3.90 \(\int \frac{\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=133 \[ -\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{3 \cot ^3(c+d x)}{a^4 d}-\frac{16 \cot (c+d x)}{a^4 d}+\frac{27 \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{a^4 d}+\frac{11 \cot (c+d x) \csc (c+d x)}{2 a^4 d}-\frac{8 \cot (c+d x)}{a^4 d (\csc (c+d x)+1)} \]

[Out]

(27*ArcTanh[Cos[c + d*x]])/(2*a^4*d) - (16*Cot[c + d*x])/(a^4*d) - (3*Cot[c + d*x]^3)/(a^4*d) - Cot[c + d*x]^5
/(5*a^4*d) + (11*Cot[c + d*x]*Csc[c + d*x])/(2*a^4*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(a^4*d) - (8*Cot[c + d*x
])/(a^4*d*(1 + Csc[c + d*x]))

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Rubi [A]  time = 0.249307, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2709, 3770, 3767, 8, 3768, 3777} \[ -\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{3 \cot ^3(c+d x)}{a^4 d}-\frac{16 \cot (c+d x)}{a^4 d}+\frac{27 \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{a^4 d}+\frac{11 \cot (c+d x) \csc (c+d x)}{2 a^4 d}-\frac{8 \cot (c+d x)}{a^4 d (\csc (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^4,x]

[Out]

(27*ArcTanh[Cos[c + d*x]])/(2*a^4*d) - (16*Cot[c + d*x])/(a^4*d) - (3*Cot[c + d*x]^3)/(a^4*d) - Cot[c + d*x]^5
/(5*a^4*d) + (11*Cot[c + d*x]*Csc[c + d*x])/(2*a^4*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(a^4*d) - (8*Cot[c + d*x
])/(a^4*d*(1 + Csc[c + d*x]))

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 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]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 3777

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Simp[(Cot[c + d*x]*(a + b*Csc[c + d*x])^n)/(d*(
2*n + 1)), x] + Dist[1/(a^2*(2*n + 1)), Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]
), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]

Rubi steps

\begin{align*} \int \frac{\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\int \left (8 a^2-8 a^2 \csc (c+d x)+8 a^2 \csc ^2(c+d x)-8 a^2 \csc ^3(c+d x)+7 a^2 \csc ^4(c+d x)-4 a^2 \csc ^5(c+d x)+a^2 \csc ^6(c+d x)-\frac{8 a^2}{1+\csc (c+d x)}\right ) \, dx}{a^6}\\ &=\frac{8 x}{a^4}+\frac{\int \csc ^6(c+d x) \, dx}{a^4}-\frac{4 \int \csc ^5(c+d x) \, dx}{a^4}+\frac{7 \int \csc ^4(c+d x) \, dx}{a^4}-\frac{8 \int \csc (c+d x) \, dx}{a^4}+\frac{8 \int \csc ^2(c+d x) \, dx}{a^4}-\frac{8 \int \csc ^3(c+d x) \, dx}{a^4}-\frac{8 \int \frac{1}{1+\csc (c+d x)} \, dx}{a^4}\\ &=\frac{8 x}{a^4}+\frac{8 \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{4 \cot (c+d x) \csc (c+d x)}{a^4 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{a^4 d}-\frac{8 \cot (c+d x)}{a^4 d (1+\csc (c+d x))}-\frac{3 \int \csc ^3(c+d x) \, dx}{a^4}-\frac{4 \int \csc (c+d x) \, dx}{a^4}+\frac{8 \int -1 \, dx}{a^4}-\frac{\operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^4 d}-\frac{7 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^4 d}-\frac{8 \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^4 d}\\ &=\frac{12 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{16 \cot (c+d x)}{a^4 d}-\frac{3 \cot ^3(c+d x)}{a^4 d}-\frac{\cot ^5(c+d x)}{5 a^4 d}+\frac{11 \cot (c+d x) \csc (c+d x)}{2 a^4 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{a^4 d}-\frac{8 \cot (c+d x)}{a^4 d (1+\csc (c+d x))}-\frac{3 \int \csc (c+d x) \, dx}{2 a^4}\\ &=\frac{27 \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}-\frac{16 \cot (c+d x)}{a^4 d}-\frac{3 \cot ^3(c+d x)}{a^4 d}-\frac{\cot ^5(c+d x)}{5 a^4 d}+\frac{11 \cot (c+d x) \csc (c+d x)}{2 a^4 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{a^4 d}-\frac{8 \cot (c+d x)}{a^4 d (1+\csc (c+d x))}\\ \end{align*}

