3.1251 \(\int \cot ^6(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=158 \[ -\frac{a^2 \cot ^7(c+d x)}{7 d}+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a b \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac{5 a b \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac{5 a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b^2 \cot ^5(c+d x)}{5 d}+\frac{b^2 \cot ^3(c+d x)}{3 d}-\frac{b^2 \cot (c+d x)}{d}-b^2 x \]

[Out]

-(b^2*x) + (5*a*b*ArcTanh[Cos[c + d*x]])/(8*d) - (b^2*Cot[c + d*x])/d + (b^2*Cot[c + d*x]^3)/(3*d) - (b^2*Cot[
c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x]^7)/(7*d) - (5*a*b*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (5*a*b*Cot[c + d*x]
^3*Csc[c + d*x])/(12*d) - (a*b*Cot[c + d*x]^5*Csc[c + d*x])/(3*d)

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Rubi [A]  time = 0.41435, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2911, 2611, 3770, 14, 203} \[ -\frac{a^2 \cot ^7(c+d x)}{7 d}+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a b \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac{5 a b \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac{5 a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b^2 \cot ^5(c+d x)}{5 d}+\frac{b^2 \cot ^3(c+d x)}{3 d}-\frac{b^2 \cot (c+d x)}{d}-b^2 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]

[Out]

-(b^2*x) + (5*a*b*ArcTanh[Cos[c + d*x]])/(8*d) - (b^2*Cot[c + d*x])/d + (b^2*Cot[c + d*x]^3)/(3*d) - (b^2*Cot[
c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x]^7)/(7*d) - (5*a*b*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (5*a*b*Cot[c + d*x]
^3*Csc[c + d*x])/(12*d) - (a*b*Cot[c + d*x]^5*Csc[c + d*x])/(3*d)

Rule 2911

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^2, x_Symbol] :> Dist[(2*a*b)/d, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e
+ f*x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 -
b^2, 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 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]]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \cot ^6(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cot ^6(c+d x) \csc (c+d x) \, dx+\int \cot ^6(c+d x) \csc ^2(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{a b \cot ^5(c+d x) \csc (c+d x)}{3 d}-\frac{1}{3} (5 a b) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac{\operatorname{Subst}\left (\int \frac{a^2+\frac{b^2 x^2}{1+x^2}}{x^8} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{5 a b \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac{a b \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac{1}{4} (5 a b) \int \cot ^2(c+d x) \csc (c+d x) \, dx+\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^8}+\frac{b^2}{x^6}-\frac{b^2}{x^4}+\frac{b^2}{x^2}-\frac{b^2}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{b^2 \cot (c+d x)}{d}+\frac{b^2 \cot ^3(c+d x)}{3 d}-\frac{b^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{5 a b \cot (c+d x) \csc (c+d x)}{8 d}+\frac{5 a b \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac{a b \cot ^5(c+d x) \csc (c+d x)}{3 d}-\frac{1}{8} (5 a b) \int \csc (c+d x) \, dx-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-b^2 x+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{b^2 \cot (c+d x)}{d}+\frac{b^2 \cot ^3(c+d x)}{3 d}-\frac{b^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{5 a b \cot (c+d x) \csc (c+d x)}{8 d}+\frac{5 a b \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac{a b \cot ^5(c+d x) \csc (c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 1.37271, size = 280, normalized size = 1.77 \[ \frac{\csc ^7(c+d x) \left (-84 \left (15 a^2-41 b^2\right ) \cos (3 (c+d x))-28 \left (15 a^2+71 b^2\right ) \cos (5 (c+d x))-60 a^2 \cos (7 (c+d x))+980 a b \sin (4 (c+d x))-1155 a b \sin (6 (c+d x))+8820 b^2 c \sin (3 (c+d x))+8820 b^2 d x \sin (3 (c+d x))-2940 b^2 c \sin (5 (c+d x))-2940 b^2 d x \sin (5 (c+d x))+420 b^2 c \sin (7 (c+d x))+420 b^2 d x \sin (7 (c+d x))+644 b^2 \cos (7 (c+d x))\right )-350 \cot (c+d x) \csc ^6(c+d x) \left (6 \left (a^2+b^2\right )+17 a b \sin (c+d x)\right )+16800 a b \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )-14700 b^2 (c+d x) \csc ^6(c+d x)}{26880 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]

[Out]

(-14700*b^2*(c + d*x)*Csc[c + d*x]^6 + 16800*a*b*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - 350*Cot[c +
d*x]*Csc[c + d*x]^6*(6*(a^2 + b^2) + 17*a*b*Sin[c + d*x]) + Csc[c + d*x]^7*(-84*(15*a^2 - 41*b^2)*Cos[3*(c +
d*x)] - 28*(15*a^2 + 71*b^2)*Cos[5*(c + d*x)] - 60*a^2*Cos[7*(c + d*x)] + 644*b^2*Cos[7*(c + d*x)] + 8820*b^2*
c*Sin[3*(c + d*x)] + 8820*b^2*d*x*Sin[3*(c + d*x)] + 980*a*b*Sin[4*(c + d*x)] - 2940*b^2*c*Sin[5*(c + d*x)] -
2940*b^2*d*x*Sin[5*(c + d*x)] - 1155*a*b*Sin[6*(c + d*x)] + 420*b^2*c*Sin[7*(c + d*x)] + 420*b^2*d*x*Sin[7*(c
+ d*x)]))/(26880*d)

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Maple [A]  time = 0.108, size = 222, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{12\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d}}-{\frac{5\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24\,d}}-{\frac{5\,ab\cos \left ( dx+c \right ) }{8\,d}}-{\frac{5\,ab\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{b}^{2}\cot \left ( dx+c \right ) }{d}}-{b}^{2}x-{\frac{{b}^{2}c}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^8*(a+b*sin(d*x+c))^2,x)

