3.749 \(\int \frac{\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=140 \[ -\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]

[Out]

(-5*ArcTanh[Cos[c + d*x]])/(16*a^3*d) - (4*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^5/(a^3*d) - Cot[c + d*x]^7
/(7*a^3*d) - (5*Cot[c + d*x]*Csc[c + d*x])/(16*a^3*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^3*d) + (Cot[c + d*x
]*Csc[c + d*x]^5)/(2*a^3*d)

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Rubi [A]  time = 0.231049, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2709, 3768, 3770, 3767} \[ -\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]

[Out]

(-5*ArcTanh[Cos[c + d*x]])/(16*a^3*d) - (4*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^5/(a^3*d) - Cot[c + d*x]^7
/(7*a^3*d) - (5*Cot[c + d*x]*Csc[c + d*x])/(16*a^3*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^3*d) + (Cot[c + d*x
]*Csc[c + d*x]^5)/(2*a^3*d)

Rule 2709

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Dist[a^p, Int[Expan
dIntegrand[(Sin[e + f*x]^p*(a + b*Sin[e + f*x])^(m - p/2))/(a - b*Sin[e + f*x])^(p/2), x], x], x] /; FreeQ[{a,
b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \frac{\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \left (a^5 \csc ^3(c+d x)-3 a^5 \csc ^4(c+d x)+2 a^5 \csc ^5(c+d x)+2 a^5 \csc ^6(c+d x)-3 a^5 \csc ^7(c+d x)+a^5 \csc ^8(c+d x)\right ) \, dx}{a^8}\\ &=\frac{\int \csc ^3(c+d x) \, dx}{a^3}+\frac{\int \csc ^8(c+d x) \, dx}{a^3}+\frac{2 \int \csc ^5(c+d x) \, dx}{a^3}+\frac{2 \int \csc ^6(c+d x) \, dx}{a^3}-\frac{3 \int \csc ^4(c+d x) \, dx}{a^3}-\frac{3 \int \csc ^7(c+d x) \, dx}{a^3}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac{\int \csc (c+d x) \, dx}{2 a^3}+\frac{3 \int \csc ^3(c+d x) \, dx}{2 a^3}-\frac{5 \int \csc ^5(c+d x) \, dx}{2 a^3}-\frac{\operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}-\frac{2 \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}+\frac{3 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{4 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac{3 \int \csc (c+d x) \, dx}{4 a^3}-\frac{15 \int \csc ^3(c+d x) \, dx}{8 a^3}\\ &=-\frac{5 \tanh ^{-1}(\cos (c+d x))}{4 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}-\frac{15 \int \csc (c+d x) \, dx}{16 a^3}\\ &=-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.962836, size = 251, normalized size = 1.79 \[ \frac{\csc ^7(c+d x) \left (4998 \sin (2 (c+d x))+504 \sin (4 (c+d x))-210 \sin (6 (c+d x))-4704 \cos (c+d x)+672 \cos (3 (c+d x))+1120 \cos (5 (c+d x))-160 \cos (7 (c+d x))+3675 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2205 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+735 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-105 \sin (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-3675 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+2205 \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-735 \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+105 \sin (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{21504 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]

[Out]

(Csc[c + d*x]^7*(-4704*Cos[c + d*x] + 672*Cos[3*(c + d*x)] + 1120*Cos[5*(c + d*x)] - 160*Cos[7*(c + d*x)] - 36
75*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 3675*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 4998*Sin[2*(c + d*x)] + 2205
*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 2205*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 504*Sin[4*(c + d*x)] -
735*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 735*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 210*Sin[6*(c + d*x)
] + 105*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] - 105*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(21504*a^3*d)

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Maple [B]  time = 0.198, size = 284, normalized size = 2. \begin{align*}{\frac{1}{896\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{1}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}+{\frac{3}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{5}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}+{\frac{13}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{3}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{29}{128\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{896\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}+{\frac{29}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{3}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{5}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{5}{16\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{1}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}-{\frac{13}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{3}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x)

[Out]

