3.1.89 \(\int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx\) [89]

Optimal. Leaf size=120 \[ \frac {14 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {9 \cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {2 \cot (c+d x) \csc (c+d x)}{a^4 d}+\frac {4 \cot (c+d x)}{3 a^4 d (1+\csc (c+d x))^2}-\frac {44 \cot (c+d x)}{3 a^4 d (1+\csc (c+d x))} \]

[Out]

14*arctanh(cos(d*x+c))/a^4/d-33*cot(d*x+c)/a^4/d-11*cot(d*x+c)^3/a^4/d+14*cot(d*x+c)*csc(d*x+c)/a^4/d+4/3*cot(
d*x+c)*csc(d*x+c)^2/a^4/d/(1+sin(d*x+c))^2+28/3*cot(d*x+c)*csc(d*x+c)^2/a^4/d/(1+sin(d*x+c))

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Rubi [A]
time = 0.19, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2788, 3855, 3852, 8, 3853, 3862, 4004, 3879} \begin {gather*} -\frac {\cot ^3(c+d x)}{3 a^4 d}-\frac {9 \cot (c+d x)}{a^4 d}+\frac {14 \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac {2 \cot (c+d x) \csc (c+d x)}{a^4 d}-\frac {44 \cot (c+d x)}{3 a^4 d (\csc (c+d x)+1)}+\frac {4 \cot (c+d x)}{3 a^4 d (\csc (c+d x)+1)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(14*ArcTanh[Cos[c + d*x]])/(a^4*d) - (9*Cot[c + d*x])/(a^4*d) - Cot[c + d*x]^3/(3*a^4*d) + (2*Cot[c + d*x]*Csc
[c + d*x])/(a^4*d) + (4*Cot[c + d*x])/(3*a^4*d*(1 + Csc[c + d*x])^2) - (44*Cot[c + d*x])/(3*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]

Rule 3879

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rubi steps

\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\int \left (16-12 \csc (c+d x)+8 \csc ^2(c+d x)-4 \csc ^3(c+d x)+\csc ^4(c+d x)+\frac {4}{(1+\csc (c+d x))^2}-\frac {20}{1+\csc (c+d x)}\right ) \, dx}{a^4}\\ &=\frac {16 x}{a^4}+\frac {\int \csc ^4(c+d x) \, dx}{a^4}-\frac {4 \int \csc ^3(c+d x) \, dx}{a^4}+\frac {4 \int \frac {1}{(1+\csc (c+d x))^2} \, dx}{a^4}+\frac {8 \int \csc ^2(c+d x) \, dx}{a^4}-\frac {12 \int \csc (c+d x) \, dx}{a^4}-\frac {20 \int \frac {1}{1+\csc (c+d x)} \, dx}{a^4}\\ &=\frac {16 x}{a^4}+\frac {12 \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac {2 \cot (c+d x) \csc (c+d x)}{a^4 d}+\frac {4 \cot (c+d x)}{3 a^4 d (1+\csc (c+d x))^2}-\frac {20 \cot (c+d x)}{a^4 d (1+\csc (c+d x))}-\frac {4 \int \frac {-3+\csc (c+d x)}{1+\csc (c+d x)} \, dx}{3 a^4}-\frac {2 \int \csc (c+d x) \, dx}{a^4}+\frac {20 \int -1 \, dx}{a^4}-\frac {\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 {14 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {9 \cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {2 \cot (c+d x) \csc (c+d x)}{a^4 d}+\frac {4 \cot (c+d x)}{3 a^4 d (1+\csc (c+d x))^2}-\frac {20 \cot (c+d x)}{a^4 d (1+\csc (c+d x))}-\frac {16 \int \frac {\csc (c+d x)}{1+\csc (c+d x)} \, dx}{3 a^4}\\ &=\frac {14 \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {9 \cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {2 \cot (c+d x) \csc (c+d x)}{a^4 d}+\frac {4 \cot (c+d x)}{3 a^4 d (1+\csc (c+d x))^2}-\frac {44 \cot (c+d x)}{3 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. \(359\) vs. \(2(120)=240\).
time = 4.61, size = 359, normalized size = 2.99 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 \left (-\cos \left (\frac {1}{2} (c+d x)\right ) \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right )^3+64 \sin \left (\frac {1}{2} (c+d x)\right )+12 \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right )^3 \sin \left (\frac {1}{2} (c+d x)\right )-32 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+640 \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 )^2-104 \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 )^3+336 \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 )^3-336 \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 )^3+104 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \tan \left (\frac {1}{2} (c+d x)\right )-12 \cos \left (\frac {1}{2} (c+d x)\right ) \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^3+\sin \left (\frac {1}{2} (c+d x)\right ) \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^3\right )}{24 a^4 d (1+\sin (c+d x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*(-(Cos[(c + d*x)/2]*(1 + Cot[(c + d*x)/2])^3) + 64*Sin[(c + d*x)/2] +
12*(1 + Cot[(c + d*x)/2])^3*Sin[(c + d*x)/2] - 32*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 640*Sin[(c + d*x)/2
]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 104*Cot[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 336
*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 336*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])^3 + 104*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*Tan[(c + d*x)/2] - 12*Cos[(c + d*x)/2]*(1 +
Tan[(c + d*x)/2])^3 + Sin[(c + d*x)/2]*(1 + Tan[(c + d*x)/2])^3))/(24*a^4*d*(1 + Sin[c + d*x])^4)

