\(\int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1225]

Optimal result

Integrand size = 29, antiderivative size = 138 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {b^2 \csc ^2(c+d x)}{2 d}-\frac {2 a b \csc ^3(c+d x)}{3 d}-\frac {\left (a^2-2 b^2\right ) \csc ^4(c+d x)}{4 d}+\frac {4 a b \csc ^5(c+d x)}{5 d}+\frac {\left (2 a^2-b^2\right ) \csc ^6(c+d x)}{6 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}-\frac {a^2 \csc ^8(c+d x)}{8 d} \]

[Out]

-1/2*b^2*csc(d*x+c)^2/d-2/3*a*b*csc(d*x+c)^3/d-1/4*(a^2-2*b^2)*csc(d*x+c)^4/d+4/5*a*b*csc(d*x+c)^5/d+1/6*(2*a^
2-b^2)*csc(d*x+c)^6/d-2/7*a*b*csc(d*x+c)^7/d-1/8*a^2*csc(d*x+c)^8/d

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2916, 12, 962} \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (2 a^2-b^2\right ) \csc ^6(c+d x)}{6 d}-\frac {\left (a^2-2 b^2\right ) \csc ^4(c+d x)}{4 d}-\frac {a^2 \csc ^8(c+d x)}{8 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}+\frac {4 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^3(c+d x)}{3 d}-\frac {b^2 \csc ^2(c+d x)}{2 d} \]

[In]

Int[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]

[Out]

-1/2*(b^2*Csc[c + d*x]^2)/d - (2*a*b*Csc[c + d*x]^3)/(3*d) - ((a^2 - 2*b^2)*Csc[c + d*x]^4)/(4*d) + (4*a*b*Csc
[c + d*x]^5)/(5*d) + ((2*a^2 - b^2)*Csc[c + d*x]^6)/(6*d) - (2*a*b*Csc[c + d*x]^7)/(7*d) - (a^2*Csc[c + d*x]^8
)/(8*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 962

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))

Rule 2916

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^9 (a+x)^2 \left (b^2-x^2\right )^2}{x^9} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {b^4 \text {Subst}\left (\int \frac {(a+x)^2 \left (b^2-x^2\right )^2}{x^9} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^4 \text {Subst}\left (\int \left (\frac {a^2 b^4}{x^9}+\frac {2 a b^4}{x^8}+\frac {-2 a^2 b^2+b^4}{x^7}-\frac {4 a b^2}{x^6}+\frac {a^2-2 b^2}{x^5}+\frac {2 a}{x^4}+\frac {1}{x^3}\right ) \, dx,x,b \sin (c+d x)\right )}{d} \\ & = -\frac {b^2 \csc ^2(c+d x)}{2 d}-\frac {2 a b \csc ^3(c+d x)}{3 d}-\frac {\left (a^2-2 b^2\right ) \csc ^4(c+d x)}{4 d}+\frac {4 a b \csc ^5(c+d x)}{5 d}+\frac {\left (2 a^2-b^2\right ) \csc ^6(c+d x)}{6 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}-\frac {a^2 \csc ^8(c+d x)}{8 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\csc ^2(c+d x) \left (420 b^2+560 a b \csc (c+d x)+210 \left (a^2-2 b^2\right ) \csc ^2(c+d x)-672 a b \csc ^3(c+d x)-140 \left (2 a^2-b^2\right ) \csc ^4(c+d x)+240 a b \csc ^5(c+d x)+105 a^2 \csc ^6(c+d x)\right )}{840 d} \]

[In]

Integrate[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]

[Out]

-1/840*(Csc[c + d*x]^2*(420*b^2 + 560*a*b*Csc[c + d*x] + 210*(a^2 - 2*b^2)*Csc[c + d*x]^2 - 672*a*b*Csc[c + d*
x]^3 - 140*(2*a^2 - b^2)*Csc[c + d*x]^4 + 240*a*b*Csc[c + d*x]^5 + 105*a^2*Csc[c + d*x]^6))/d

