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

Optimal. Leaf size=135 \[ -\frac{4}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{\csc ^5(c+d x)}{5 a^4 d}+\frac{\csc ^4(c+d x)}{a^4 d}-\frac{8 \csc ^3(c+d x)}{3 a^4 d}+\frac{6 \csc ^2(c+d x)}{a^4 d}-\frac{16 \csc (c+d x)}{a^4 d}-\frac{20 \log (\sin (c+d x))}{a^4 d}+\frac{20 \log (\sin (c+d x)+1)}{a^4 d} \]

[Out]

(-16*Csc[c + d*x])/(a^4*d) + (6*Csc[c + d*x]^2)/(a^4*d) - (8*Csc[c + d*x]^3)/(3*a^4*d) + Csc[c + d*x]^4/(a^4*d
) - Csc[c + d*x]^5/(5*a^4*d) - (20*Log[Sin[c + d*x]])/(a^4*d) + (20*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 = 0.132366, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ -\frac{4}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{\csc ^5(c+d x)}{5 a^4 d}+\frac{\csc ^4(c+d x)}{a^4 d}-\frac{8 \csc ^3(c+d x)}{3 a^4 d}+\frac{6 \csc ^2(c+d x)}{a^4 d}-\frac{16 \csc (c+d x)}{a^4 d}-\frac{20 \log (\sin (c+d x))}{a^4 d}+\frac{20 \log (\sin (c+d x)+1)}{a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*Csc[c + d*x])/(a^4*d) + (6*Csc[c + d*x]^2)/(a^4*d) - (8*Csc[c + d*x]^3)/(3*a^4*d) + Csc[c + d*x]^4/(a^4*d
) - Csc[c + d*x]^5/(5*a^4*d) - (20*Log[Sin[c + d*x]])/(a^4*d) + (20*Log[1 + Sin[c + d*x]])/(a^4*d) - 4/(d*(a^4
+ a^4*Sin[c + d*x]))

Rule 2836

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rule 12

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

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) \csc (c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^6 (a-x)^2}{x^6 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{(a-x)^2}{x^6 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (\frac{1}{x^6}-\frac{4}{a x^5}+\frac{8}{a^2 x^4}-\frac{12}{a^3 x^3}+\frac{16}{a^4 x^2}-\frac{20}{a^5 x}+\frac{4}{a^4 (a+x)^2}+\frac{20}{a^5 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{16 \csc (c+d x)}{a^4 d}+\frac{6 \csc ^2(c+d x)}{a^4 d}-\frac{8 \csc ^3(c+d x)}{3 a^4 d}+\frac{\csc ^4(c+d x)}{a^4 d}-\frac{\csc ^5(c+d x)}{5 a^4 d}-\frac{20 \log (\sin (c+d x))}{a^4 d}+\frac{20 \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 = 0.289052, size = 91, normalized size = 0.67 \[ -\frac{\frac{60}{\sin (c+d x)+1}+3 \csc ^5(c+d x)-15 \csc ^4(c+d x)+40 \csc ^3(c+d x)-90 \csc ^2(c+d x)+240 \csc (c+d x)+300 \log (\sin (c+d x))-300 \log (\sin (c+d x)+1)}{15 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(240*Csc[c + d*x] - 90*Csc[c + d*x]^2 + 40*Csc[c + d*x]^3 - 15*Csc[c + d*x]^4 + 3*Csc[c + d*x]^5 + 300*Log[Si
n[c + d*x]] - 300*Log[1 + Sin[c + d*x]] + 60/(1 + Sin[c + d*x]))/(15*a^4*d)

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Maple [A]  time = 0.208, size = 131, normalized size = 1. \begin{align*} -4\,{\frac{1}{d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+20\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{4}}}-{\frac{1}{5\,d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{8}{3\,d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+6\,{\frac{1}{d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-16\,{\frac{1}{d{a}^{4}\sin \left ( dx+c \right ) }}-20\,{\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)^6/(a+a*sin(d*x+c))^4,x)

[Out]

-4/d/a^4/(1+sin(d*x+c))+20*ln(1+sin(d*x+c))/a^4/d-1/5/d/a^4/sin(d*x+c)^5+1/d/a^4/sin(d*x+c)^4-8/3/d/a^4/sin(d*
x+c)^3+6/d/a^4/sin(d*x+c)^2-16/d/a^4/sin(d*x+c)-20*ln(sin(d*x+c))/a^4/d

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Maxima [A]  time = 1.2236, size = 149, normalized size = 1.1 \begin{align*} -\frac{\frac{300 \, \sin \left (d x + c\right )^{5} + 150 \, \sin \left (d x + c\right )^{4} - 50 \, \sin \left (d x + c\right )^{3} + 25 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) + 3}{a^{4} \sin \left (d x + c\right )^{6} + a^{4} \sin \left (d x + c\right )^{5}} - \frac{300 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac{300 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^6/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/15*((300*sin(d*x + c)^5 + 150*sin(d*x + c)^4 - 50*sin(d*x + c)^3 + 25*sin(d*x + c)^2 - 12*sin(d*x + c) + 3)
/(a^4*sin(d*x + c)^6 + a^4*sin(d*x + c)^5) - 300*log(sin(d*x + c) + 1)/a^4 + 300*log(sin(d*x + c))/a^4)/d

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Fricas [B]  time = 1.18772, size = 749, normalized size = 5.55 \begin{align*} \frac{150 \, \cos \left (d x + c\right )^{4} - 325 \, \cos \left (d x + c\right )^{2} - 300 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \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 ) + 300 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \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 ) + 2 \,{\left (150 \, \cos \left (d x + c\right )^{4} - 275 \, \cos \left (d x + c\right )^{2} + 119\right )} \sin \left (d x + c\right ) + 178}{15 \,{\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, 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)^6/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/15*(150*cos(d*x + c)^4 - 325*cos(d*x + c)^2 - 300*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - (c
os(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*sin(d*x + c) - 1)*log(1/2*sin(d*x + c)) + 300*(cos(d*x + c)^6 - 3*cos(d*
x + c)^4 + 3*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)
+ 2*(150*cos(d*x + c)^4 - 275*cos(d*x + c)^2 + 119)*sin(d*x + c) + 178)/(a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*
x + c)^4 + 3*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)**6/(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.29281, size = 335, normalized size = 2.48 \begin{align*} \frac{\frac{19200 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{9600 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac{1920 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 28 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15\right )}}{a^{4}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{2}} + \frac{21920 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 4350 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 840 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 175 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 30 \, \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 )^{5}} - \frac{3 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 30 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 175 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 840 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 4350 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{20}}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^6/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/480*(19200*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 9600*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 - 1920*(15*tan(1
/2*d*x + 1/2*c)^2 + 28*tan(1/2*d*x + 1/2*c) + 15)/(a^4*(tan(1/2*d*x + 1/2*c) + 1)^2) + (21920*tan(1/2*d*x + 1/
2*c)^5 - 4350*tan(1/2*d*x + 1/2*c)^4 + 840*tan(1/2*d*x + 1/2*c)^3 - 175*tan(1/2*d*x + 1/2*c)^2 + 30*tan(1/2*d*
x + 1/2*c) - 3)/(a^4*tan(1/2*d*x + 1/2*c)^5) - (3*a^16*tan(1/2*d*x + 1/2*c)^5 - 30*a^16*tan(1/2*d*x + 1/2*c)^4
+ 175*a^16*tan(1/2*d*x + 1/2*c)^3 - 840*a^16*tan(1/2*d*x + 1/2*c)^2 + 4350*a^16*tan(1/2*d*x + 1/2*c))/a^20)/d