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Maple [A]
time = 0.29, size = 143, normalized size = 1.19

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {128}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {64}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {256}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {35}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-112 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(143\)
default	\(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {128}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {64}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {256}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {35}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-112 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(143\)
risch	\(-\frac {4 \left (-119 \,{\mathrm e}^{6 i \left (d x +c \right )}+63 i {\mathrm e}^{7 i \left (d x +c \right )}+204 \,{\mathrm e}^{4 i \left (d x +c \right )}-192 i {\mathrm e}^{5 i \left (d x +c \right )}+21 \,{\mathrm e}^{8 i \left (d x +c \right )}-135 \,{\mathrm e}^{2 i \left (d x +c \right )}+211 i {\mathrm e}^{3 i \left (d x +c \right )}+33-78 i {\mathrm e}^{i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} d \,a^{4}}+\frac {14 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{4} d}-\frac {14 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{4} d}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/8/d/a^4*(1/3*tan(1/2*d*x+1/2*c)^3-4*tan(1/2*d*x+1/2*c)^2+35*tan(1/2*d*x+1/2*c)-128/3/(tan(1/2*d*x+1/2*c)+1)^
3+64/(tan(1/2*d*x+1/2*c)+1)^2-256/(tan(1/2*d*x+1/2*c)+1)-1/3/tan(1/2*d*x+1/2*c)^3+4/tan(1/2*d*x+1/2*c)^2-35/ta
n(1/2*d*x+1/2*c)-112*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. 285 vs. \(2 (130) = 260\).
time = 0.29, size = 285, normalized size = 2.38 \begin {gather*} \frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {72 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {984 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1647 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {873 \, \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 )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3 \, 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 {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{4}} - \frac {336 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((9*sin(d*x + c)/(cos(d*x + c) + 1) - 72*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 984*sin(d*x + c)^3/(cos(d*
x + c) + 1)^3 - 1647*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 873*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1)/(a^4*s
in(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*a^4*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 3*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) + (105*sin(d*x + c)/(cos(d*x + c) + 1) - 12*sin(d*x +
c)^2/(cos(d*x + c) + 1)^2 + sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^4 - 336*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. 445 vs. \(2 (130) = 260\).
time = 0.37, size = 445, normalized size = 3.71 \begin {gather*} -\frac {66 \, \cos \left (d x + c\right )^{5} - 24 \, \cos \left (d x + c\right )^{4} - 147 \, \cos \left (d x + c\right )^{3} + 29 \, \cos \left (d x + c\right )^{2} - 21 \, {\left (\cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 2\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 21 \, {\left (\cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 2\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (66 \, \cos \left (d x + c\right )^{4} + 90 \, \cos \left (d x + c\right )^{3} - 57 \, \cos \left (d x + c\right )^{2} - 86 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) + 82 \, \cos \left (d x + c\right ) - 4}{3 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 2 \, a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{3} - 4 \, a^{4} d \cos \left (d x + c\right )^{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} - a^{4} d \cos \left (d x + c\right )^{3} - 3 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right ) + 2 \, 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)^4/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/3*(66*cos(d*x + c)^5 - 24*cos(d*x + c)^4 - 147*cos(d*x + c)^3 + 29*cos(d*x + c)^2 - 21*(cos(d*x + c)^5 + 2*
cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 4*cos(d*x + c)^2 + (cos(d*x + c)^4 - cos(d*x + c)^3 - 3*cos(d*x + c)^2 + c
os(d*x + c) + 2)*sin(d*x + c) + cos(d*x + c) + 2)*log(1/2*cos(d*x + c) + 1/2) + 21*(cos(d*x + c)^5 + 2*cos(d*x
+ c)^4 - 2*cos(d*x + c)^3 - 4*cos(d*x + c)^2 + (cos(d*x + c)^4 - cos(d*x + c)^3 - 3*cos(d*x + c)^2 + cos(d*x
+ c) + 2)*sin(d*x + c) + cos(d*x + c) + 2)*log(-1/2*cos(d*x + c) + 1/2) - (66*cos(d*x + c)^4 + 90*cos(d*x + c)
^3 - 57*cos(d*x + c)^2 - 86*cos(d*x + c) - 4)*sin(d*x + c) + 82*cos(d*x + c) - 4)/(a^4*d*cos(d*x + c)^5 + 2*a^
4*d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c)^3 - 4*a^4*d*cos(d*x + c)^2 + a^4*d*cos(d*x + c) + 2*a^4*d + (a^4*d*c
os(d*x + c)^4 - a^4*d*cos(d*x + c)^3 - 3*a^4*d*cos(d*x + c)^2 + a^4*d*cos(d*x + c) + 2*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 ^{4}{\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)**4/(a+a*sin(d*x+c))**4,x)

