3.1.42 \(\int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx\) [42]

Optimal. Leaf size=140 \[ -\frac {61 a^4 x}{8}+\frac {2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {5 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {19 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d} \]

[Out]

-61/8*a^4*x+2*a^4*arctanh(cos(d*x+c))/d+4/3*a^4*cos(d*x+c)^3/d-5*a^4*cot(d*x+c)/d-1/3*a^4*cot(d*x+c)^3/d-2*a^4
*cot(d*x+c)*csc(d*x+c)/d-19/8*a^4*cos(d*x+c)*sin(d*x+c)/d-1/4*a^4*cos(d*x+c)*sin(d*x+c)^3/d

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Rubi [A]
time = 0.16, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 17, 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, 2718, 2715, 2713} \begin {gather*} \frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {5 a^4 \cot (c+d x)}{d}-\frac {a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {19 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {61 a^4 x}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-61*a^4*x)/8 + (2*a^4*ArcTanh[Cos[c + d*x]])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) - (5*a^4*Cot[c + d*x])/d - (a^4
*Cot[c + d*x]^3)/(3*d) - (2*a^4*Cot[c + d*x]*Csc[c + d*x])/d - (19*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4
*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)

Rule 8

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

Rule 2713

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

Rule 2715

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

Rule 2718

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

Rubi steps

\begin {align*} \int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac {\int \left (-10 a^8-4 a^8 \csc (c+d x)+4 a^8 \csc ^2(c+d x)+4 a^8 \csc ^3(c+d x)+a^8 \csc ^4(c+d x)-4 a^8 \sin (c+d x)+4 a^8 \sin ^2(c+d x)+4 a^8 \sin ^3(c+d x)+a^8 \sin ^4(c+d x)\right ) \, dx}{a^4}\\ &=-10 a^4 x+a^4 \int \csc ^4(c+d x) \, dx+a^4 \int \sin ^4(c+d x) \, dx-\left (4 a^4\right ) \int \csc (c+d x) \, dx+\left (4 a^4\right ) \int \csc ^2(c+d x) \, dx+\left (4 a^4\right ) \int \csc ^3(c+d x) \, dx-\left (4 a^4\right ) \int \sin (c+d x) \, dx+\left (4 a^4\right ) \int \sin ^2(c+d x) \, dx+\left (4 a^4\right ) \int \sin ^3(c+d x) \, dx\\ &=-10 a^4 x+\frac {4 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {4 a^4 \cos (c+d x)}{d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {2 a^4 \cos (c+d x) \sin (c+d x)}{d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{4} \left (3 a^4\right ) \int \sin ^2(c+d x) \, dx+\left (2 a^4\right ) \int 1 \, dx+\left (2 a^4\right ) \int \csc (c+d x) \, dx-\frac {a^4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (4 a^4\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac {\left (4 a^4\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-8 a^4 x+\frac {2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {5 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {19 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{8} \left (3 a^4\right ) \int 1 \, dx\\ &=-\frac {61 a^4 x}{8}+\frac {2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {5 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {19 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(685\) vs. \(2(140)=280\).
time = 6.23, size = 685, normalized size = 4.89 \begin {gather*} -\frac {61 (c+d x) (a+a \sin (c+d x))^4}{8 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}+\frac {\cos (c+d x) (a+a \sin (c+d x))^4}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}+\frac {\cos (3 (c+d x)) (a+a \sin (c+d x))^4}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}-\frac {7 \cot \left (\frac {1}{2} (c+d x)\right ) (a+a \sin (c+d x))^4}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sin (c+d x))^4}{2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sin (c+d x))^4}{24 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}+\frac {2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a+a \sin (c+d x))^4}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}-\frac {2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+a \sin (c+d x))^4}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sin (c+d x))^4}{2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}-\frac {5 (a+a \sin (c+d x))^4 \sin (2 (c+d x))}{4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}+\frac {(a+a \sin (c+d x))^4 \sin (4 (c+d x))}{32 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}+\frac {7 (a+a \sin (c+d x))^4 \tan \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sin (c+d x))^4 \tan \left (\frac {1}{2} (c+d x)\right )}{24 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-61*(c + d*x)*(a + a*Sin[c + d*x])^4)/(8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8) + (Cos[c + d*x]*(a + a*Si
n[c + d*x])^4)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8) + (Cos[3*(c + d*x)]*(a + a*Sin[c + d*x])^4)/(3*d*(C
os[(c + d*x)/2] + Sin[(c + d*x)/2])^8) - (7*Cot[(c + d*x)/2]*(a + a*Sin[c + d*x])^4)/(3*d*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])^8) - (Csc[(c + d*x)/2]^2*(a + a*Sin[c + d*x])^4)/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^
8) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(a + a*Sin[c + d*x])^4)/(24*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^
8) + (2*Log[Cos[(c + d*x)/2]]*(a + a*Sin[c + d*x])^4)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8) - (2*Log[Sin
[(c + d*x)/2]]*(a + a*Sin[c + d*x])^4)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8) + (Sec[(c + d*x)/2]^2*(a +
a*Sin[c + d*x])^4)/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8) - (5*(a + a*Sin[c + d*x])^4*Sin[2*(c + d*x)])
/(4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8) + ((a + a*Sin[c + d*x])^4*Sin[4*(c + d*x)])/(32*d*(Cos[(c + d*x
)/2] + Sin[(c + d*x)/2])^8) + (7*(a + a*Sin[c + d*x])^4*Tan[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*
x)/2])^8) + (Sec[(c + d*x)/2]^2*(a + a*Sin[c + d*x])^4*Tan[(c + d*x)/2])/(24*d*(Cos[(c + d*x)/2] + Sin[(c + d*
x)/2])^8)