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(-\frac {\frac {\left (\csc ^{8}\left (d x +c \right )\right ) a^{2}}{8}+\frac {2 a b \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (-2 a^{2}+b^{2}\right ) \left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {4 a b \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (a^{2}-2 b^{2}\right ) \left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {2 a b \left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {b^{2} \left (\csc ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(107\)
default	\(-\frac {\frac {\left (\csc ^{8}\left (d x +c \right )\right ) a^{2}}{8}+\frac {2 a b \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (-2 a^{2}+b^{2}\right ) \left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {4 a b \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (a^{2}-2 b^{2}\right ) \left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {2 a b \left (\csc ^{3}\left (d x +c \right )\right )}{3}+\frac {b^{2} \left (\csc ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(107\)
parallelrisch	\(-\frac {853 \left (\left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} \left (\cos \left (2 d x +2 c \right )+\frac {2561 \cos \left (4 d x +4 c \right )}{5118}+\frac {73 \cos \left (6 d x +6 c \right )}{2559}-\frac {73 \cos \left (8 d x +8 c \right )}{20472}+\frac {5975}{6824}\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {32768 b \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (2 d x +2 c \right )+\frac {5 \cos \left (4 d x +4 c \right )}{4}+\frac {57}{28}\right ) a \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{12795}+\frac {3520 b^{2} \left (\cos \left (2 d x +2 c \right )+\frac {42 \cos \left (4 d x +4 c \right )}{55}+\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {14}{11}\right )}{853}\right ) \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4194304 d}\)	\(181\)
risch	\(\frac {2 b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-4 a^{2} {\mathrm e}^{12 i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+\frac {1376 i a b \,{\mathrm e}^{9 i \left (d x +c \right )}}{105}-\frac {16 a^{2} {\mathrm e}^{10 i \left (d x +c \right )}}{3}+\frac {26 b^{2} {\mathrm e}^{10 i \left (d x +c \right )}}{3}+\frac {16 i a b \,{\mathrm e}^{13 i \left (d x +c \right )}}{3}-\frac {40 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}}{3}-\frac {40 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}}{3}-\frac {16 i a b \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}-\frac {16 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {26 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {16 i a b \,{\mathrm e}^{5 i \left (d x +c \right )}}{15}-4 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-\frac {1376 i a b \,{\mathrm e}^{7 i \left (d x +c \right )}}{105}+2 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-\frac {16 i a b \,{\mathrm e}^{11 i \left (d x +c \right )}}{15}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}\)	\(272\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^9*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/d*(1/8*csc(d*x+c)^8*a^2+2/7*a*b*csc(d*x+c)^7+1/6*(-2*a^2+b^2)*csc(d*x+c)^6-4/5*a*b*csc(d*x+c)^5+1/4*(a^2-2*
b^2)*csc(d*x+c)^4+2/3*a*b*csc(d*x+c)^3+1/2*b^2*csc(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.07 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {420 \, b^{2} \cos \left (d x + c\right )^{6} - 210 \, {\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 35 \, a^{2} - 140 \, b^{2} - 16 \, {\left (35 \, a b \cos \left (d x + c\right )^{4} - 28 \, a b \cos \left (d x + c\right )^{2} + 8 \, a b\right )} \sin \left (d x + c\right )}{840 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^9*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/840*(420*b^2*cos(d*x + c)^6 - 210*(a^2 + 4*b^2)*cos(d*x + c)^4 + 140*(a^2 + 4*b^2)*cos(d*x + c)^2 - 35*a^2 -
140*b^2 - 16*(35*a*b*cos(d*x + c)^4 - 28*a*b*cos(d*x + c)^2 + 8*a*b)*sin(d*x + c))/(d*cos(d*x + c)^8 - 4*d*co
s(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**9*(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.77 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {420 \, b^{2} \sin \left (d x + c\right )^{6} + 560 \, a b \sin \left (d x + c\right )^{5} - 672 \, a b \sin \left (d x + c\right )^{3} + 210 \, {\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{4} + 240 \, a b \sin \left (d x + c\right ) - 140 \, {\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} + 105 \, a^{2}}{840 \, d \sin \left (d x + c\right )^{8}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^9*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/840*(420*b^2*sin(d*x + c)^6 + 560*a*b*sin(d*x + c)^5 - 672*a*b*sin(d*x + c)^3 + 210*(a^2 - 2*b^2)*sin(d*x +
c)^4 + 240*a*b*sin(d*x + c) - 140*(2*a^2 - b^2)*sin(d*x + c)^2 + 105*a^2)/(d*sin(d*x + c)^8)

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {420 \, b^{2} \sin \left (d x + c\right )^{6} + 560 \, a b \sin \left (d x + c\right )^{5} + 210 \, a^{2} \sin \left (d x + c\right )^{4} - 420 \, b^{2} \sin \left (d x + c\right )^{4} - 672 \, a b \sin \left (d x + c\right )^{3} - 280 \, a^{2} \sin \left (d x + c\right )^{2} + 140 \, b^{2} \sin \left (d x + c\right )^{2} + 240 \, a b \sin \left (d x + c\right ) + 105 \, a^{2}}{840 \, d \sin \left (d x + c\right )^{8}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^9*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/840*(420*b^2*sin(d*x + c)^6 + 560*a*b*sin(d*x + c)^5 + 210*a^2*sin(d*x + c)^4 - 420*b^2*sin(d*x + c)^4 - 67
2*a*b*sin(d*x + c)^3 - 280*a^2*sin(d*x + c)^2 + 140*b^2*sin(d*x + c)^2 + 240*a*b*sin(d*x + c) + 105*a^2)/(d*si
n(d*x + c)^8)

Mupad [B] (verification not implemented)

Time = 12.51 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\frac {a^2}{8}+{\sin \left (c+d\,x\right )}^4\,\left (\frac {a^2}{4}-\frac {b^2}{2}\right )-{\sin \left (c+d\,x\right )}^2\,\left (\frac {a^2}{3}-\frac {b^2}{6}\right )+\frac {b^2\,{\sin \left (c+d\,x\right )}^6}{2}+\frac {2\,a\,b\,\sin \left (c+d\,x\right )}{7}-\frac {4\,a\,b\,{\sin \left (c+d\,x\right )}^3}{5}+\frac {2\,a\,b\,{\sin \left (c+d\,x\right )}^5}{3}}{d\,{\sin \left (c+d\,x\right )}^8} \]

[In]

int((cos(c + d*x)^5*(a + b*sin(c + d*x))^2)/sin(c + d*x)^9,x)

[Out]

-(a^2/8 + sin(c + d*x)^4*(a^2/4 - b^2/2) - sin(c + d*x)^2*(a^2/3 - b^2/6) + (b^2*sin(c + d*x)^6)/2 + (2*a*b*si
n(c + d*x))/7 - (4*a*b*sin(c + d*x)^3)/5 + (2*a*b*sin(c + d*x)^5)/3)/(d*sin(c + d*x)^8)