3.563 \(\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=120 \[ \frac{4}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{\csc ^4(c+d x)}{4 a^4 d}+\frac{4 \csc ^3(c+d x)}{3 a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{12 \csc (c+d x)}{a^4 d}+\frac{16 \log (\sin (c+d x))}{a^4 d}-\frac{16 \log (\sin (c+d x)+1)}{a^4 d} \]

[Out]

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

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Rubi [A]  time = 0.0826289, antiderivative size = 120, 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} \[ \frac{4}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{\csc ^4(c+d x)}{4 a^4 d}+\frac{4 \csc ^3(c+d x)}{3 a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{12 \csc (c+d x)}{a^4 d}+\frac{16 \log (\sin (c+d x))}{a^4 d}-\frac{16 \log (\sin (c+d x)+1)}{a^4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^4,x]

[Out]

(12*Csc[c + d*x])/(a^4*d) - (4*Csc[c + d*x]^2)/(a^4*d) + (4*Csc[c + d*x]^3)/(3*a^4*d) - Csc[c + d*x]^4/(4*a^4*
d) + (16*Log[Sin[c + d*x]])/(a^4*d) - (16*Log[1 + Sin[c + d*x]])/(a^4*d) + 4/(d*(a^4 + a^4*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 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))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{x^5 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^5}-\frac{4}{a x^4}+\frac{8}{a^2 x^3}-\frac{12}{a^3 x^2}+\frac{16}{a^4 x}-\frac{4}{a^3 (a+x)^2}-\frac{16}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{12 \csc (c+d x)}{a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{4 \csc ^3(c+d x)}{3 a^4 d}-\frac{\csc ^4(c+d x)}{4 a^4 d}+\frac{16 \log (\sin (c+d x))}{a^4 d}-\frac{16 \log (1+\sin (c+d x))}{a^4 d}+\frac{4}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.729173, size = 81, normalized size = 0.68 \[ \frac{\frac{48}{\sin (c+d x)+1}-3 \csc ^4(c+d x)+16 \csc ^3(c+d x)-48 \csc ^2(c+d x)+144 \csc (c+d x)+192 \log (\sin (c+d x))-192 \log (\sin (c+d x)+1)}{12 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^4,x]

[Out]

(144*Csc[c + d*x] - 48*Csc[c + d*x]^2 + 16*Csc[c + d*x]^3 - 3*Csc[c + d*x]^4 + 192*Log[Sin[c + d*x]] - 192*Log
[1 + Sin[c + d*x]] + 48/(1 + Sin[c + d*x]))/(12*a^4*d)

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Maple [A]  time = 0.197, size = 116, normalized size = 1. \begin{align*} 4\,{\frac{1}{d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-16\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{4}}}-{\frac{1}{4\,d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{4}{3\,d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-4\,{\frac{1}{d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+12\,{\frac{1}{d{a}^{4}\sin \left ( dx+c \right ) }}+16\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{4}}} \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))^4,x)

[Out]

4/d/a^4/(1+sin(d*x+c))-16*ln(1+sin(d*x+c))/a^4/d-1/4/d/a^4/sin(d*x+c)^4+4/3/d/a^4/sin(d*x+c)^3-4/d/a^4/sin(d*x
+c)^2+12/d/a^4/sin(d*x+c)+16*ln(sin(d*x+c))/a^4/d

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Maxima [A]  time = 1.0906, size = 135, normalized size = 1.12 \begin{align*} \frac{\frac{192 \, \sin \left (d x + c\right )^{4} + 96 \, \sin \left (d x + c\right )^{3} - 32 \, \sin \left (d x + c\right )^{2} + 13 \, \sin \left (d x + c\right ) - 3}{a^{4} \sin \left (d x + c\right )^{5} + a^{4} \sin \left (d x + c\right )^{4}} - \frac{192 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac{192 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}}}{12 \, 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))^4,x, algorithm="maxima")

[Out]

1/12*((192*sin(d*x + c)^4 + 96*sin(d*x + c)^3 - 32*sin(d*x + c)^2 + 13*sin(d*x + c) - 3)/(a^4*sin(d*x + c)^5 +
a^4*sin(d*x + c)^4) - 192*log(sin(d*x + c) + 1)/a^4 + 192*log(sin(d*x + c))/a^4)/d

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Fricas [B]  time = 1.19727, size = 632, normalized size = 5.27 \begin{align*} \frac{192 \, \cos \left (d x + c\right )^{4} - 352 \, \cos \left (d x + c\right )^{2} + 192 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 192 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (96 \, \cos \left (d x + c\right )^{2} - 109\right )} \sin \left (d x + c\right ) + 157}{12 \,{\left (a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d +{\left (a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} 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)^5*csc(d*x+c)^5/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/12*(192*cos(d*x + c)^4 - 352*cos(d*x + c)^2 + 192*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + (cos(d*x + c)^4 - 2*c
os(d*x + c)^2 + 1)*sin(d*x + c) + 1)*log(1/2*sin(d*x + c)) - 192*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + (cos(d*x
+ c)^4 - 2*cos(d*x + c)^2 + 1)*sin(d*x + c) + 1)*log(sin(d*x + c) + 1) - (96*cos(d*x + c)^2 - 109)*sin(d*x +
c) + 157)/(a^4*d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c)^2 + a^4*d + (a^4*d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c
)^2 + a^4*d)*sin(d*x + c))

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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))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.34919, size = 294, normalized size = 2.45 \begin{align*} -\frac{\frac{6144 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{3072 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac{1536 \,{\left (6 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 11 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6\right )}}{a^{4}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{2}} + \frac{6400 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1248 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 204 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 32 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3}{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}} + \frac{3 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 32 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 204 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1248 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{16}}}{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))^4,x, algorithm="giac")

[Out]

-1/192*(6144*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 3072*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 - 1536*(6*tan(1/
2*d*x + 1/2*c)^2 + 11*tan(1/2*d*x + 1/2*c) + 6)/(a^4*(tan(1/2*d*x + 1/2*c) + 1)^2) + (6400*tan(1/2*d*x + 1/2*c
)^4 - 1248*tan(1/2*d*x + 1/2*c)^3 + 204*tan(1/2*d*x + 1/2*c)^2 - 32*tan(1/2*d*x + 1/2*c) + 3)/(a^4*tan(1/2*d*x
+ 1/2*c)^4) + (3*a^12*tan(1/2*d*x + 1/2*c)^4 - 32*a^12*tan(1/2*d*x + 1/2*c)^3 + 204*a^12*tan(1/2*d*x + 1/2*c)
^2 - 1248*a^12*tan(1/2*d*x + 1/2*c))/a^16)/d