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Maple [A]
time = 0.23, size = 222, normalized size = 1.59

method	result	size
derivativedivides	\(\frac {a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+4 a^{4} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{4} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\)	\(222\)
default	\(\frac {a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+4 a^{4} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{4} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\)	\(222\)
risch	\(-\frac {61 a^{4} x}{8}-\frac {i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}+\frac {5 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {5 i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {i a^{4} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}+\frac {4 a^{4} \left (-6 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}+15 i {\mathrm e}^{2 i \left (d x +c \right )}-7 i-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {2 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {2 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a^{4} \cos \left (3 d x +3 c \right )}{3 d}\)	\(241\)

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/d*(a^4*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4*a^4*(1/3*cos(d*x+c)^3+cos(d*x+c)+ln(cs
c(d*x+c)-cot(d*x+c)))+6*a^4*(-1/sin(d*x+c)*cos(d*x+c)^5-(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)-3/2*d*x-3/2*c
)+4*a^4*(-1/2/sin(d*x+c)^2*cos(d*x+c)^5-1/2*cos(d*x+c)^3-3/2*cos(d*x+c)-3/2*ln(csc(d*x+c)-cot(d*x+c)))+a^4*(-1
/3*cot(d*x+c)^3+cot(d*x+c)+d*x+c))

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Maxima [A]
time = 0.50, size = 218, normalized size = 1.56 \begin {gather*} \frac {64 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 288 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{4} + 32 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{4} + 96 \, a^{4} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{96 \, 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]

1/96*(64*(2*cos(d*x + c)^3 + 6*cos(d*x + c) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1))*a^4 + 3*(12*d
*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^4 - 288*(3*d*x + 3*c + (3*tan(d*x + c)^2 + 2)/(tan(d*x +
c)^3 + tan(d*x + c)))*a^4 + 32*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^4 + 96*a^4*(2*cos(d*x +
c)/(cos(d*x + c)^2 - 1) - 4*cos(d*x + c) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)))/d

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Fricas [A]
time = 0.39, size = 219, normalized size = 1.56 \begin {gather*} -\frac {6 \, a^{4} \cos \left (d x + c\right )^{7} - 75 \, a^{4} \cos \left (d x + c\right )^{5} + 244 \, a^{4} \cos \left (d x + c\right )^{3} - 183 \, a^{4} \cos \left (d x + c\right ) - 24 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 24 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - {\left (32 \, a^{4} \cos \left (d x + c\right )^{5} - 183 \, a^{4} d x \cos \left (d x + c\right )^{2} - 32 \, a^{4} \cos \left (d x + c\right )^{3} + 183 \, a^{4} d x + 48 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\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]

