3.408 \(\int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx\)

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

[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)

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Rubi [A]  time = 0.228222, antiderivative size = 140, normalized size of antiderivative = 1., 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 = {2709, 3770, 3767, 8, 3768, 2638, 2635, 2633} \[ \frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{19 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{61 a^4 x}{8} \]

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 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 2638

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

Rule 2635

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] && Integer
Q[2*n]

Rule 2633

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]

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

Mathematica [A]  time = 5.29066, size = 209, normalized size = 1.49 \[ \frac{a^4 (\sin (c+d x)+1)^4 \left (-732 (c+d x)-120 \sin (2 (c+d x))+3 \sin (4 (c+d x))+96 \cos (c+d x)+32 \cos (3 (c+d x))+224 \tan \left (\frac{1}{2} (c+d x)\right )-224 \cot \left (\frac{1}{2} (c+d x)\right )-48 \csc ^2\left (\frac{1}{2} (c+d x)\right )+48 \sec ^2\left (\frac{1}{2} (c+d x)\right )-192 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+192 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+32 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-2 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )\right )}{96 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(1 + Sin[c + d*x])^4*(-732*(c + d*x) + 96*Cos[c + d*x] + 32*Cos[3*(c + d*x)] - 224*Cot[(c + d*x)/2] - 48*
Csc[(c + d*x)/2]^2 + 192*Log[Cos[(c + d*x)/2]] - 192*Log[Sin[(c + d*x)/2]] + 48*Sec[(c + d*x)/2]^2 + 32*Csc[c
+ d*x]^3*Sin[(c + d*x)/2]^4 - 2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 120*Sin[2*(c + d*x)] + 3*Sin[4*(c + d*x)] +
224*Tan[(c + d*x)/2]))/(96*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)

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Maple [A]  time = 0.091, size = 190, normalized size = 1.4 \begin{align*} -{\frac{23\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}-{\frac{69\,{a}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}-{\frac{61\,{a}^{4}x}{8}}-{\frac{61\,{a}^{4}c}{8\,d}}-{\frac{2\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-2\,{\frac{{a}^{4}\cos \left ( dx+c \right ) }{d}}-2\,{\frac{{a}^{4}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}-6\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d\sin \left ( dx+c \right ) }}-2\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{4}\cot \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x)

[Out]

-23/4*a^4*cos(d*x+c)^3*sin(d*x+c)/d-69/8*a^4*cos(d*x+c)*sin(d*x+c)/d-61/8*a^4*x-61/8/d*a^4*c-2/3*a^4*cos(d*x+c
)^3/d-2*a^4*cos(d*x+c)/d-2/d*a^4*ln(csc(d*x+c)-cot(d*x+c))-6/d*a^4/sin(d*x+c)*cos(d*x+c)^5-2/d*a^4/sin(d*x+c)^
2*cos(d*x+c)^5-1/3*a^4*cot(d*x+c)^3/d+a^4*cot(d*x+c)/d

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Maxima [A]  time = 1.64189, size = 294, normalized size = 2.1 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(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 = 1.26403, size = 560, normalized size = 4. \begin{align*} -\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(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(-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)**4*csc(d*x+c)**4*(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.50484, size = 370, normalized size = 2.64 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(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