3.561 \(\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=96 \[ -\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{a^3 d}-\frac{2 \csc ^2(c+d x)}{a^3 d}+\frac{4 \csc (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]

[Out]

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

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Rubi [A]  time = 0.0677812, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 88} \[ -\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{a^3 d}-\frac{2 \csc ^2(c+d x)}{a^3 d}+\frac{4 \csc (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]

[Out]

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

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{x^5 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^5}-\frac{3}{x^4}+\frac{4}{a x^3}-\frac{4}{a^2 x^2}+\frac{4}{a^3 x}-\frac{4}{a^3 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{4 \csc (c+d x)}{a^3 d}-\frac{2 \csc ^2(c+d x)}{a^3 d}+\frac{\csc ^3(c+d x)}{a^3 d}-\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (1+\sin (c+d x))}{a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.318515, size = 69, normalized size = 0.72 \[ \frac{-\csc ^4(c+d x)+4 \csc ^3(c+d x)-8 \csc ^2(c+d x)+16 \csc (c+d x)+16 \log (\sin (c+d x))-16 \log (\sin (c+d x)+1)}{4 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]

[Out]

(16*Csc[c + d*x] - 8*Csc[c + d*x]^2 + 4*Csc[c + d*x]^3 - Csc[c + d*x]^4 + 16*Log[Sin[c + d*x]] - 16*Log[1 + Si
n[c + d*x]])/(4*a^3*d)

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Maple [A]  time = 0.181, size = 97, normalized size = 1. \begin{align*} -4\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{1}{4\,d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-2\,{\frac{1}{d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{1}{d{a}^{3}\sin \left ( dx+c \right ) }}+4\,{\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(cos(d*x+c)^5*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.19002, size = 101, normalized size = 1.05 \begin{align*} -\frac{\frac{16 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac{16 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac{16 \, \sin \left (d x + c\right )^{3} - 8 \, \sin \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) - 1}{a^{3} \sin \left (d x + c\right )^{4}}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/4*(16*log(sin(d*x + c) + 1)/a^3 - 16*log(sin(d*x + c))/a^3 - (16*sin(d*x + c)^3 - 8*sin(d*x + c)^2 + 4*sin(
d*x + c) - 1)/(a^3*sin(d*x + c)^4))/d

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Fricas [A]  time = 1.13131, size = 348, normalized size = 3.62 \begin{align*} \frac{8 \, \cos \left (d x + c\right )^{2} + 16 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 16 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \,{\left (4 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 9}{4 \,{\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/4*(8*cos(d*x + c)^2 + 16*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(1/2*sin(d*x + c)) - 16*(cos(d*x + c)^4
- 2*cos(d*x + c)^2 + 1)*log(sin(d*x + c) + 1) - 4*(4*cos(d*x + c)^2 - 5)*sin(d*x + c) - 9)/(a^3*d*cos(d*x + c)
^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*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)**5/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.25489, size = 235, normalized size = 2.45 \begin{align*} -\frac{\frac{1536 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{768 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac{1600 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 456 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 108 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}} + \frac{3 \,{\left (a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 8 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 36 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 152 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{12}}}{192 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/192*(1536*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^3 - 768*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 + (1600*tan(1/2*d
*x + 1/2*c)^4 - 456*tan(1/2*d*x + 1/2*c)^3 + 108*tan(1/2*d*x + 1/2*c)^2 - 24*tan(1/2*d*x + 1/2*c) + 3)/(a^3*ta
n(1/2*d*x + 1/2*c)^4) + 3*(a^9*tan(1/2*d*x + 1/2*c)^4 - 8*a^9*tan(1/2*d*x + 1/2*c)^3 + 36*a^9*tan(1/2*d*x + 1/
2*c)^2 - 152*a^9*tan(1/2*d*x + 1/2*c))/a^12)/d