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

Optimal. Leaf size=75 \[ \frac{1}{a^2 d (a+b \sin (c+d x))}-\frac{\log (a+b \sin (c+d x))}{a^3 d}+\frac{\log (\sin (c+d x))}{a^3 d}+\frac{1}{2 a d (a+b \sin (c+d x))^2} \]

[Out]

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

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Rubi [A]  time = 0.060479, antiderivative size = 75, 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 = {2721, 44} \[ \frac{1}{a^2 d (a+b \sin (c+d x))}-\frac{\log (a+b \sin (c+d x))}{a^3 d}+\frac{\log (\sin (c+d x))}{a^3 d}+\frac{1}{2 a d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2721

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)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[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+b \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+x)^3} \, dx,x,b \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,b \sin (c+d x)\right )}{d}\\ &=\frac{\log (\sin (c+d x))}{a^3 d}-\frac{\log (a+b \sin (c+d x))}{a^3 d}+\frac{1}{2 a d (a+b \sin (c+d x))^2}+\frac{1}{a^2 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.261819, size = 60, normalized size = 0.8 \[ \frac{\frac{a (3 a+2 b \sin (c+d x))}{(a+b \sin (c+d x))^2}-2 \log (a+b \sin (c+d x))+2 \log (\sin (c+d x))}{2 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 74, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}-{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}+{\frac{1}{2\,da \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{{a}^{2}d \left ( a+b\sin \left ( dx+c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.54714, size = 109, normalized size = 1.45 \begin{align*} \frac{\frac{2 \, b \sin \left (d x + c\right ) + 3 \, a}{a^{2} b^{2} \sin \left (d x + c\right )^{2} + 2 \, a^{3} b \sin \left (d x + c\right ) + a^{4}} - \frac{2 \, \log \left (b \sin \left (d x + c\right ) + a\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+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/2*((2*b*sin(d*x + c) + 3*a)/(a^2*b^2*sin(d*x + c)^2 + 2*a^3*b*sin(d*x + c) + a^4) - 2*log(b*sin(d*x + c) + a
)/a^3 + 2*log(sin(d*x + c))/a^3)/d

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Fricas [B]  time = 2.00358, size = 365, normalized size = 4.87 \begin{align*} -\frac{2 \, a b \sin \left (d x + c\right ) + 3 \, a^{2} + 2 \,{\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \,{\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \log \left (-\frac{1}{2} \, \sin \left (d x + c\right )\right )}{2 \,{\left (a^{3} b^{2} d \cos \left (d x + c\right )^{2} - 2 \, a^{4} b d \sin \left (d x + c\right ) -{\left (a^{5} + a^{3} b^{2}\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*b*sin(d*x + c) + 3*a^2 + 2*(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)*log(b*sin(d*x + c)
+ a) - 2*(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)*log(-1/2*sin(d*x + c)))/(a^3*b^2*d*cos(d*x + c)
^2 - 2*a^4*b*d*sin(d*x + c) - (a^5 + a^3*b^2)*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (c + d x \right )}}{\left (a + b \sin{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.74614, size = 93, normalized size = 1.24 \begin{align*} -\frac{\frac{2 \, \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{3}} - \frac{2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{2 \, a b \sin \left (d x + c\right ) + 3 \, a^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2} a^{3}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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