3.1544 \(\int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=315 \[ \frac{\left (a^2 (-B)+a A b+3 b^2 B\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac{\left (a^2-3 b^2\right ) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left (a^3 A b+3 a^2 b^2 B+a^4 (-B)-3 a A b^3-3 b^4 B\right ) \sin ^3(c+d x)}{3 b^5 d}-\frac{\left (-3 a^2 b^2+a^4+3 b^4\right ) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac{\left (-3 a^3 A b^3+a^5 A b+3 a^4 b^2 B-3 a^2 b^4 B+a^6 (-B)+3 a A b^5+b^6 B\right ) \sin (c+d x)}{b^7 d}-\frac{\left (a^2-b^2\right )^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}-\frac{(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac{B \sin ^7(c+d x)}{7 b d} \]

[Out]

-(((a^2 - b^2)^3*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^8*d)) + ((a^5*A*b - 3*a^3*A*b^3 + 3*a*A*b^5 - a^6*B +
3*a^4*b^2*B - 3*a^2*b^4*B + b^6*B)*Sin[c + d*x])/(b^7*d) - ((a^4 - 3*a^2*b^2 + 3*b^4)*(A*b - a*B)*Sin[c + d*x
]^2)/(2*b^6*d) + ((a^3*A*b - 3*a*A*b^3 - a^4*B + 3*a^2*b^2*B - 3*b^4*B)*Sin[c + d*x]^3)/(3*b^5*d) - ((a^2 - 3*
b^2)*(A*b - a*B)*Sin[c + d*x]^4)/(4*b^4*d) + ((a*A*b - a^2*B + 3*b^2*B)*Sin[c + d*x]^5)/(5*b^3*d) - ((A*b - a*
B)*Sin[c + d*x]^6)/(6*b^2*d) - (B*Sin[c + d*x]^7)/(7*b*d)

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Rubi [A]  time = 0.356067, antiderivative size = 315, 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} \[ \frac{\left (a^2 (-B)+a A b+3 b^2 B\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac{\left (a^2-3 b^2\right ) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left (a^3 A b+3 a^2 b^2 B+a^4 (-B)-3 a A b^3-3 b^4 B\right ) \sin ^3(c+d x)}{3 b^5 d}-\frac{\left (-3 a^2 b^2+a^4+3 b^4\right ) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac{\left (-3 a^3 A b^3+a^5 A b+3 a^4 b^2 B-3 a^2 b^4 B+a^6 (-B)+3 a A b^5+b^6 B\right ) \sin (c+d x)}{b^7 d}-\frac{\left (a^2-b^2\right )^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}-\frac{(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac{B \sin ^7(c+d x)}{7 b d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]

[Out]

-(((a^2 - b^2)^3*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^8*d)) + ((a^5*A*b - 3*a^3*A*b^3 + 3*a*A*b^5 - a^6*B +
3*a^4*b^2*B - 3*a^2*b^4*B + b^6*B)*Sin[c + d*x])/(b^7*d) - ((a^4 - 3*a^2*b^2 + 3*b^4)*(A*b - a*B)*Sin[c + d*x
]^2)/(2*b^6*d) + ((a^3*A*b - 3*a*A*b^3 - a^4*B + 3*a^2*b^2*B - 3*b^4*B)*Sin[c + d*x]^3)/(3*b^5*d) - ((a^2 - 3*
b^2)*(A*b - a*B)*Sin[c + d*x]^4)/(4*b^4*d) + ((a*A*b - a^2*B + 3*b^2*B)*Sin[c + d*x]^5)/(5*b^3*d) - ((A*b - a*
B)*Sin[c + d*x]^6)/(6*b^2*d) - (B*Sin[c + d*x]^7)/(7*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 ^7(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 )^3}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^5 A b-3 a^3 A b^3+3 a A b^5-a^6 B+3 a^4 b^2 B-3 a^2 b^4 B+b^6 B}{b}-\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) (A b-a B) x}{b}-\frac{\left (-a^3 A b+3 a A b^3+a^4 B-3 a^2 b^2 B+3 b^4 B\right ) x^2}{b}+\frac{\left (-a^2+3 b^2\right ) (A b-a B) x^3}{b}+\frac{\left (a A b-a^2 B+3 b^2 B\right ) x^4}{b}-\frac{(A b-a B) x^5}{b}-\frac{B x^6}{b}+\frac{\left (-a^2+b^2\right )^3 (A b-a B)}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=-\frac{\left (a^2-b^2\right )^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}+\frac{\left (a^5 A b-3 a^3 A b^3+3 a A b^5-a^6 B+3 a^4 b^2 B-3 a^2 b^4 B+b^6 B\right ) \sin (c+d x)}{b^7 d}-\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac{\left (a^3 A b-3 a A b^3-a^4 B+3 a^2 b^2 B-3 b^4 B\right ) \sin ^3(c+d x)}{3 b^5 d}-\frac{\left (a^2-3 b^2\right ) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left (a A b-a^2 B+3 b^2 B\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac{(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac{B \sin ^7(c+d x)}{7 b d}\\ \end{align*}

