3.748 \(\int \frac{\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.363856, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2875, 2873, 2607, 30, 2611, 3768, 3770, 14} \[ \frac{3 \cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{7 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac{17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}+\frac{7 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^3,x]

[Out]

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

Rule 2875

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[((g*Cos[e + f*x])^(2*m + 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] && ILtQ[m, 0]

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

Mathematica [A]  time = 0.971315, size = 242, normalized size = 1.95 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6 \left (704 \tan \left (\frac{1}{2} (c+d x)\right )-704 \cot \left (\frac{1}{2} (c+d x)\right )+210 \csc ^2\left (\frac{1}{2} (c+d x)\right )+5 \sec ^6\left (\frac{1}{2} (c+d x)\right )+90 \sec ^4\left (\frac{1}{2} (c+d x)\right )-210 \sec ^2\left (\frac{1}{2} (c+d x)\right )-840 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+840 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-544 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)+(18 \sin (c+d x)-5) \csc ^6\left (\frac{1}{2} (c+d x)\right )+(34 \sin (c+d x)-90) \csc ^4\left (\frac{1}{2} (c+d x)\right )-36 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )\right )}{1920 a^3 d (\sin (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^3,x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(-704*Cot[(c + d*x)/2] + 210*Csc[(c + d*x)/2]^2 + 840*Log[Cos[(c + d*
x)/2]] - 840*Log[Sin[(c + d*x)/2]] - 210*Sec[(c + d*x)/2]^2 + 90*Sec[(c + d*x)/2]^4 + 5*Sec[(c + d*x)/2]^6 - 5
44*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*(-90 + 3
4*Sin[c + d*x]) + 704*Tan[(c + d*x)/2] - 36*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(1920*a^3*d*(1 + Sin[c + d*x
])^3)

________________________________________________________________________________________

Maple [B]  time = 0.187, size = 246, normalized size = 2. \begin{align*}{\frac{1}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}-{\frac{3}{160\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{7}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{7}{96\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{1}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{5}{16\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{5}{16\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{3}{160\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}-{\frac{7}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}-{\frac{7}{16\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}+{\frac{7}{96\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^3,x)

[Out]

1/384/d/a^3*tan(1/2*d*x+1/2*c)^6-3/160/d/a^3*tan(1/2*d*x+1/2*c)^5+7/128/d/a^3*tan(1/2*d*x+1/2*c)^4-7/96/d/a^3*
tan(1/2*d*x+1/2*c)^3-1/128/d/a^3*tan(1/2*d*x+1/2*c)^2+5/16/d/a^3*tan(1/2*d*x+1/2*c)-5/16/d/a^3/tan(1/2*d*x+1/2
*c)+3/160/d/a^3/tan(1/2*d*x+1/2*c)^5-7/128/d/a^3/tan(1/2*d*x+1/2*c)^4-7/16/d/a^3*ln(tan(1/2*d*x+1/2*c))-1/384/
d/a^3/tan(1/2*d*x+1/2*c)^6+7/96/d/a^3/tan(1/2*d*x+1/2*c)^3+1/128/d/a^3/tan(1/2*d*x+1/2*c)^2

________________________________________________________________________________________

Maxima [B]  time = 1.06775, size = 370, normalized size = 2.98 \begin{align*} \frac{\frac{\frac{600 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{36 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a^{3}} - \frac{840 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac{{\left (\frac{36 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{105 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{600 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{3} \sin \left (d x + c\right )^{6}}}{1920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/1920*((600*sin(d*x + c)/(cos(d*x + c) + 1) - 15*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 140*sin(d*x + c)^3/(co
s(d*x + c) + 1)^3 + 105*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 36*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d
*x + c)^6/(cos(d*x + c) + 1)^6)/a^3 - 840*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 + (36*sin(d*x + c)/(cos(d*x
+ c) + 1) - 105*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 140*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x +
c)^4/(cos(d*x + c) + 1)^4 - 600*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5)*(cos(d*x + c) + 1)^6/(a^3*sin(d*x + c
)^6))/d

________________________________________________________________________________________

Fricas [A]  time = 1.13023, size = 532, normalized size = 4.29 \begin{align*} -\frac{210 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 105 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 105 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 32 \,{\left (11 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) - 210 \, \cos \left (d x + c\right )}{480 \,{\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/480*(210*cos(d*x + c)^5 - 80*cos(d*x + c)^3 - 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1
)*log(1/2*cos(d*x + c) + 1/2) + 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*
x + c) + 1/2) - 32*(11*cos(d*x + c)^5 - 20*cos(d*x + c)^3)*sin(d*x + c) - 210*cos(d*x + c))/(a^3*d*cos(d*x + c
)^6 - 3*a^3*d*cos(d*x + c)^4 + 3*a^3*d*cos(d*x + c)^2 - a^3*d)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.34516, size = 292, normalized size = 2.35 \begin{align*} -\frac{\frac{840 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac{2058 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 600 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 140 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 36 \, \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}} - \frac{5 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 36 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 105 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 140 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 600 \, a^{15} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{18}}}{1920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/1920*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (2058*tan(1/2*d*x + 1/2*c)^6 - 600*tan(1/2*d*x + 1/2*c)^5 +
15*tan(1/2*d*x + 1/2*c)^4 + 140*tan(1/2*d*x + 1/2*c)^3 - 105*tan(1/2*d*x + 1/2*c)^2 + 36*tan(1/2*d*x + 1/2*c)
- 5)/(a^3*tan(1/2*d*x + 1/2*c)^6) - (5*a^15*tan(1/2*d*x + 1/2*c)^6 - 36*a^15*tan(1/2*d*x + 1/2*c)^5 + 105*a^15
*tan(1/2*d*x + 1/2*c)^4 - 140*a^15*tan(1/2*d*x + 1/2*c)^3 - 15*a^15*tan(1/2*d*x + 1/2*c)^2 + 600*a^15*tan(1/2*
d*x + 1/2*c))/a^18)/d