Mathematica [B]  time = 6.14507, size = 733, normalized size = 5.51 \[ \frac{16 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^7}{d (a \sin (c+d x)+a)^4}+\frac{27 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{2 d (a \sin (c+d x)+a)^4}-\frac{27 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{2 d (a \sin (c+d x)+a)^4}+\frac{33 \tan \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{5 d (a \sin (c+d x)+a)^4}-\frac{33 \cot \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{5 d (a \sin (c+d x)+a)^4}+\frac{\csc ^4\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{16 d (a \sin (c+d x)+a)^4}+\frac{11 \csc ^2\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{8 d (a \sin (c+d x)+a)^4}-\frac{\sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{16 d (a \sin (c+d x)+a)^4}-\frac{11 \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{8 d (a \sin (c+d x)+a)^4}-\frac{\cot \left (\frac{1}{2} (c+d x)\right ) \csc ^4\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{160 d (a \sin (c+d x)+a)^4}-\frac{53 \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{160 d (a \sin (c+d x)+a)^4}+\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{160 d (a \sin (c+d x)+a)^4}+\frac{53 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8}{160 d (a \sin (c+d x)+a)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^4,x]

[Out]

(16*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)/(d*(a + a*Sin[c + d*x])^4) - (33*Cot[(c + d*x)/2
]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(5*d*(a + a*Sin[c + d*x])^4) + (11*Csc[(c + d*x)/2]^2*(Cos[(c + d*x
)/2] + Sin[(c + d*x)/2])^8)/(8*d*(a + a*Sin[c + d*x])^4) - (53*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(Cos[(c + d
*x)/2] + Sin[(c + d*x)/2])^8)/(160*d*(a + a*Sin[c + d*x])^4) + (Csc[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2])^8)/(16*d*(a + a*Sin[c + d*x])^4) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2])^8)/(160*d*(a + a*Sin[c + d*x])^4) + (27*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])
^8)/(2*d*(a + a*Sin[c + d*x])^4) - (27*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(2*d*(a
+ a*Sin[c + d*x])^4) - (11*Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(8*d*(a + a*Sin[c + d*x
])^4) - (Sec[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(16*d*(a + a*Sin[c + d*x])^4) + (33*(Cos[
(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d*x)/2])/(5*d*(a + a*Sin[c + d*x])^4) + (53*Sec[(c + d*x)/2]^2*(Co
s[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d*x)/2])/(160*d*(a + a*Sin[c + d*x])^4) + (Sec[(c + d*x)/2]^4*(C
os[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d*x)/2])/(160*d*(a + a*Sin[c + d*x])^4)

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Maple [A]  time = 0.149, size = 229, normalized size = 1.7 \begin{align*}{\frac{1}{160\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{16\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}+{\frac{11}{32\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{3}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{111}{16\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-16\,{\frac{1}{d{a}^{4} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}-{\frac{1}{160\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{1}{16\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}-{\frac{11}{32\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{3}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{111}{16\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{27}{2\,d{a}^{4}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6/(a+a*sin(d*x+c))^4,x)

[Out]

