) }{bd}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) A{a}^{4}}{d{b}^{5}}}-2\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) A{a}^{2}}{d{b}^{3}}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) A}{bd}}-{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) B{a}^{5}}{d{b}^{6}}}+2\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) B{a}^{3}}{d{b}^{4}}}-{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) Ba}{{b}^{2}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c)),x)

[Out]

1/5*B*sin(d*x+c)^5/b/d+1/4/d/b*A*sin(d*x+c)^4-1/4/d/b^2*B*sin(d*x+c)^4*a-1/3/d/b^2*A*sin(d*x+c)^3*a+1/3/d/b^3*
B*sin(d*x+c)^3*a^2-2/3*B*sin(d*x+c)^3/b/d+1/2/d/b^3*A*sin(d*x+c)^2*a^2-1/d/b*A*sin(d*x+c)^2-1/2/d/b^4*B*sin(d*
x+c)^2*a^3+1/d/b^2*B*sin(d*x+c)^2*a-1/d/b^4*A*a^3*sin(d*x+c)+2/d/b^2*A*a*sin(d*x+c)+1/d/b^5*B*a^4*sin(d*x+c)-2
/d/b^3*B*a^2*sin(d*x+c)+B*sin(d*x+c)/b/d+1/d/b^5*ln(a+b*sin(d*x+c))*A*a^4-2/d/b^3*ln(a+b*sin(d*x+c))*A*a^2+1/d
/b*ln(a+b*sin(d*x+c))*A-1/d/b^6*ln(a+b*sin(d*x+c))*B*a^5+2/d/b^4*ln(a+b*sin(d*x+c))*B*a^3-1/d/b^2*ln(a+b*sin(d
*x+c))*B*a

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Maxima [A]  time = 0.997679, size = 297, normalized size = 1.47 \begin{align*} \frac{\frac{12 \, B b^{4} \sin \left (d x + c\right )^{5} - 15 \,{\left (B a b^{3} - A b^{4}\right )} \sin \left (d x + c\right )^{4} + 20 \,{\left (B a^{2} b^{2} - A a b^{3} - 2 \, B b^{4}\right )} \sin \left (d x + c\right )^{3} - 30 \,{\left (B a^{3} b - A a^{2} b^{2} - 2 \, B a b^{3} + 2 \, A b^{4}\right )} \sin \left (d x + c\right )^{2} + 60 \,{\left (B a^{4} - A a^{3} b - 2 \, B a^{2} b^{2} + 2 \, A a b^{3} + B b^{4}\right )} \sin \left (d x + c\right )}{b^{5}} - \frac{60 \,{\left (B a^{5} - A a^{4} b - 2 \, B a^{3} b^{2} + 2 \, A a^{2} b^{3} + B a b^{4} - A b^{5}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{6}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/60*((12*B*b^4*sin(d*x + c)^5 - 15*(B*a*b^3 - A*b^4)*sin(d*x + c)^4 + 20*(B*a^2*b^2 - A*a*b^3 - 2*B*b^4)*sin(
d*x + c)^3 - 30*(B*a^3*b - A*a^2*b^2 - 2*B*a*b^3 + 2*A*b^4)*sin(d*x + c)^2 + 60*(B*a^4 - A*a^3*b - 2*B*a^2*b^2
+ 2*A*a*b^3 + B*b^4)*sin(d*x + c))/b^5 - 60*(B*a^5 - A*a^4*b - 2*B*a^3*b^2 + 2*A*a^2*b^3 + B*a*b^4 - A*b^5)*l
og(b*sin(d*x + c) + a)/b^6)/d