[Out]

-1/7/d*a^2/sin(d*x+c)^7*cos(d*x+c)^7-1/3/d*a*b/sin(d*x+c)^6*cos(d*x+c)^7+1/12/d*a*b/sin(d*x+c)^4*cos(d*x+c)^7-
1/8/d*a*b/sin(d*x+c)^2*cos(d*x+c)^7-1/8*a*b*cos(d*x+c)^5/d-5/24*a*b*cos(d*x+c)^3/d-5/8*a*b*cos(d*x+c)/d-5/8/d*
a*b*ln(csc(d*x+c)-cot(d*x+c))-1/5*b^2*cot(d*x+c)^5/d+1/3*b^2*cot(d*x+c)^3/d-b^2*cot(d*x+c)/d-b^2*x-1/d*b^2*c

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Maxima [A]  time = 1.46273, size = 207, normalized size = 1.31 \begin{align*} -\frac{112 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} b^{2} - 35 \, a b{\left (\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \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} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{240 \, a^{2}}{\tan \left (d x + c\right )^{7}}}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/1680*(112*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*b^2 - 35*a*b*(2*(33*c
os(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1
) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 240*a^2/tan(d*x + c)^7)/d

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Fricas [B]  time = 1.86217, size = 849, normalized size = 5.37 \begin{align*} \frac{16 \,{\left (15 \, a^{2} - 161 \, b^{2}\right )} \cos \left (d x + c\right )^{7} + 6496 \, b^{2} \cos \left (d x + c\right )^{5} - 5600 \, b^{2} \cos \left (d x + c\right )^{3} + 1680 \, b^{2} \cos \left (d x + c\right ) + 525 \,{\left (a b \cos \left (d x + c\right )^{6} - 3 \, a b \cos \left (d x + c\right )^{4} + 3 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 525 \,{\left (a b \cos \left (d x + c\right )^{6} - 3 \, a b \cos \left (d x + c\right )^{4} + 3 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 70 \,{\left (24 \, b^{2} d x \cos \left (d x + c\right )^{6} - 72 \, b^{2} d x \cos \left (d x + c\right )^{4} - 33 \, a b \cos \left (d x + c\right )^{5} + 72 \, b^{2} d x \cos \left (d x + c\right )^{2} + 40 \, a b \cos \left (d x + c\right )^{3} - 24 \, b^{2} d x - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \,{\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)^6*csc(d*x+c)^8*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/1680*(16*(15*a^2 - 161*b^2)*cos(d*x + c)^7 + 6496*b^2*cos(d*x + c)^5 - 5600*b^2*cos(d*x + c)^3 + 1680*b^2*co
s(d*x + c) + 525*(a*b*cos(d*x + c)^6 - 3*a*b*cos(d*x + c)^4 + 3*a*b*cos(d*x + c)^2 - a*b)*log(1/2*cos(d*x + c)
+ 1/2)*sin(d*x + c) - 525*(a*b*cos(d*x + c)^6 - 3*a*b*cos(d*x + c)^4 + 3*a*b*cos(d*x + c)^2 - a*b)*log(-1/2*c
os(d*x + c) + 1/2)*sin(d*x + c) - 70*(24*b^2*d*x*cos(d*x + c)^6 - 72*b^2*d*x*cos(d*x + c)^4 - 33*a*b*cos(d*x +
c)^5 + 72*b^2*d*x*cos(d*x + c)^2 + 40*a*b*cos(d*x + c)^3 - 24*b^2*d*x - 15*a*b*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)**6*csc(d*x+c)**8*(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.24021, size = 481, normalized size = 3.04 \begin{align*} \frac{15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 70 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 105 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 84 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 630 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 315 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 980 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3150 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 13440 \,{\left (d x + c\right )} b^{2} - 8400 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 525 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 9240 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{21780 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 525 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 9240 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 3150 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 315 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 980 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 630 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 105 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 84 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 70 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/13440*(15*a^2*tan(1/2*d*x + 1/2*c)^7 + 70*a*b*tan(1/2*d*x + 1/2*c)^6 - 105*a^2*tan(1/2*d*x + 1/2*c)^5 + 84*b
^2*tan(1/2*d*x + 1/2*c)^5 - 630*a*b*tan(1/2*d*x + 1/2*c)^4 + 315*a^2*tan(1/2*d*x + 1/2*c)^3 - 980*b^2*tan(1/2*
d*x + 1/2*c)^3 + 3150*a*b*tan(1/2*d*x + 1/2*c)^2 - 13440*(d*x + c)*b^2 - 8400*a*b*log(abs(tan(1/2*d*x + 1/2*c)
)) - 525*a^2*tan(1/2*d*x + 1/2*c) + 9240*b^2*tan(1/2*d*x + 1/2*c) + (21780*a*b*tan(1/2*d*x + 1/2*c)^7 + 525*a^
2*tan(1/2*d*x + 1/2*c)^6 - 9240*b^2*tan(1/2*d*x + 1/2*c)^6 - 3150*a*b*tan(1/2*d*x + 1/2*c)^5 - 315*a^2*tan(1/2
*d*x + 1/2*c)^4 + 980*b^2*tan(1/2*d*x + 1/2*c)^4 + 630*a*b*tan(1/2*d*x + 1/2*c)^3 + 105*a^2*tan(1/2*d*x + 1/2*
c)^2 - 84*b^2*tan(1/2*d*x + 1/2*c)^2 - 70*a*b*tan(1/2*d*x + 1/2*c) - 15*a^2)/tan(1/2*d*x + 1/2*c)^7)/d