3.536 \(\int \frac{\cos ^4(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=65 \[ \frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d} \]

[Out]

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

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Rubi [A]  time = 0.0891006, antiderivative size = 65, 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 = {2836, 12, 75} \[ \frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

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

Mathematica [A]  time = 0.0497562, size = 49, normalized size = 0.75 \[ \frac{2 \sin ^3(c+d x)-3 \sin ^2(c+d x)-6 \sin (c+d x)+6 \log (\sin (c+d x))-2}{6 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

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

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Maple [A]  time = 0.089, size = 62, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}-{\frac{\sin \left ( dx+c \right ) }{da}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,da}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.1146, size = 69, normalized size = 1.06 \begin{align*} \frac{\frac{2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right )}{a} + \frac{6 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.31532, size = 127, normalized size = 1.95 \begin{align*} \frac{3 \, \cos \left (d x + c\right )^{2} - 2 \,{\left (\cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) + 6 \, \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right )}{6 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.21895, size = 82, normalized size = 1.26 \begin{align*} \frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac{2 \, a^{2} \sin \left (d x + c\right )^{3} - 3 \, a^{2} \sin \left (d x + c\right )^{2} - 6 \, a^{2} \sin \left (d x + c\right )}{a^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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