1/896/d/a^3*tan(1/2*d*x+1/2*c)^7-1/128/d/a^3*tan(1/2*d*x+1/2*c)^6+3/128/d/a^3*tan(1/2*d*x+1/2*c)^5-5/128/d/a^3
*tan(1/2*d*x+1/2*c)^4+13/384/d/a^3*tan(1/2*d*x+1/2*c)^3+3/128/d/a^3*tan(1/2*d*x+1/2*c)^2-29/128/d/a^3*tan(1/2*
d*x+1/2*c)-1/896/d/a^3/tan(1/2*d*x+1/2*c)^7+29/128/d/a^3/tan(1/2*d*x+1/2*c)-3/128/d/a^3/tan(1/2*d*x+1/2*c)^5+5
/128/d/a^3/tan(1/2*d*x+1/2*c)^4+5/16/d/a^3*ln(tan(1/2*d*x+1/2*c))+1/128/d/a^3/tan(1/2*d*x+1/2*c)^6-13/384/d/a^
3/tan(1/2*d*x+1/2*c)^3-3/128/d/a^3/tan(1/2*d*x+1/2*c)^2

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Maxima [B]  time = 1.01462, size = 425, normalized size = 3.04 \begin{align*} -\frac{\frac{\frac{609 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{91 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{21 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3}} - \frac{840 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac{{\left (\frac{21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{105 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{91 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{609 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 3\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{3} \sin \left (d x + c\right )^{7}}}{2688 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2688*((609*sin(d*x + c)/(cos(d*x + c) + 1) - 63*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 91*sin(d*x + c)^3/(co
s(d*x + c) + 1)^3 + 105*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 21*sin(
d*x + c)^6/(cos(d*x + c) + 1)^6 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^3 - 840*log(sin(d*x + c)/(cos(d*x +
c) + 1))/a^3 - (21*sin(d*x + c)/(cos(d*x + c) + 1) - 63*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 105*sin(d*x + c
)^3/(cos(d*x + c) + 1)^3 - 91*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 6
09*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 3)*(cos(d*x + c) + 1)^7/(a^3*sin(d*x + c)^7))/d

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Fricas [A]  time = 1.17624, size = 612, normalized size = 4.37 \begin{align*} \frac{320 \, \cos \left (d x + c\right )^{7} - 1120 \, \cos \left (d x + c\right )^{5} + 896 \, \cos \left (d x + c\right )^{3} - 105 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 105 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 42 \,{\left (5 \, \cos \left (d x + c\right )^{5} - 8 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \,{\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/672*(320*cos(d*x + c)^7 - 1120*cos(d*x + c)^5 + 896*cos(d*x + c)^3 - 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4
+ 3*cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*
cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 42*(5*cos(d*x + c)^5 - 8*cos(d*x + c)^3 - 5*co
s(d*x + c))*sin(d*x + c))/((a^3*d*cos(d*x + c)^6 - 3*a^3*d*cos(d*x + c)^4 + 3*a^3*d*cos(d*x + c)^2 - a^3*d)*si
n(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)**8*csc(d*x+c)**8/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.37335, size = 329, normalized size = 2.35 \begin{align*} \frac{\frac{840 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac{2178 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 609 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 91 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 21 \, \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 )^{7}} + \frac{3 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 21 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 63 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 105 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 91 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 63 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 609 \, a^{18} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{21}}}{2688 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/2688*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (2178*tan(1/2*d*x + 1/2*c)^7 - 609*tan(1/2*d*x + 1/2*c)^6 + 6
3*tan(1/2*d*x + 1/2*c)^5 + 91*tan(1/2*d*x + 1/2*c)^4 - 105*tan(1/2*d*x + 1/2*c)^3 + 63*tan(1/2*d*x + 1/2*c)^2
- 21*tan(1/2*d*x + 1/2*c) + 3)/(a^3*tan(1/2*d*x + 1/2*c)^7) + (3*a^18*tan(1/2*d*x + 1/2*c)^7 - 21*a^18*tan(1/2
*d*x + 1/2*c)^6 + 63*a^18*tan(1/2*d*x + 1/2*c)^5 - 105*a^18*tan(1/2*d*x + 1/2*c)^4 + 91*a^18*tan(1/2*d*x + 1/2
*c)^3 + 63*a^18*tan(1/2*d*x + 1/2*c)^2 - 609*a^18*tan(1/2*d*x + 1/2*c))/a^21)/d