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

Optimal. Leaf size=90 \[ -\frac{2}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{\csc (c+d x)}{a^3 d}-\frac{3 \log (\sin (c+d x))}{a^3 d}+\frac{3 \log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]

[Out]

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

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Rubi [A]  time = 0.0836685, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 44} \[ -\frac{2}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{\csc (c+d x)}{a^3 d}-\frac{3 \log (\sin (c+d x))}{a^3 d}+\frac{3 \log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^2}{x^2 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{1}{x^2 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^2}-\frac{3}{a^4 x}+\frac{1}{a^2 (a+x)^3}+\frac{2}{a^3 (a+x)^2}+\frac{3}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{\csc (c+d x)}{a^3 d}-\frac{3 \log (\sin (c+d x))}{a^3 d}+\frac{3 \log (1+\sin (c+d x))}{a^3 d}-\frac{1}{2 a d (a+a \sin (c+d x))^2}-\frac{2}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.301958, size = 61, normalized size = 0.68 \[ -\frac{\frac{4}{\sin (c+d x)+1}+\frac{1}{(\sin (c+d x)+1)^2}+2 \csc (c+d x)+6 \log (\sin (c+d x))-6 \log (\sin (c+d x)+1)}{2 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*Csc[c + d*x] + 6*Log[Sin[c + d*x]] - 6*Log[1 + Sin[c + d*x]] + (1 + Sin[c + d*x])^(-2) + 4/(1 + Sin[c + d*
x]))/(2*a^3*d)

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Maple [A]  time = 0.049, size = 86, normalized size = 1. \begin{align*} -{\frac{1}{2\,{a}^{3}d \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-2\,{\frac{1}{{a}^{3}d \left ( 1+\sin \left ( dx+c \right ) \right ) }}+3\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}-{\frac{1}{{a}^{3}d\sin \left ( dx+c \right ) }}-3\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{{a}^{3}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^2/(a+a*sin(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.01597, size = 123, normalized size = 1.37 \begin{align*} -\frac{\frac{6 \, \sin \left (d x + c\right )^{2} + 9 \, \sin \left (d x + c\right ) + 2}{a^{3} \sin \left (d x + c\right )^{3} + 2 \, a^{3} \sin \left (d x + c\right )^{2} + a^{3} \sin \left (d x + c\right )} - \frac{6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{6 \, \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(cos(d*x+c)*csc(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2*((6*sin(d*x + c)^2 + 9*sin(d*x + c) + 2)/(a^3*sin(d*x + c)^3 + 2*a^3*sin(d*x + c)^2 + a^3*sin(d*x + c)) -
6*log(sin(d*x + c) + 1)/a^3 + 6*log(sin(d*x + c))/a^3)/d

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Fricas [A]  time = 1.51805, size = 404, normalized size = 4.49 \begin{align*} -\frac{6 \, \cos \left (d x + c\right )^{2} + 6 \,{\left (2 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 2\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 6 \,{\left (2 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, \sin \left (d x + c\right ) - 8}{2 \,{\left (2 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d +{\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/2*(6*cos(d*x + c)^2 + 6*(2*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2)*sin(d*x + c) - 2)*log(1/2*sin(d*x + c)) -
6*(2*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2)*sin(d*x + c) - 2)*log(sin(d*x + c) + 1) - 9*sin(d*x + c) - 8)/(2*a^
3*d*cos(d*x + c)^2 - 2*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d)*sin(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos{\left (c + d x \right )} \csc ^{2}{\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(cos(d*x+c)*csc(d*x+c)**2/(a+a*sin(d*x+c))**3,x)

[Out]

Integral(cos(c + d*x)*csc(c + d*x)**2/(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 = 1.32079, size = 104, normalized size = 1.16 \begin{align*} \frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{6 \, \sin \left (d x + c\right )^{2} + 9 \, \sin \left (d x + c\right ) + 2}{a^{3}{\left (\sin \left (d x + c\right ) + 1\right )}^{2} \sin \left (d x + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/2*(6*log(abs(sin(d*x + c) + 1))/a^3 - 6*log(abs(sin(d*x + c)))/a^3 - (6*sin(d*x + c)^2 + 9*sin(d*x + c) + 2)
/(a^3*(sin(d*x + c) + 1)^2*sin(d*x + c)))/d