1/160/d/a^4*tan(1/2*d*x+1/2*c)^5-1/16/d/a^4*tan(1/2*d*x+1/2*c)^4+11/32/d/a^4*tan(1/2*d*x+1/2*c)^3-3/2/d/a^4*ta
n(1/2*d*x+1/2*c)^2+111/16/d/a^4*tan(1/2*d*x+1/2*c)-16/d/a^4/(tan(1/2*d*x+1/2*c)+1)-1/160/d/a^4/tan(1/2*d*x+1/2
*c)^5+1/16/d/a^4/tan(1/2*d*x+1/2*c)^4-11/32/d/a^4/tan(1/2*d*x+1/2*c)^3+3/2/d/a^4/tan(1/2*d*x+1/2*c)^2-111/16/d
/a^4/tan(1/2*d*x+1/2*c)-27/2/d/a^4*ln(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 3.61884, size = 377, normalized size = 2.83 \begin{align*} \frac{\frac{\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{45 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{185 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{870 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{3670 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1}{\frac{a^{4} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac{\frac{1110 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{240 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{55 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{10 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{4}} - \frac{2160 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{160 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/160*((9*sin(d*x + c)/(cos(d*x + c) + 1) - 45*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 185*sin(d*x + c)^3/(cos(d
*x + c) + 1)^3 - 870*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3670*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1)/(a^4*
sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + a^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (1110*sin(d*x + c)/(cos(d*x +
c) + 1) - 240*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 55*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 10*sin(d*x + c)^
4/(cos(d*x + c) + 1)^4 + sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^4 - 2160*log(sin(d*x + c)/(cos(d*x + c) + 1))/
a^4)/d

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Fricas [B]  time = 1.78661, size = 1191, normalized size = 8.95 \begin{align*} \frac{424 \, \cos \left (d x + c\right )^{6} + 154 \, \cos \left (d x + c\right )^{5} - 1060 \, \cos \left (d x + c\right )^{4} - 340 \, \cos \left (d x + c\right )^{3} + 800 \, \cos \left (d x + c\right )^{2} + 135 \,{\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 )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 135 \,{\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 )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \,{\left (212 \, \cos \left (d x + c\right )^{5} + 135 \, \cos \left (d x + c\right )^{4} - 395 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 175 \, \cos \left (d x + c\right ) + 80\right )} \sin \left (d x + c\right ) + 190 \, \cos \left (d x + c\right ) - 160}{20 \,{\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 )^{5} + a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{3} - 2 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right ) + 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(cot(d*x+c)^6/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/20*(424*cos(d*x + c)^6 + 154*cos(d*x + c)^5 - 1060*cos(d*x + c)^4 - 340*cos(d*x + c)^3 + 800*cos(d*x + c)^2
+ 135*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - (cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c
)^3 - 2*cos(d*x + c)^2 + cos(d*x + c) + 1)*sin(d*x + c) - 1)*log(1/2*cos(d*x + c) + 1/2) - 135*(cos(d*x + c)^6
- 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - (cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^
2 + cos(d*x + c) + 1)*sin(d*x + c) - 1)*log(-1/2*cos(d*x + c) + 1/2) + 2*(212*cos(d*x + c)^5 + 135*cos(d*x + c
)^4 - 395*cos(d*x + c)^3 - 225*cos(d*x + c)^2 + 175*cos(d*x + c) + 80)*sin(d*x + c) + 190*cos(d*x + c) - 160)/
(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)^5 + a^4*
d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c)^3 - 2*a^4*d*cos(d*x + c)^2 + a^4*d*cos(d*x + c) + 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(cot(d*x+c)**6/(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.73458, size = 275, normalized size = 2.07 \begin{align*} -\frac{\frac{2160 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{4}} + \frac{2560}{a^{4}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}} - \frac{4932 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1110 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 240 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 55 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1}{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}} - \frac{a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 10 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 55 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 240 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1110 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{20}}}{160 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/160*(2160*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 + 2560/(a^4*(tan(1/2*d*x + 1/2*c) + 1)) - (4932*tan(1/2*d*x +
1/2*c)^5 - 1110*tan(1/2*d*x + 1/2*c)^4 + 240*tan(1/2*d*x + 1/2*c)^3 - 55*tan(1/2*d*x + 1/2*c)^2 + 10*tan(1/2*d
*x + 1/2*c) - 1)/(a^4*tan(1/2*d*x + 1/2*c)^5) - (a^16*tan(1/2*d*x + 1/2*c)^5 - 10*a^16*tan(1/2*d*x + 1/2*c)^4
+ 55*a^16*tan(1/2*d*x + 1/2*c)^3 - 240*a^16*tan(1/2*d*x + 1/2*c)^2 + 1110*a^16*tan(1/2*d*x + 1/2*c))/a^20)/d