3.1.90 \(\int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx\) [90]

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

[Out]

27/2*arctanh(cos(d*x+c))/a^4/d-40*cot(d*x+c)/a^4/d-27*cot(d*x+c)^3/a^4/d-41/5*cot(d*x+c)^5/a^4/d+27/2*cot(d*x+
c)*csc(d*x+c)/a^4/d+9*cot(d*x+c)*csc(d*x+c)^3/a^4/d+8*cot(d*x+c)*csc(d*x+c)^4/a^4/d/(1+sin(d*x+c))

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Rubi [A]
time = 0.20, antiderivative size = 133, normalized size of antiderivative = 1.00, 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 = {2788, 3855, 3852, 8, 3853, 3862} \begin {gather*} -\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)} \end {gather*}

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 8

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

Rule 2788

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 3852

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 3853

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 3855

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

Rule 3862

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 {\text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^4 d}-\frac {7 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^4 d}-\frac {8 \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(733\) vs. \(2(133)=266\).
time = 6.07, size = 733, normalized size = 5.51 \begin {gather*} \frac {16 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^7}{d (a+a \sin (c+d x))^4}-\frac {33 \cot \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{5 d (a+a \sin (c+d x))^4}+\frac {11 \csc ^2\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{8 d (a+a \sin (c+d x))^4}-\frac {53 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{160 d (a+a \sin (c+d x))^4}+\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{16 d (a+a \sin (c+d x))^4}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{160 d (a+a \sin (c+d x))^4}+\frac {27 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{2 d (a+a \sin (c+d x))^4}-\frac {27 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{2 d (a+a \sin (c+d x))^4}-\frac {11 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{8 d (a+a \sin (c+d x))^4}-\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}{16 d (a+a \sin (c+d x))^4}+\frac {33 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8 \tan \left (\frac {1}{2} (c+d x)\right )}{5 d (a+a \sin (c+d x))^4}+\frac {53 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8 \tan \left (\frac {1}{2} (c+d x)\right )}{160 d (a+a \sin (c+d x))^4}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8 \tan \left (\frac {1}{2} (c+d x)\right )}{160 d (a+a \sin (c+d x))^4} \end {gather*}

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 = 0.33, size = 165, normalized size = 1.24

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+222 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {512}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {11}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {48}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {222}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-432 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d \,a^{4}}\)	\(165\)
default	\(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+222 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {512}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {11}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {48}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {222}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-432 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d \,a^{4}}\)	\(165\)
risch	\(-\frac {135 i {\mathrm e}^{9 i \left (d x +c \right )}-630 \,{\mathrm e}^{8 i \left (d x +c \right )}+135 \,{\mathrm e}^{10 i \left (d x +c \right )}-610 i {\mathrm e}^{7 i \left (d x +c \right )}+1260 \,{\mathrm e}^{6 i \left (d x +c \right )}+860 i {\mathrm e}^{5 i \left (d x +c \right )}-1510 \,{\mathrm e}^{4 i \left (d x +c \right )}-430 i {\mathrm e}^{3 i \left (d x +c \right )}+925 \,{\mathrm e}^{2 i \left (d x +c \right )}+77 i {\mathrm e}^{i \left (d x +c \right )}-212}{5 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) d \,a^{4}}-\frac {27 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a^{4} d}+\frac {27 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a^{4} d}\)	\(194\)

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,method=_RETURNVERBOSE)

[Out]

1/32/d/a^4*(1/5*tan(1/2*d*x+1/2*c)^5-2*tan(1/2*d*x+1/2*c)^4+11*tan(1/2*d*x+1/2*c)^3-48*tan(1/2*d*x+1/2*c)^2+22
2*tan(1/2*d*x+1/2*c)-512/(tan(1/2*d*x+1/2*c)+1)-1/5/tan(1/2*d*x+1/2*c)^5+2/tan(1/2*d*x+1/2*c)^4-11/tan(1/2*d*x
+1/2*c)^3+48/tan(1/2*d*x+1/2*c)^2-222/tan(1/2*d*x+1/2*c)-432*ln(tan(1/2*d*x+1/2*c)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (136) = 272\).
time = 0.30, size = 279, normalized size = 2.10 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (136) = 272\).
time = 0.36, size = 439, normalized size = 3.30 \begin {gather*} \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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\cot ^{6}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \end {gather*}

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]

Integral(cot(c + d*x)**6/(sin(c + d*x)**4 + 4*sin(c + d*x)**3 + 6*sin(c + d*x)**2 + 4*sin(c + d*x) + 1), x)/a*
*4

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Giac [A]
time = 9.18, size = 204, normalized size = 1.53 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 7.67, size = 209, normalized size = 1.57 \begin {gather*} \frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,a^4\,d}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16\,a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a^4\,d}-\frac {27\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^4\,d}+\frac {111\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a^4\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {367\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {87\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16}-\frac {37\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{160}+\frac {1}{160}\right )}{a^4\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11*tan(c/2 + (d*x)/2)^3)/(32*a^4*d) - (3*tan(c/2 + (d*x)/2)^2)/(2*a^4*d) - tan(c/2 + (d*x)/2)^4/(16*a^4*d) +
tan(c/2 + (d*x)/2)^5/(160*a^4*d) - (27*log(tan(c/2 + (d*x)/2)))/(2*a^4*d) + (111*tan(c/2 + (d*x)/2))/(16*a^4*d
) - (cot(c/2 + (d*x)/2)^5*((9*tan(c/2 + (d*x)/2)^2)/32 - (9*tan(c/2 + (d*x)/2))/160 - (37*tan(c/2 + (d*x)/2)^3
)/32 + (87*tan(c/2 + (d*x)/2)^4)/16 + (367*tan(c/2 + (d*x)/2)^5)/16 + 1/160))/(a^4*d*(tan(c/2 + (d*x)/2) + 1))

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