Mathematica [A]  time = 0.875837, size = 218, normalized size = 0.69 \[ \frac{\frac{(A b-a B) \left (20 a b^3 \left (a^2-3 b^2\right ) \sin ^3(c+d x)-30 b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)+60 a b \left (-3 a^2 b^2+a^4+3 b^4\right ) \sin (c+d x)+15 b^4 \left (b^2-a^2\right ) \cos ^4(c+d x)-60 \left (a^2-b^2\right )^3 \log (a+b \sin (c+d x))+12 a b^5 \sin ^5(c+d x)+10 b^6 \cos ^6(c+d x)\right )}{60 b}+\frac{b^6 B (1225 \sin (c+d x)+245 \sin (3 (c+d x))+49 \sin (5 (c+d x))+5 \sin (7 (c+d x)))}{2240}}{b^7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]

[Out]

(((A*b - a*B)*(15*b^4*(-a^2 + b^2)*Cos[c + d*x]^4 + 10*b^6*Cos[c + d*x]^6 - 60*(a^2 - b^2)^3*Log[a + b*Sin[c +
d*x]] + 60*a*b*(a^4 - 3*a^2*b^2 + 3*b^4)*Sin[c + d*x] - 30*b^2*(a^2 - b^2)^2*Sin[c + d*x]^2 + 20*a*b^3*(a^2 -
3*b^2)*Sin[c + d*x]^3 + 12*a*b^5*Sin[c + d*x]^5))/(60*b) + (b^6*B*(1225*Sin[c + d*x] + 245*Sin[3*(c + d*x)] +
49*Sin[5*(c + d*x)] + 5*Sin[7*(c + d*x)]))/2240)/(b^7*d)

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Maple [B]  time = 0.075, size = 689, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c)),x)

[Out]

3/d/b^2*A*a*sin(d*x+c)-1/d/b^2*A*sin(d*x+c)^3*a-1/d/b^7*B*a^6*sin(d*x+c)+1/5/d/b^2*A*sin(d*x+c)^5*a-1/5/d/b^3*
B*sin(d*x+c)^5*a^2-3/2/d/b^4*B*sin(d*x+c)^2*a^3-3/4/d/b^2*B*sin(d*x+c)^4*a-3/d/b^3*ln(a+b*sin(d*x+c))*A*a^2-3/
d/b^4*A*a^3*sin(d*x+c)+3/4/d/b*A*sin(d*x+c)^4+1/d/b*ln(a+b*sin(d*x+c))*A-3/2/d/b*A*sin(d*x+c)^2-1/6/d/b*A*sin(
d*x+c)^6+3/2/d/b^3*A*sin(d*x+c)^2*a^2-1/d/b^2*ln(a+b*sin(d*x+c))*B*a-1/d/b^7*ln(a+b*sin(d*x+c))*A*a^6+3/d/b^5*
ln(a+b*sin(d*x+c))*A*a^4+3/d/b^5*B*a^4*sin(d*x+c)-3/d/b^3*B*a^2*sin(d*x+c)+1/6/d/b^2*B*sin(d*x+c)^6*a+1/d/b^8*
ln(a+b*sin(d*x+c))*B*a^7+1/3/d/b^4*A*sin(d*x+c)^3*a^3+3/d/b^4*ln(a+b*sin(d*x+c))*B*a^3+1/d/b^6*A*a^5*sin(d*x+c
)+1/2/d/b^6*B*sin(d*x+c)^2*a^5-1/4/d/b^3*A*sin(d*x+c)^4*a^2-1/2/d/b^5*A*sin(d*x+c)^2*a^4+1/d/b^3*B*sin(d*x+c)^
3*a^2-3/d/b^6*ln(a+b*sin(d*x+c))*B*a^5-1/3/d/b^5*B*sin(d*x+c)^3*a^4+1/4/d/b^4*B*sin(d*x+c)^4*a^3+3/2/d/b^2*B*s
in(d*x+c)^2*a+B*sin(d*x+c)/b/d-B*sin(d*x+c)^3/b/d-1/7*B*sin(d*x+c)^7/b/d+3/5*B*sin(d*x+c)^5/b/d