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Fricas [A]  time = 1.69814, size = 498, normalized size = 2.47 \begin{align*} -\frac{15 \,{\left (B a b^{4} - A b^{5}\right )} \cos \left (d x + c\right )^{4} - 30 \,{\left (B a^{3} b^{2} - A a^{2} b^{3} - B a b^{4} + A b^{5}\right )} \cos \left (d x + c\right )^{2} + 60 \,{\left (B a^{5} - A a^{4} b - 2 \, B a^{3} b^{2} + 2 \, A a^{2} b^{3} + B a b^{4} - A b^{5}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \,{\left (3 \, B b^{5} \cos \left (d x + c\right )^{4} + 15 \, B a^{4} b - 15 \, A a^{3} b^{2} - 25 \, B a^{2} b^{3} + 25 \, A a b^{4} + 8 \, B b^{5} -{\left (5 \, B a^{2} b^{3} - 5 \, A a b^{4} - 4 \, B b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, b^{6} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/60*(15*(B*a*b^4 - A*b^5)*cos(d*x + c)^4 - 30*(B*a^3*b^2 - A*a^2*b^3 - B*a*b^4 + A*b^5)*cos(d*x + c)^2 + 60*
(B*a^5 - A*a^4*b - 2*B*a^3*b^2 + 2*A*a^2*b^3 + B*a*b^4 - A*b^5)*log(b*sin(d*x + c) + a) - 4*(3*B*b^5*cos(d*x +
c)^4 + 15*B*a^4*b - 15*A*a^3*b^2 - 25*B*a^2*b^3 + 25*A*a*b^4 + 8*B*b^5 - (5*B*a^2*b^3 - 5*A*a*b^4 - 4*B*b^5)*
cos(d*x + c)^2)*sin(d*x + c))/(b^6*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(cos(d*x+c)**5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.33364, size = 386, normalized size = 1.91 \begin{align*} \frac{\frac{12 \, B b^{4} \sin \left (d x + c\right )^{5} - 15 \, B a b^{3} \sin \left (d x + c\right )^{4} + 15 \, A b^{4} \sin \left (d x + c\right )^{4} + 20 \, B a^{2} b^{2} \sin \left (d x + c\right )^{3} - 20 \, A a b^{3} \sin \left (d x + c\right )^{3} - 40 \, B b^{4} \sin \left (d x + c\right )^{3} - 30 \, B a^{3} b \sin \left (d x + c\right )^{2} + 30 \, A a^{2} b^{2} \sin \left (d x + c\right )^{2} + 60 \, B a b^{3} \sin \left (d x + c\right )^{2} - 60 \, A b^{4} \sin \left (d x + c\right )^{2} + 60 \, B a^{4} \sin \left (d x + c\right ) - 60 \, A a^{3} b \sin \left (d x + c\right ) - 120 \, B a^{2} b^{2} \sin \left (d x + c\right ) + 120 \, A a b^{3} \sin \left (d x + c\right ) + 60 \, B b^{4} \sin \left (d x + c\right )}{b^{5}} - \frac{60 \,{\left (B a^{5} - A a^{4} b - 2 \, B a^{3} b^{2} + 2 \, A a^{2} b^{3} + B a b^{4} - A b^{5}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{6}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/60*((12*B*b^4*sin(d*x + c)^5 - 15*B*a*b^3*sin(d*x + c)^4 + 15*A*b^4*sin(d*x + c)^4 + 20*B*a^2*b^2*sin(d*x +
c)^3 - 20*A*a*b^3*sin(d*x + c)^3 - 40*B*b^4*sin(d*x + c)^3 - 30*B*a^3*b*sin(d*x + c)^2 + 30*A*a^2*b^2*sin(d*x
+ c)^2 + 60*B*a*b^3*sin(d*x + c)^2 - 60*A*b^4*sin(d*x + c)^2 + 60*B*a^4*sin(d*x + c) - 60*A*a^3*b*sin(d*x + c)
- 120*B*a^2*b^2*sin(d*x + c) + 120*A*a*b^3*sin(d*x + c) + 60*B*b^4*sin(d*x + c))/b^5 - 60*(B*a^5 - A*a^4*b -
2*B*a^3*b^2 + 2*A*a^2*b^3 + B*a*b^4 - A*b^5)*log(abs(b*sin(d*x + c) + a))/b^6)/d