3.1297 \(\int \frac{\cos ^3(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=89 \[ -\frac{\left (a^2-b^2\right ) \sin (c+d x)}{b^3 d}+\frac{a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^4 d}+\frac{a \sin ^2(c+d x)}{2 b^2 d}-\frac{\sin ^3(c+d x)}{3 b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.114042, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2837, 12, 772} \[ -\frac{\left (a^2-b^2\right ) \sin (c+d x)}{b^3 d}+\frac{a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^4 d}+\frac{a \sin ^2(c+d x)}{2 b^2 d}-\frac{\sin ^3(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^4*d) - ((a^2 - b^2)*Sin[c + d*x])/(b^3*d) + (a*Sin[c + d*x]^2)/(2*b
^2*d) - Sin[c + d*x]^3/(3*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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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) \sin (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x \left (b^2-x^2\right )}{b (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x \left (b^2-x^2\right )}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a^2 \left (1-\frac{b^2}{a^2}\right )+a x-x^2+\frac{a^3-a b^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac{a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac{\left (a^2-b^2\right ) \sin (c+d x)}{b^3 d}+\frac{a \sin ^2(c+d x)}{2 b^2 d}-\frac{\sin ^3(c+d x)}{3 b d}\\ \end{align*}

Mathematica [A]  time = 0.200605, size = 79, normalized size = 0.89 \[ \frac{6 b \left (b^2-a^2\right ) \sin (c+d x)+6 a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))+3 a b^2 \sin ^2(c+d x)-2 b^3 \sin ^3(c+d x)}{6 b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 106, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,bd}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{2\,{b}^{2}d}}-{\frac{{a}^{2}\sin \left ( dx+c \right ) }{{b}^{3}d}}+{\frac{\sin \left ( dx+c \right ) }{bd}}+{\frac{{a}^{3}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{4}d}}-{\frac{a\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{2}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.98536, size = 107, normalized size = 1.2 \begin{align*} -\frac{\frac{2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \,{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}{b^{3}} - \frac{6 \,{\left (a^{3} - a b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{4}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*((2*b^2*sin(d*x + c)^3 - 3*a*b*sin(d*x + c)^2 + 6*(a^2 - b^2)*sin(d*x + c))/b^3 - 6*(a^3 - a*b^2)*log(b*s
in(d*x + c) + a)/b^4)/d

________________________________________________________________________________________

Fricas [A]  time = 1.49913, size = 185, normalized size = 2.08 \begin{align*} -\frac{3 \, a b^{2} \cos \left (d x + c\right )^{2} - 6 \,{\left (a^{3} - a b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \,{\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )}{6 \, b^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a*b^2*cos(d*x + c)^2 - 6*(a^3 - a*b^2)*log(b*sin(d*x + c) + a) - 2*(b^3*cos(d*x + c)^2 - 3*a^2*b + 2*b
^3)*sin(d*x + c))/(b^4*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(cos(d*x+c)**3*sin(d*x+c)/(a+b*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.17481, size = 115, normalized size = 1.29 \begin{align*} -\frac{\frac{2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \, a^{2} \sin \left (d x + c\right ) - 6 \, b^{2} \sin \left (d x + c\right )}{b^{3}} - \frac{6 \,{\left (a^{3} - a b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{4}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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