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Maxima [A]  time = 1.01263, size = 494, normalized size = 1.57 \begin{align*} -\frac{\frac{60 \, B b^{6} \sin \left (d x + c\right )^{7} - 70 \,{\left (B a b^{5} - A b^{6}\right )} \sin \left (d x + c\right )^{6} + 84 \,{\left (B a^{2} b^{4} - A a b^{5} - 3 \, B b^{6}\right )} \sin \left (d x + c\right )^{5} - 105 \,{\left (B a^{3} b^{3} - A a^{2} b^{4} - 3 \, B a b^{5} + 3 \, A b^{6}\right )} \sin \left (d x + c\right )^{4} + 140 \,{\left (B a^{4} b^{2} - A a^{3} b^{3} - 3 \, B a^{2} b^{4} + 3 \, A a b^{5} + 3 \, B b^{6}\right )} \sin \left (d x + c\right )^{3} - 210 \,{\left (B a^{5} b - A a^{4} b^{2} - 3 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4} + 3 \, B a b^{5} - 3 \, A b^{6}\right )} \sin \left (d x + c\right )^{2} + 420 \,{\left (B a^{6} - A a^{5} b - 3 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3} + 3 \, B a^{2} b^{4} - 3 \, A a b^{5} - B b^{6}\right )} \sin \left (d x + c\right )}{b^{7}} - \frac{420 \,{\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/420*((60*B*b^6*sin(d*x + c)^7 - 70*(B*a*b^5 - A*b^6)*sin(d*x + c)^6 + 84*(B*a^2*b^4 - A*a*b^5 - 3*B*b^6)*si
n(d*x + c)^5 - 105*(B*a^3*b^3 - A*a^2*b^4 - 3*B*a*b^5 + 3*A*b^6)*sin(d*x + c)^4 + 140*(B*a^4*b^2 - A*a^3*b^3 -
3*B*a^2*b^4 + 3*A*a*b^5 + 3*B*b^6)*sin(d*x + c)^3 - 210*(B*a^5*b - A*a^4*b^2 - 3*B*a^3*b^3 + 3*A*a^2*b^4 + 3*
B*a*b^5 - 3*A*b^6)*sin(d*x + c)^2 + 420*(B*a^6 - A*a^5*b - 3*B*a^4*b^2 + 3*A*a^3*b^3 + 3*B*a^2*b^4 - 3*A*a*b^5
- B*b^6)*sin(d*x + c))/b^7 - 420*(B*a^7 - A*a^6*b - 3*B*a^5*b^2 + 3*A*a^4*b^3 + 3*B*a^3*b^4 - 3*A*a^2*b^5 - B
*a*b^6 + A*b^7)*log(b*sin(d*x + c) + a)/b^8)/d

