3.354 \(\int \cos ^2(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=56 \[ -\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0552297, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2836, 12, 75} \[ -\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]

[Out]

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

Rule 2836

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 + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[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 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

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

Mathematica [A]  time = 0.0290646, size = 56, normalized size = 1. \[ -\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]

[Out]

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

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Maple [A]  time = 0.051, size = 60, normalized size = 1.1 \begin{align*}{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) a}{3\,d}}+{\frac{2\,a\sin \left ( dx+c \right ) }{3\,d}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/d*cos(d*x+c)^2*sin(d*x+c)*a+2/3*a*sin(d*x+c)/d+1/2*a*cos(d*x+c)^2/d+a*ln(sin(d*x+c))/d

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Maxima [A]  time = 1.01705, size = 63, normalized size = 1.12 \begin{align*} -\frac{2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - 6 \, a \log \left (\sin \left (d x + c\right )\right ) - 6 \, a \sin \left (d x + c\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.92589, size = 132, normalized size = 2.36 \begin{align*} \frac{3 \, a \cos \left (d x + c\right )^{2} + 6 \, a \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 2 \,{\left (a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.27741, size = 65, normalized size = 1.16 \begin{align*} -\frac{2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - 6 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 6 \, a \sin \left (d x + c\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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