[Out]

Integral(cot(c + d*x)**4/(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 = 7.79, size = 179, normalized size = 1.49 \begin {gather*} -\frac {\frac {336 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {308 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 51 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 723 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 676 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3} a^{4}} - \frac {a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 105 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(336*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 - (308*tan(1/2*d*x + 1/2*c)^6 + 51*tan(1/2*d*x + 1/2*c)^5 - 723*
tan(1/2*d*x + 1/2*c)^4 - 676*tan(1/2*d*x + 1/2*c)^3 - 72*tan(1/2*d*x + 1/2*c)^2 + 9*tan(1/2*d*x + 1/2*c) - 1)/
((tan(1/2*d*x + 1/2*c)^2 + tan(1/2*d*x + 1/2*c))^3*a^4) - (a^8*tan(1/2*d*x + 1/2*c)^3 - 12*a^8*tan(1/2*d*x + 1
/2*c)^2 + 105*a^8*tan(1/2*d*x + 1/2*c))/a^12)/d

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Mupad [B]
time = 7.83, size = 171, normalized size = 1.42 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,a^4\,d}-\frac {14\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}+\frac {35\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^4\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {291\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8}+\frac {549\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{8}+41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {1}{24}\right )}{a^4\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(c/2 + (d*x)/2)^3/(24*a^4*d) - tan(c/2 + (d*x)/2)^2/(2*a^4*d) - (14*log(tan(c/2 + (d*x)/2)))/(a^4*d) + (35*
tan(c/2 + (d*x)/2))/(8*a^4*d) - (cot(c/2 + (d*x)/2)^3*(3*tan(c/2 + (d*x)/2)^2 - (3*tan(c/2 + (d*x)/2))/8 + 41*
tan(c/2 + (d*x)/2)^3 + (549*tan(c/2 + (d*x)/2)^4)/8 + (291*tan(c/2 + (d*x)/2)^5)/8 + 1/24))/(a^4*d*(tan(c/2 +
(d*x)/2) + 1)^3)

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