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Fricas [A]  time = 1.90702, size = 833, normalized size = 2.64 \begin{align*} -\frac{70 \,{\left (B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{6} - 105 \,{\left (B a^{3} b^{4} - A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \cos \left (d x + c\right )^{4} + 210 \,{\left (B a^{5} b^{2} - A a^{4} b^{3} - 2 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5} + B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{2} - 420 \,{\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \,{\left (15 \, B b^{7} \cos \left (d x + c\right )^{6} - 105 \, B a^{6} b + 105 \, A a^{5} b^{2} + 280 \, B a^{4} b^{3} - 280 \, A a^{3} b^{4} - 231 \, B a^{2} b^{5} + 231 \, A a b^{6} + 48 \, B b^{7} - 3 \,{\left (7 \, B a^{2} b^{5} - 7 \, A a b^{6} - 6 \, B b^{7}\right )} \cos \left (d x + c\right )^{4} +{\left (35 \, B a^{4} b^{3} - 35 \, A a^{3} b^{4} - 63 \, B a^{2} b^{5} + 63 \, A a b^{6} + 24 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{420 \, b^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/420*(70*(B*a*b^6 - A*b^7)*cos(d*x + c)^6 - 105*(B*a^3*b^4 - A*a^2*b^5 - B*a*b^6 + A*b^7)*cos(d*x + c)^4 + 2
10*(B*a^5*b^2 - A*a^4*b^3 - 2*B*a^3*b^4 + 2*A*a^2*b^5 + B*a*b^6 - A*b^7)*cos(d*x + c)^2 - 420*(B*a^7 - A*a^6*b
- 3*B*a^5*b^2 + 3*A*a^4*b^3 + 3*B*a^3*b^4 - 3*A*a^2*b^5 - B*a*b^6 + A*b^7)*log(b*sin(d*x + c) + a) - 4*(15*B*
b^7*cos(d*x + c)^6 - 105*B*a^6*b + 105*A*a^5*b^2 + 280*B*a^4*b^3 - 280*A*a^3*b^4 - 231*B*a^2*b^5 + 231*A*a*b^6
+ 48*B*b^7 - 3*(7*B*a^2*b^5 - 7*A*a*b^6 - 6*B*b^7)*cos(d*x + c)^4 + (35*B*a^4*b^3 - 35*A*a^3*b^4 - 63*B*a^2*b
^5 + 63*A*a*b^6 + 24*B*b^7)*cos(d*x + c)^2)*sin(d*x + c))/(b^8*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)**7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.30407, size = 690, normalized size = 2.19 \begin{align*} -\frac{\frac{60 \, B b^{6} \sin \left (d x + c\right )^{7} - 70 \, B a b^{5} \sin \left (d x + c\right )^{6} + 70 \, A b^{6} \sin \left (d x + c\right )^{6} + 84 \, B a^{2} b^{4} \sin \left (d x + c\right )^{5} - 84 \, A a b^{5} \sin \left (d x + c\right )^{5} - 252 \, B b^{6} \sin \left (d x + c\right )^{5} - 105 \, B a^{3} b^{3} \sin \left (d x + c\right )^{4} + 105 \, A a^{2} b^{4} \sin \left (d x + c\right )^{4} + 315 \, B a b^{5} \sin \left (d x + c\right )^{4} - 315 \, A b^{6} \sin \left (d x + c\right )^{4} + 140 \, B a^{4} b^{2} \sin \left (d x + c\right )^{3} - 140 \, A a^{3} b^{3} \sin \left (d x + c\right )^{3} - 420 \, B a^{2} b^{4} \sin \left (d x + c\right )^{3} + 420 \, A a b^{5} \sin \left (d x + c\right )^{3} + 420 \, B b^{6} \sin \left (d x + c\right )^{3} - 210 \, B a^{5} b \sin \left (d x + c\right )^{2} + 210 \, A a^{4} b^{2} \sin \left (d x + c\right )^{2} + 630 \, B a^{3} b^{3} \sin \left (d x + c\right )^{2} - 630 \, A a^{2} b^{4} \sin \left (d x + c\right )^{2} - 630 \, B a b^{5} \sin \left (d x + c\right )^{2} + 630 \, A b^{6} \sin \left (d x + c\right )^{2} + 420 \, B a^{6} \sin \left (d x + c\right ) - 420 \, A a^{5} b \sin \left (d x + c\right ) - 1260 \, B a^{4} b^{2} \sin \left (d x + c\right ) + 1260 \, A a^{3} b^{3} \sin \left (d x + c\right ) + 1260 \, B a^{2} b^{4} \sin \left (d x + c\right ) - 1260 \, A a b^{5} \sin \left (d x + c\right ) - 420 \, B b^{6} \sin \left (d x + c\right )}{b^{7}} - \frac{420 \,{\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/420*((60*B*b^6*sin(d*x + c)^7 - 70*B*a*b^5*sin(d*x + c)^6 + 70*A*b^6*sin(d*x + c)^6 + 84*B*a^2*b^4*sin(d*x
+ c)^5 - 84*A*a*b^5*sin(d*x + c)^5 - 252*B*b^6*sin(d*x + c)^5 - 105*B*a^3*b^3*sin(d*x + c)^4 + 105*A*a^2*b^4*s
in(d*x + c)^4 + 315*B*a*b^5*sin(d*x + c)^4 - 315*A*b^6*sin(d*x + c)^4 + 140*B*a^4*b^2*sin(d*x + c)^3 - 140*A*a
^3*b^3*sin(d*x + c)^3 - 420*B*a^2*b^4*sin(d*x + c)^3 + 420*A*a*b^5*sin(d*x + c)^3 + 420*B*b^6*sin(d*x + c)^3 -
210*B*a^5*b*sin(d*x + c)^2 + 210*A*a^4*b^2*sin(d*x + c)^2 + 630*B*a^3*b^3*sin(d*x + c)^2 - 630*A*a^2*b^4*sin(
d*x + c)^2 - 630*B*a*b^5*sin(d*x + c)^2 + 630*A*b^6*sin(d*x + c)^2 + 420*B*a^6*sin(d*x + c) - 420*A*a^5*b*sin(
d*x + c) - 1260*B*a^4*b^2*sin(d*x + c) + 1260*A*a^3*b^3*sin(d*x + c) + 1260*B*a^2*b^4*sin(d*x + c) - 1260*A*a*
b^5*sin(d*x + c) - 420*B*b^6*sin(d*x + c))/b^7 - 420*(B*a^7 - A*a^6*b - 3*B*a^5*b^2 + 3*A*a^4*b^3 + 3*B*a^3*b^
4 - 3*A*a^2*b^5 - B*a*b^6 + A*b^7)*log(abs(b*sin(d*x + c) + a))/b^8)/d