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

Optimal. Leaf size=150 \[ -\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}-\frac{a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{16 d} \]

[Out]

(-9*a^3*ArcTanh[Cos[c + d*x]])/(16*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) + (3*a^3*Cot
[c + d*x]*Csc[c + d*x])/(16*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)
/(8*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(2*d)

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Rubi [A]  time = 0.284279, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2611, 3770, 2607, 30, 3768, 14} \[ -\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}-\frac{a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{16 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*a^3*ArcTanh[Cos[c + d*x]])/(16*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) + (3*a^3*Cot
[c + d*x]*Csc[c + d*x])/(16*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)
/(8*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(2*d)

Rule 2873

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

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3770

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

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^4(c+d x) \csc (c+d x)+3 a^3 \cot ^4(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^3(c+d x)+a^3 \cot ^4(c+d x) \csc ^4(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \csc (c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx\\ &=-\frac{a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}-\frac{1}{4} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx-\frac{1}{2} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac{1}{8} \left (3 a^3\right ) \int \csc (c+d x) \, dx+\frac{1}{8} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac{1}{16} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^7(c+d x)}{7 d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}\\ \end{align*}

Mathematica [B]  time = 0.131935, size = 363, normalized size = 2.42 \[ a^3 \left (\frac{23 \tan \left (\frac{1}{2} (c+d x)\right )}{70 d}-\frac{23 \cot \left (\frac{1}{2} (c+d x)\right )}{70 d}-\frac{\csc ^6\left (\frac{1}{2} (c+d x)\right )}{128 d}+\frac{\csc ^4\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{7 \csc ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{\sec ^6\left (\frac{1}{2} (c+d x)\right )}{128 d}-\frac{\sec ^4\left (\frac{1}{2} (c+d x)\right )}{32 d}-\frac{7 \sec ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{9 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}-\frac{9 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}-\frac{\cot \left (\frac{1}{2} (c+d x)\right ) \csc ^6\left (\frac{1}{2} (c+d x)\right )}{896 d}-\frac{31 \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^4\left (\frac{1}{2} (c+d x)\right )}{2240 d}+\frac{297 \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )}{2240 d}+\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^6\left (\frac{1}{2} (c+d x)\right )}{896 d}+\frac{31 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )}{2240 d}-\frac{297 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{2240 d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*((-23*Cot[(c + d*x)/2])/(70*d) + (7*Csc[(c + d*x)/2]^2)/(64*d) + (297*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)
/(2240*d) + Csc[(c + d*x)/2]^4/(32*d) - (31*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(2240*d) - Csc[(c + d*x)/2]^6
/(128*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^6)/(896*d) - (9*Log[Cos[(c + d*x)/2]])/(16*d) + (9*Log[Sin[(c +
d*x)/2]])/(16*d) - (7*Sec[(c + d*x)/2]^2)/(64*d) - Sec[(c + d*x)/2]^4/(32*d) + Sec[(c + d*x)/2]^6/(128*d) + (2
3*Tan[(c + d*x)/2])/(70*d) - (297*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(2240*d) + (31*Sec[(c + d*x)/2]^4*Tan[(
c + d*x)/2])/(2240*d) + (Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2])/(896*d))

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Maple [A]  time = 0.089, size = 176, normalized size = 1.2 \begin{align*} -{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{16\,d}}+{\frac{9\,{a}^{3}\cos \left ( dx+c \right ) }{16\,d}}+{\frac{9\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{23\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x)

[Out]

-3/8/d*a^3/sin(d*x+c)^4*cos(d*x+c)^5+3/16/d*a^3/sin(d*x+c)^2*cos(d*x+c)^5+3/16*a^3*cos(d*x+c)^3/d+9/16*a^3*cos
(d*x+c)/d+9/16/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-23/35/d*a^3/sin(d*x+c)^5*cos(d*x+c)^5-1/2/d*a^3/sin(d*x+c)^6*co
s(d*x+c)^5-1/7/d*a^3/sin(d*x+c)^7*cos(d*x+c)^5

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Maxima [A]  time = 1.00123, size = 278, normalized size = 1.85 \begin{align*} \frac{35 \, a^{3}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \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 )} - 70 \, 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 )} - \frac{672 \, a^{3}}{\tan \left (d x + c\right )^{5}} - \frac{32 \,{\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}}}{1120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/1120*(35*a^3*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 +
3*cos(d*x + c)^2 - 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 70*a^3*(2*(5*cos(d*x + c)^3 - 3*c
os(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)) - 67
2*a^3/tan(d*x + c)^5 - 32*(7*tan(d*x + c)^2 + 5)*a^3/tan(d*x + c)^7)/d

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Fricas [A]  time = 1.34851, size = 629, normalized size = 4.19 \begin{align*} -\frac{736 \, a^{3} \cos \left (d x + c\right )^{7} - 896 \, a^{3} \cos \left (d x + c\right )^{5} + 315 \,{\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, 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 ) - 315 \,{\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, 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 ) + 70 \,{\left (7 \, a^{3} \cos \left (d x + c\right )^{5} - 24 \, a^{3} \cos \left (d x + c\right )^{3} + 9 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1120 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, 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)^8*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/1120*(736*a^3*cos(d*x + c)^7 - 896*a^3*cos(d*x + c)^5 + 315*(a^3*cos(d*x + c)^6 - 3*a^3*cos(d*x + c)^4 + 3*
a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 315*(a^3*cos(d*x + c)^6 - 3*a^3*cos(d*x +
c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 70*(7*a^3*cos(d*x + c)^5 - 24*
a^3*cos(d*x + c)^3 + 9*a^3*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*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)**8*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.80796, size = 352, normalized size = 2.35 \begin{align*} \frac{5 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 35 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 77 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 35 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 455 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 665 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 2520 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 945 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{6534 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 945 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 665 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 455 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 35 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 77 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 35 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{4480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/4480*(5*a^3*tan(1/2*d*x + 1/2*c)^7 + 35*a^3*tan(1/2*d*x + 1/2*c)^6 + 77*a^3*tan(1/2*d*x + 1/2*c)^5 - 35*a^3*
tan(1/2*d*x + 1/2*c)^4 - 455*a^3*tan(1/2*d*x + 1/2*c)^3 - 665*a^3*tan(1/2*d*x + 1/2*c)^2 + 2520*a^3*log(abs(ta
n(1/2*d*x + 1/2*c))) + 945*a^3*tan(1/2*d*x + 1/2*c) - (6534*a^3*tan(1/2*d*x + 1/2*c)^7 + 945*a^3*tan(1/2*d*x +
1/2*c)^6 - 665*a^3*tan(1/2*d*x + 1/2*c)^5 - 455*a^3*tan(1/2*d*x + 1/2*c)^4 - 35*a^3*tan(1/2*d*x + 1/2*c)^3 +
77*a^3*tan(1/2*d*x + 1/2*c)^2 + 35*a^3*tan(1/2*d*x + 1/2*c) + 5*a^3)/tan(1/2*d*x + 1/2*c)^7)/d