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

Optimal. Leaf size=168 \[ -\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{19 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac{3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+a^3 x \]

[Out]

a^3*x - (19*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot
[c + d*x]^5)/(5*d) + (17*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (
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)/(6*d)

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Rubi [A]  time = 0.274657, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2873, 3473, 8, 2611, 3770, 2607, 30, 3768} \[ -\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{19 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac{3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+a^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*x - (19*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot
[c + d*x]^5)/(5*d) + (17*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (
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)/(6*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 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

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

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]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^4(c+d x)+3 a^3 \cot ^4(c+d x) \csc (c+d x)+3 a^3 \cot ^4(c+d x) \csc ^2(c+d x)+a^3 \cot ^4(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{3 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)}{6 d}-\frac{1}{2} a^3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx-a^3 \int \cot ^2(c+d x) \, dx-\frac{1}{4} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{9 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{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)}{6 d}+\frac{1}{8} a^3 \int \csc ^3(c+d x) \, dx+a^3 \int 1 \, dx+\frac{1}{8} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=a^3 x-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{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)}{6 d}+\frac{1}{16} a^3 \int \csc (c+d x) \, dx\\ &=a^3 x-\frac{19 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{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)}{6 d}\\ \end{align*}

Mathematica [A]  time = 0.766006, size = 217, normalized size = 1.29 \[ \frac{a^3 \left (-704 \tan \left (\frac{1}{2} (c+d x)\right )+704 \cot \left (\frac{1}{2} (c+d x)\right )+870 \csc ^2\left (\frac{1}{2} (c+d x)\right )+5 \sec ^6\left (\frac{1}{2} (c+d x)\right )+60 \sec ^4\left (\frac{1}{2} (c+d x)\right )-870 \sec ^2\left (\frac{1}{2} (c+d x)\right )+2280 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2280 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-1376 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-(18 \sin (c+d x)+5) \csc ^6\left (\frac{1}{2} (c+d x)\right )+(86 \sin (c+d x)-60) \csc ^4\left (\frac{1}{2} (c+d x)\right )+36 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )+1920 c+1920 d x\right )}{1920 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1920*c + 1920*d*x + 704*Cot[(c + d*x)/2] + 870*Csc[(c + d*x)/2]^2 - 2280*Log[Cos[(c + d*x)/2]] + 2280*Lo
g[Sin[(c + d*x)/2]] - 870*Sec[(c + d*x)/2]^2 + 60*Sec[(c + d*x)/2]^4 + 5*Sec[(c + d*x)/2]^6 - 1376*Csc[c + d*x
]^3*Sin[(c + d*x)/2]^4 - Csc[(c + d*x)/2]^6*(5 + 18*Sin[c + d*x]) + Csc[(c + d*x)/2]^4*(-60 + 86*Sin[c + d*x])
- 704*Tan[(c + d*x)/2] + 36*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(1920*d)

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Maple [A]  time = 0.09, size = 194, normalized size = 1.2 \begin{align*} -{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}+{a}^{3}x+{\frac{{a}^{3}c}{d}}-{\frac{19\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{19\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{19\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{48\,d}}+{\frac{19\,{a}^{3}\cos \left ( dx+c \right ) }{16\,d}}+{\frac{19\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x)

[Out]

-1/3*a^3*cot(d*x+c)^3/d+a^3*cot(d*x+c)/d+a^3*x+1/d*a^3*c-19/24/d*a^3/sin(d*x+c)^4*cos(d*x+c)^5+19/48/d*a^3/sin
(d*x+c)^2*cos(d*x+c)^5+19/48*a^3*cos(d*x+c)^3/d+19/16*a^3*cos(d*x+c)/d+19/16/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-3
/5/d*a^3/sin(d*x+c)^5*cos(d*x+c)^5-1/6/d*a^3/sin(d*x+c)^6*cos(d*x+c)^5

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Maxima [A]  time = 1.69469, size = 290, normalized size = 1.73 \begin{align*} \frac{160 \,{\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} + 5 \, 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 )} - 90 \, 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{288 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/480*(160*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^3 + 5*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)) - 90*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)) - 288*a^3/tan(d*x + c)^5)/d

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Fricas [A]  time = 1.17591, size = 745, normalized size = 4.43 \begin{align*} \frac{480 \, a^{3} d x \cos \left (d x + c\right )^{6} - 1440 \, a^{3} d x \cos \left (d x + c\right )^{4} - 870 \, a^{3} \cos \left (d x + c\right )^{5} + 1440 \, a^{3} d x \cos \left (d x + c\right )^{2} + 1520 \, a^{3} \cos \left (d x + c\right )^{3} - 480 \, a^{3} d x - 570 \, a^{3} \cos \left (d x + c\right ) - 285 \,{\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 ) + 285 \,{\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 ) - 32 \,{\left (11 \, a^{3} \cos \left (d x + c\right )^{5} - 35 \, a^{3} \cos \left (d x + c\right )^{3} + 15 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/480*(480*a^3*d*x*cos(d*x + c)^6 - 1440*a^3*d*x*cos(d*x + c)^4 - 870*a^3*cos(d*x + c)^5 + 1440*a^3*d*x*cos(d*
x + c)^2 + 1520*a^3*cos(d*x + c)^3 - 480*a^3*d*x - 570*a^3*cos(d*x + c) - 285*(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) + 285*(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) - 32*(11*a^3*cos(d*x + c)^5 - 35*a^3*cos(d*
x + c)^3 + 15*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)

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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)**7*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.51949, size = 323, normalized size = 1.92 \begin{align*} \frac{5 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 36 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 75 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 100 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 735 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1920 \,{\left (d x + c\right )} a^{3} + 2280 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 840 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{5586 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 840 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 735 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 100 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 75 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 36 \, 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 )^{6}}}{1920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/1920*(5*a^3*tan(1/2*d*x + 1/2*c)^6 + 36*a^3*tan(1/2*d*x + 1/2*c)^5 + 75*a^3*tan(1/2*d*x + 1/2*c)^4 - 100*a^3
*tan(1/2*d*x + 1/2*c)^3 - 735*a^3*tan(1/2*d*x + 1/2*c)^2 + 1920*(d*x + c)*a^3 + 2280*a^3*log(abs(tan(1/2*d*x +
1/2*c))) - 840*a^3*tan(1/2*d*x + 1/2*c) - (5586*a^3*tan(1/2*d*x + 1/2*c)^6 - 840*a^3*tan(1/2*d*x + 1/2*c)^5 -
735*a^3*tan(1/2*d*x + 1/2*c)^4 - 100*a^3*tan(1/2*d*x + 1/2*c)^3 + 75*a^3*tan(1/2*d*x + 1/2*c)^2 + 36*a^3*tan(
1/2*d*x + 1/2*c) + 5*a^3)/tan(1/2*d*x + 1/2*c)^6)/d