3.547 \(\int \frac{\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=47 \[ \frac{\sin ^2(c+d x)}{2 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d} \]

[Out]

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

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Rubi [A]  time = 0.0818374, antiderivative size = 47, 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, 43} \[ \frac{\sin ^2(c+d x)}{2 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]

[Out]

Log[Sin[c + d*x]]/(a^2*d) - (2*Sin[c + d*x])/(a^2*d) + Sin[c + d*x]^2/(2*a^2*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0397337, size = 36, normalized size = 0.77 \[ \frac{\sin ^2(c+d x)-4 \sin (c+d x)+2 \log (\sin (c+d x))}{2 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]

[Out]

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

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Maple [A]  time = 0.094, size = 46, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}-2\,{\frac{\sin \left ( dx+c \right ) }{d{a}^{2}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d{a}^{2}}} \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))^2,x)

[Out]

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

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Maxima [A]  time = 1.01191, size = 53, normalized size = 1.13 \begin{align*} \frac{\frac{\sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{a^{2}} + \frac{2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{2 \, 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.10183, size = 100, normalized size = 2.13 \begin{align*} -\frac{\cos \left (d x + c\right )^{2} - 2 \, \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, \sin \left (d x + c\right )}{2 \, a^{2} 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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.1984, size = 63, normalized size = 1.34 \begin{align*} \frac{\frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac{a^{2} \sin \left (d x + c\right )^{2} - 4 \, a^{2} \sin \left (d x + c\right )}{a^{4}}}{2 \, 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))^2,x, algorithm="giac")

[Out]

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