3.75 \(\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=74 \[ \frac{1}{d \left (a^3 \sin (c+d x)+a^3\right )}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{\log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]

[Out]

Log[Sin[c + d*x]]/(a^3*d) - Log[1 + Sin[c + d*x]]/(a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 1/(d*(a^3 + a^3
*Sin[c + d*x]))

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Rubi [A]  time = 0.0582604, antiderivative size = 74, 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} \[ \frac{1}{d \left (a^3 \sin (c+d x)+a^3\right )}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{\log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.175268, size = 52, normalized size = 0.7 \[ \frac{\frac{2 \sin (c+d x)+3}{(\sin (c+d x)+1)^2}+2 \log (\sin (c+d x))-2 \log (\sin (c+d x)+1)}{2 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Log[Sin[c + d*x]] - 2*Log[1 + Sin[c + d*x]] + (3 + 2*Sin[c + d*x])/(1 + Sin[c + d*x])^2)/(2*a^3*d)

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Maple [A]  time = 0.037, size = 68, normalized size = 0.9 \begin{align*}{\frac{1}{2\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{3}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.76862, size = 97, normalized size = 1.31 \begin{align*} \frac{\frac{2 \, \sin \left (d x + c\right ) + 3}{a^{3} \sin \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) + a^{3}} - \frac{2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{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))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.6075, size = 284, normalized size = 3.84 \begin{align*} \frac{2 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 2 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, \sin \left (d x + c\right ) - 3}{2 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(cos(d*x + c)^2 - 2*sin(d*x + c) - 2)*log(1/2*sin(d*x + c)) - 2*(cos(d*x + c)^2 - 2*sin(d*x + c) - 2)*l
og(sin(d*x + c) + 1) - 2*sin(d*x + c) - 3)/(a^3*d*cos(d*x + c)^2 - 2*a^3*d*sin(d*x + c) - 2*a^3*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 ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+a*sin(d*x+c))**3,x)

[Out]

Integral(cot(c + d*x)/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3

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Giac [A]  time = 1.9801, size = 80, normalized size = 1.08 \begin{align*} -\frac{\frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{2 \, \sin \left (d x + c\right ) + 3}{a^{3}{\left (\sin \left (d x + c\right ) + 1\right )}^{2}}}{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))^3,x, algorithm="giac")

[Out]

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