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

Optimal. Leaf size=52 \[ \frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{\log (\sin (c+d x)+1)}{a^2 d} \]

[Out]

Log[Sin[c + d*x]]/(a^2*d) - Log[1 + Sin[c + d*x]]/(a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))

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Rubi [A]  time = 0.0504909, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 44} \[ \frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{\log (\sin (c+d x)+1)}{a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Sin[c + d*x]]/(a^2*d) - Log[1 + Sin[c + d*x]]/(a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0548377, size = 36, normalized size = 0.69 \[ \frac{\frac{1}{\sin (c+d x)+1}+\log (\sin (c+d x))-\log (\sin (c+d x)+1)}{a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[Sin[c + d*x]] - Log[1 + Sin[c + d*x]] + (1 + Sin[c + d*x])^(-1))/(a^2*d)

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Maple [A]  time = 0.034, size = 50, normalized size = 1. \begin{align*}{\frac{1}{d{a}^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)/(a+a*sin(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 1.11919, size = 62, normalized size = 1.19 \begin{align*} \frac{\frac{1}{a^{2} \sin \left (d x + c\right ) + a^{2}} - \frac{\log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac{\log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.50947, size = 162, normalized size = 3.12 \begin{align*} \frac{{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) -{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 1}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+a*sin(d*x+c))**2,x)

[Out]

Integral(cot(c + d*x)/(sin(c + d*x)**2 + 2*sin(c + d*x) + 1), x)/a**2

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Giac [A]  time = 1.7951, size = 61, normalized size = 1.17 \begin{align*} \frac{a{\left (\frac{\log \left ({\left | -\frac{a}{a \sin \left (d x + c\right ) + a} + 1 \right |}\right )}{a^{3}} + \frac{1}{{\left (a \sin \left (d x + c\right ) + a\right )} a^{2}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

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