3.1554 \(\int \frac{\cos ^3(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=113 \[ \frac{\left (a^2-b^2\right ) (A b-a B)}{b^4 d (a+b \sin (c+d x))}+\frac{\left (-3 a^2 B+2 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac{(A b-2 a B) \sin (c+d x)}{b^3 d}-\frac{B \sin ^2(c+d x)}{2 b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.168508, antiderivative size = 113, 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} \[ \frac{\left (a^2-b^2\right ) (A b-a B)}{b^4 d (a+b \sin (c+d x))}+\frac{\left (-3 a^2 B+2 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac{(A b-2 a B) \sin (c+d x)}{b^3 d}-\frac{B \sin ^2(c+d x)}{2 b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]

[Out]

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

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

Mathematica [A]  time = 0.504945, size = 111, normalized size = 0.98 \[ \frac{\frac{B \left (b^2-a^2\right ) \log (a+b \sin (c+d x))}{b}+\left (A-\frac{a B}{b}\right ) \left (\frac{(a-b) (a+b)}{a+b \sin (c+d x)}+2 a \log (a+b \sin (c+d x))-b \sin (c+d x)\right )+a B \sin (c+d x)-\frac{1}{2} b B \sin ^2(c+d x)}{b^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]

[Out]

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

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Maple [A]  time = 0.121, size = 202, normalized size = 1.8 \begin{align*} -{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,{b}^{2}d}}-{\frac{A\sin \left ( dx+c \right ) }{{b}^{2}d}}+2\,{\frac{aB\sin \left ( dx+c \right ) }{d{b}^{3}}}+2\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) Aa}{d{b}^{3}}}-3\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) B{a}^{2}}{d{b}^{4}}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) B}{{b}^{2}d}}+{\frac{{a}^{2}A}{d{b}^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) }}-{\frac{A}{bd \left ( a+b\sin \left ( dx+c \right ) \right ) }}-{\frac{B{a}^{3}}{d{b}^{4} \left ( a+b\sin \left ( dx+c \right ) \right ) }}+{\frac{aB}{{b}^{2}d \left ( a+b\sin \left ( dx+c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 0.976019, size = 159, normalized size = 1.41 \begin{align*} -\frac{\frac{2 \,{\left (B a^{3} - A a^{2} b - B a b^{2} + A b^{3}\right )}}{b^{5} \sin \left (d x + c\right ) + a b^{4}} + \frac{B b \sin \left (d x + c\right )^{2} - 2 \,{\left (2 \, B a - A b\right )} \sin \left (d x + c\right )}{b^{3}} + \frac{2 \,{\left (3 \, B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{4}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.55436, size = 408, normalized size = 3.61 \begin{align*} -\frac{4 \, B a^{3} - 4 \, A a^{2} b - 11 \, B a b^{2} + 8 \, A b^{3} + 2 \,{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (3 \, B a^{3} - 2 \, A a^{2} b - B a b^{2} +{\left (3 \, B a^{2} b - 2 \, A a b^{2} - B b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) -{\left (2 \, B b^{3} \cos \left (d x + c\right )^{2} + 8 \, B a^{2} b - 4 \, A a b^{2} - B b^{3}\right )} \sin \left (d x + c\right )}{4 \,{\left (b^{5} d \sin \left (d x + c\right ) + a b^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/4*(4*B*a^3 - 4*A*a^2*b - 11*B*a*b^2 + 8*A*b^3 + 2*(3*B*a*b^2 - 2*A*b^3)*cos(d*x + c)^2 + 4*(3*B*a^3 - 2*A*a
^2*b - B*a*b^2 + (3*B*a^2*b - 2*A*a*b^2 - B*b^3)*sin(d*x + c))*log(b*sin(d*x + c) + a) - (2*B*b^3*cos(d*x + c)
^2 + 8*B*a^2*b - 4*A*a*b^2 - B*b^3)*sin(d*x + c))/(b^5*d*sin(d*x + c) + a*b^4*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)**3*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.18877, size = 254, normalized size = 2.25 \begin{align*} -\frac{\frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}{\left (B - \frac{2 \,{\left (3 \, B a b - A b^{2}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b}\right )}}{b^{4}} - \frac{2 \,{\left (3 \, B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\frac{{\left | b \sin \left (d x + c\right ) + a \right |}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{2 \,{\left (\frac{B a^{3} b^{2}}{b \sin \left (d x + c\right ) + a} - \frac{A a^{2} b^{3}}{b \sin \left (d x + c\right ) + a} - \frac{B a b^{4}}{b \sin \left (d x + c\right ) + a} + \frac{A b^{5}}{b \sin \left (d x + c\right ) + a}\right )}}{b^{6}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

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