3.401 \(\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=132 \[ -\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{d}+\frac{3 a^3 \cot (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+3 a^3 x \]

[Out]

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

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Rubi [A]  time = 0.216112, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2872, 3770, 3767, 8, 3768, 2638} \[ -\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{d}+\frac{3 a^3 \cot (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+3 a^3 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

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

Rule 2872

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 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]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\int \left (3 a^7+a^7 \csc (c+d x)-5 a^7 \csc ^2(c+d x)-5 a^7 \csc ^3(c+d x)+a^7 \csc ^4(c+d x)+3 a^7 \csc ^5(c+d x)+a^7 \csc ^6(c+d x)+a^7 \sin (c+d x)\right ) \, dx}{a^4}\\ &=3 a^3 x+a^3 \int \csc (c+d x) \, dx+a^3 \int \csc ^4(c+d x) \, dx+a^3 \int \csc ^6(c+d x) \, dx+a^3 \int \sin (c+d x) \, dx+\left (3 a^3\right ) \int \csc ^5(c+d x) \, dx-\left (5 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (5 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=3 a^3 x-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^3 \cos (c+d x)}{d}+\frac{5 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{1}{4} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{2} \left (5 a^3\right ) \int \csc (c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac{a^3 \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac{\left (5 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=3 a^3 x+\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cos (c+d x)}{d}+\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{1}{8} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=3 a^3 x+\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cos (c+d x)}{d}+\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.495615, size = 216, normalized size = 1.64 \[ \frac{a^3 \left (-320 \cos (c+d x)-608 \tan \left (\frac{1}{2} (c+d x)\right )+608 \cot \left (\frac{1}{2} (c+d x)\right )-15 \csc ^4\left (\frac{1}{2} (c+d x)\right )+110 \csc ^2\left (\frac{1}{2} (c+d x)\right )+15 \sec ^4\left (\frac{1}{2} (c+d x)\right )-110 \sec ^2\left (\frac{1}{2} (c+d x)\right )-120 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+120 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+64 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)+208 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-\sin (c+d x) \csc ^6\left (\frac{1}{2} (c+d x)\right )-13 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )+960 c+960 d x\right )}{320 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(960*c + 960*d*x - 320*Cos[c + d*x] + 608*Cot[(c + d*x)/2] + 110*Csc[(c + d*x)/2]^2 - 15*Csc[(c + d*x)/2]
^4 + 120*Log[Cos[(c + d*x)/2]] - 120*Log[Sin[(c + d*x)/2]] - 110*Sec[(c + d*x)/2]^2 + 15*Sec[(c + d*x)/2]^4 +
208*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 64*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - 13*Csc[(c + d*x)/2]^4*Sin[c + d
*x] - Csc[(c + d*x)/2]^6*Sin[c + d*x] - 608*Tan[(c + d*x)/2]))/(320*d)

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Maple [A]  time = 0.092, size = 173, normalized size = 1.3 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d}}-{\frac{3\,{a}^{3}\cos \left ( dx+c \right ) }{8\,d}}-{\frac{3\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}+3\,{a}^{3}x+3\,{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}+3\,{\frac{{a}^{3}c}{d}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.62311, size = 243, normalized size = 1.84 \begin{align*} \frac{80 \,{\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^{3} - 15 \, a^{3}{\left (\frac{2 \,{\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 20 \, a^{3}{\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 )} - \frac{16 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{80 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/80*(80*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^3 - 15*a^3*(2*(5*cos(d*x + c)^3 - 3*cos(d*x +
c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) + 20*a^3*(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)) - 16*a
^3/tan(d*x + c)^5)/d

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Fricas [B]  time = 1.24622, size = 655, normalized size = 4.96 \begin{align*} \frac{304 \, a^{3} \cos \left (d x + c\right )^{5} - 560 \, a^{3} \cos \left (d x + c\right )^{3} + 240 \, a^{3} \cos \left (d x + c\right ) + 15 \,{\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 15 \,{\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 10 \,{\left (24 \, a^{3} d x \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{5} - 48 \, a^{3} d x \cos \left (d x + c\right )^{2} + 5 \, a^{3} \cos \left (d x + c\right )^{3} + 24 \, a^{3} d x - 3 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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)^6*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/80*(304*a^3*cos(d*x + c)^5 - 560*a^3*cos(d*x + c)^3 + 240*a^3*cos(d*x + c) + 15*(a^3*cos(d*x + c)^4 - 2*a^3*
cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 15*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2
+ a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 10*(24*a^3*d*x*cos(d*x + c)^4 - 8*a^3*cos(d*x + c)^5 - 48*
a^3*d*x*cos(d*x + c)^2 + 5*a^3*cos(d*x + c)^3 + 24*a^3*d*x - 3*a^3*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c
)^4 - 2*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)**6*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.37016, size = 305, normalized size = 2.31 \begin{align*} \frac{2 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 30 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 80 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 960 \,{\left (d x + c\right )} a^{3} - 120 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 580 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{640 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + \frac{274 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 580 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 80 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 30 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{320 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/320*(2*a^3*tan(1/2*d*x + 1/2*c)^5 + 15*a^3*tan(1/2*d*x + 1/2*c)^4 + 30*a^3*tan(1/2*d*x + 1/2*c)^3 - 80*a^3*t
an(1/2*d*x + 1/2*c)^2 + 960*(d*x + c)*a^3 - 120*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 580*a^3*tan(1/2*d*x + 1/2
*c) - 640*a^3/(tan(1/2*d*x + 1/2*c)^2 + 1) + (274*a^3*tan(1/2*d*x + 1/2*c)^5 + 580*a^3*tan(1/2*d*x + 1/2*c)^4
+ 80*a^3*tan(1/2*d*x + 1/2*c)^3 - 30*a^3*tan(1/2*d*x + 1/2*c)^2 - 15*a^3*tan(1/2*d*x + 1/2*c) - 2*a^3)/tan(1/2
*d*x + 1/2*c)^5)/d