-1/24*(6*a^4*cos(d*x + c)^7 - 75*a^4*cos(d*x + c)^5 + 244*a^4*cos(d*x + c)^3 - 183*a^4*cos(d*x + c) - 24*(a^4*
cos(d*x + c)^2 - a^4)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 24*(a^4*cos(d*x + c)^2 - a^4)*log(-1/2*cos(d*
x + c) + 1/2)*sin(d*x + c) - (32*a^4*cos(d*x + c)^5 - 183*a^4*d*x*cos(d*x + c)^2 - 32*a^4*cos(d*x + c)^3 + 183
*a^4*d*x + 48*a^4*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (130) = 260\).
time = 12.88, size = 274, normalized size = 1.96 \begin {gather*} \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 183 \, {\left (d x + c\right )} a^{4} - 48 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {88 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {2 \, {\left (57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 81 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 96 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 81 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 57 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 32 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{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="giac")

[Out]

1/24*(a^4*tan(1/2*d*x + 1/2*c)^3 + 12*a^4*tan(1/2*d*x + 1/2*c)^2 - 183*(d*x + c)*a^4 - 48*a^4*log(abs(tan(1/2*
d*x + 1/2*c))) + 57*a^4*tan(1/2*d*x + 1/2*c) + (88*a^4*tan(1/2*d*x + 1/2*c)^3 - 57*a^4*tan(1/2*d*x + 1/2*c)^2
- 12*a^4*tan(1/2*d*x + 1/2*c) - a^4)/tan(1/2*d*x + 1/2*c)^3 + 2*(57*a^4*tan(1/2*d*x + 1/2*c)^7 + 96*a^4*tan(1/
2*d*x + 1/2*c)^6 + 81*a^4*tan(1/2*d*x + 1/2*c)^5 + 96*a^4*tan(1/2*d*x + 1/2*c)^4 - 81*a^4*tan(1/2*d*x + 1/2*c)
^3 + 32*a^4*tan(1/2*d*x + 1/2*c)^2 - 57*a^4*tan(1/2*d*x + 1/2*c) + 32*a^4)/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d

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Mupad [B]
time = 6.71, size = 384, normalized size = 2.74 \begin {gather*} \frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d}+\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {2\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {61\,a^4\,\mathrm {atan}\left (\frac {3721\,a^8}{16\,\left (61\,a^8-\frac {3721\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {61\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{61\,a^8-\frac {3721\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}}\right )}{4\,d}+\frac {19\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-19\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-60\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {67\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-48\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {508\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {8\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+116\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {61\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+4\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \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]

(a^4*tan(c/2 + (d*x)/2)^2)/(2*d) + (a^4*tan(c/2 + (d*x)/2)^3)/(24*d) - (2*a^4*log(tan(c/2 + (d*x)/2)))/d - (61
*a^4*atan((3721*a^8)/(16*(61*a^8 - (3721*a^8*tan(c/2 + (d*x)/2))/16)) + (61*a^8*tan(c/2 + (d*x)/2))/(61*a^8 -
(3721*a^8*tan(c/2 + (d*x)/2))/16)))/(4*d) + (19*a^4*tan(c/2 + (d*x)/2))/(8*d) - ((61*a^4*tan(c/2 + (d*x)/2)^2)
/3 - (16*a^4*tan(c/2 + (d*x)/2)^3)/3 + 116*a^4*tan(c/2 + (d*x)/2)^4 + (8*a^4*tan(c/2 + (d*x)/2)^5)/3 + (508*a^
4*tan(c/2 + (d*x)/2)^6)/3 - 48*a^4*tan(c/2 + (d*x)/2)^7 + (67*a^4*tan(c/2 + (d*x)/2)^8)/3 - 60*a^4*tan(c/2 + (
d*x)/2)^9 - 19*a^4*tan(c/2 + (d*x)/2)^10 + a^4/3 + 4*a^4*tan(c/2 + (d*x)/2))/(d*(8*tan(c/2 + (d*x)/2)^3 + 32*t
an(c/2 + (d*x)/2)^5 + 48*tan(c/2 + (d*x)/2)^7 + 32*tan(c/2 + (d*x)/2)^9 + 8*tan(c/2 + (d*x)/2)^11))

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