3.293 \(\int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=124 \[ -\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{7 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{7 a^3 \cot (c+d x) \csc (c+d x)}{16 d} \]

[Out]

(7*a^3*ArcTanh[Cos[c + d*x]])/(16*d) - (4*a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (7*a^3*Co
t[c + d*x]*Csc[c + d*x])/(16*d) - (17*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]
^5)/(6*d)

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Rubi [A]  time = 0.281918, antiderivative size = 124, 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, 2607, 30, 2611, 3768, 3770, 14} \[ -\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{7 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{7 a^3 \cot (c+d x) \csc (c+d x)}{16 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

(7*a^3*ArcTanh[Cos[c + d*x]])/(16*d) - (4*a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (7*a^3*Co
t[c + d*x]*Csc[c + d*x])/(16*d) - (17*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]
^5)/(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 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 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 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 3770

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

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 ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^2(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^4(c+d x)+a^3 \cot ^2(c+d x) \csc ^5(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx+a^3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{1}{6} a^3 \int \csc ^5(c+d x) \, dx-\frac{1}{4} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{1}{8} a^3 \int \csc ^3(c+d x) \, dx-\frac{1}{8} \left (3 a^3\right ) \int \csc (c+d x) \, dx+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (x^2+x^4\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 ^3(c+d x)}{3 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{7 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{1}{16} a^3 \int \csc (c+d x) \, dx\\ &=\frac{7 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{4 a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{7 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}\\ \end{align*}

Mathematica [B]  time = 3.53536, size = 252, normalized size = 2.03 \[ -\frac{a^3 \sin (c+d x) (\sin (c+d x)+1)^3 \left (\csc ^6\left (\frac{1}{2} (c+d x)\right ) (5 \csc (c+d x)+18)+\csc ^4\left (\frac{1}{2} (c+d x)\right ) (90 \csc (c+d x)+34)-2 \csc ^2\left (\frac{1}{2} (c+d x)\right ) (105 \csc (c+d x)+176)+(159 \cos (c+d x)+44 \cos (2 (c+d x))+97) \sec ^6\left (\frac{1}{2} (c+d x)\right )-320 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^7(c+d x)-1440 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)+840 \sin ^2\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-840 \csc (c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{1920 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

-(a^3*(Csc[(c + d*x)/2]^6*(18 + 5*Csc[c + d*x]) + Csc[(c + d*x)/2]^4*(34 + 90*Csc[c + d*x]) - 2*Csc[(c + d*x)/
2]^2*(176 + 105*Csc[c + d*x]) - 840*Csc[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + (97 + 159*C
os[c + d*x] + 44*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^6 + 840*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 - 1440*Csc[c + d
*x]^5*Sin[(c + d*x)/2]^4 - 320*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6)*Sin[c + d*x]*(1 + Sin[c + d*x])^3)/(1920*d*(
Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

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Maple [A]  time = 0.086, size = 160, normalized size = 1.3 \begin{align*} -{\frac{11\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{15\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{7\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{7\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{7\,{a}^{3}\cos \left ( dx+c \right ) }{16\,d}}-{\frac{7\,{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 ) ^{3}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{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)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x)

[Out]

-11/15/d*a^3/sin(d*x+c)^3*cos(d*x+c)^3-7/8/d*a^3/sin(d*x+c)^4*cos(d*x+c)^3-7/16/d*a^3/sin(d*x+c)^2*cos(d*x+c)^
3-7/16*a^3*cos(d*x+c)/d-7/16/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)^3-1/6/d*a^3/sin
(d*x+c)^6*cos(d*x+c)^3

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Maxima [A]  time = 1.09389, size = 270, normalized size = 2.18 \begin{align*} -\frac{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 (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{160 \, a^{3}}{\tan \left (d x + c\right )^{3}} + \frac{96 \,{\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} 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)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/480*(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*(cos(d*x + c)^3 + cos(d*
x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) + 160*a^3/tan
(d*x + c)^3 + 96*(5*tan(d*x + c)^2 + 3)*a^3/tan(d*x + c)^5)/d

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Fricas [B]  time = 1.81423, size = 575, normalized size = 4.64 \begin{align*} -\frac{210 \, a^{3} \cos \left (d x + c\right )^{5} - 80 \, a^{3} \cos \left (d x + c\right )^{3} - 210 \, a^{3} \cos \left (d x + c\right ) - 105 \,{\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 ) + 105 \,{\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} - 20 \, a^{3} \cos \left (d x + c\right )^{3}\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)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/480*(210*a^3*cos(d*x + c)^5 - 80*a^3*cos(d*x + c)^3 - 210*a^3*cos(d*x + c) - 105*(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) + 105*(a^3*cos(d*x + c)^6 - 3*a^3*c
os(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 - 20*a^3*
cos(d*x + c)^3)*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)**2*csc(d*x+c)**7*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.42961, size = 308, normalized size = 2.48 \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} + 105 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 140 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 840 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{2058 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 600 \, 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} - 140 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 105 \, 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)^2*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 + 105*a^3*tan(1/2*d*x + 1/2*c)^4 + 140*a^
3*tan(1/2*d*x + 1/2*c)^3 - 15*a^3*tan(1/2*d*x + 1/2*c)^2 - 840*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 600*a^3*ta
n(1/2*d*x + 1/2*c) + (2058*a^3*tan(1/2*d*x + 1/2*c)^6 + 600*a^3*tan(1/2*d*x + 1/2*c)^5 + 15*a^3*tan(1/2*d*x +
1/2*c)^4 - 140*a^3*tan(1/2*d*x + 1/2*c)^3 - 105*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