\(\int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1252]

Optimal result

Integrand size = 29, antiderivative size = 159 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 \left (a^2+8 b^2\right ) \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a b \cot ^7(c+d x)}{7 d}+\frac {\left (5 a^2-88 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {\left (59 a^2-104 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{192 d}+\frac {\left (17 a^2-8 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{48 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{8 d} \]

[Out]

5/128*(a^2+8*b^2)*arctanh(cos(d*x+c))/d-2/7*a*b*cot(d*x+c)^7/d+1/128*(5*a^2-88*b^2)*cot(d*x+c)*csc(d*x+c)/d-1/
192*(59*a^2-104*b^2)*cot(d*x+c)*csc(d*x+c)^3/d+1/48*(17*a^2-8*b^2)*cot(d*x+c)*csc(d*x+c)^5/d-1/8*a^2*cot(d*x+c
)*csc(d*x+c)^7/d

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2990, 2687, 30, 4451, 466, 1828, 1171, 393, 212} \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 \left (a^2+8 b^2\right ) \text {arctanh}(\cos (c+d x))}{128 d}+\frac {\left (17 a^2-8 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{48 d}-\frac {\left (59 a^2-104 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{192 d}+\frac {\left (5 a^2-88 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{8 d}-\frac {2 a b \cot ^7(c+d x)}{7 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]

[Out]

(5*(a^2 + 8*b^2)*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a*b*Cot[c + d*x]^7)/(7*d) + ((5*a^2 - 88*b^2)*Cot[c + d*x
]*Csc[c + d*x])/(128*d) - ((59*a^2 - 104*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(192*d) + ((17*a^2 - 8*b^2)*Cot[c +
d*x]*Csc[c + d*x]^5)/(48*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(8*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 212

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

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 466

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1828

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*
g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 2687

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 2990

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 4451

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, Dist
[-d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d]
, x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] &&
(EqQ[F, Sin] || EqQ[F, sin])

Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+\int \cot ^6(c+d x) \csc ^3(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {x^6 \left (a^2+b^2-b^2 x^2\right )}{\left (1-x^2\right )^5} \, dx,x,\cos (c+d x)\right )}{d}+\frac {(2 a b) \text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{8 d}+\frac {\text {Subst}\left (\int \frac {a^2+8 a^2 x^2+8 a^2 x^4-8 b^2 x^6}{\left (1-x^2\right )^4} \, dx,x,\cos (c+d x)\right )}{8 d} \\ & = -\frac {2 a b \cot ^7(c+d x)}{7 d}+\frac {\left (17 a^2-8 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{48 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{8 d}-\frac {\text {Subst}\left (\int \frac {11 a^2-8 b^2+48 \left (a^2-b^2\right ) x^2-48 b^2 x^4}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{48 d} \\ & = -\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {\left (59 a^2-104 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{192 d}+\frac {\left (17 a^2-8 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{48 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{8 d}+\frac {\text {Subst}\left (\int \frac {3 \left (5 a^2-24 b^2\right )-192 b^2 x^2}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{192 d} \\ & = -\frac {2 a b \cot ^7(c+d x)}{7 d}+\frac {\left (5 a^2-88 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {\left (59 a^2-104 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{192 d}+\frac {\left (17 a^2-8 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{48 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{8 d}+\frac {\left (5 \left (a^2+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{128 d} \\ & = \frac {5 \left (a^2+8 b^2\right ) \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a b \cot ^7(c+d x)}{7 d}+\frac {\left (5 a^2-88 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {\left (59 a^2-104 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{192 d}+\frac {\left (17 a^2-8 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{48 d}-\frac {a^2 \cot (c+d x) \csc ^7(c+d x)}{8 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.77 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {7 \left (895 a^2-904 b^2\right ) \cos (3 (c+d x)) \csc ^8(c+d x)+2779 a^2 \cos (5 (c+d x)) \csc ^8(c+d x)+3416 b^2 \cos (5 (c+d x)) \csc ^8(c+d x)+105 a^2 \cos (7 (c+d x)) \csc ^8(c+d x)-1848 b^2 \cos (7 (c+d x)) \csc ^8(c+d x)-6720 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-53760 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+6720 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+53760 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+7 \cot (c+d x) \csc ^7(c+d x) \left (1765 a^2+680 b^2+1536 a b \sin (c+d x)\right )+5376 a b \csc ^8(c+d x) \sin (4 (c+d x))+2304 a b \csc ^8(c+d x) \sin (6 (c+d x))+384 a b \csc ^8(c+d x) \sin (8 (c+d x))}{172032 d} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]

[Out]

-1/172032*(7*(895*a^2 - 904*b^2)*Cos[3*(c + d*x)]*Csc[c + d*x]^8 + 2779*a^2*Cos[5*(c + d*x)]*Csc[c + d*x]^8 +
3416*b^2*Cos[5*(c + d*x)]*Csc[c + d*x]^8 + 105*a^2*Cos[7*(c + d*x)]*Csc[c + d*x]^8 - 1848*b^2*Cos[7*(c + d*x)]
*Csc[c + d*x]^8 - 6720*a^2*Log[Cos[(c + d*x)/2]] - 53760*b^2*Log[Cos[(c + d*x)/2]] + 6720*a^2*Log[Sin[(c + d*x
)/2]] + 53760*b^2*Log[Sin[(c + d*x)/2]] + 7*Cot[c + d*x]*Csc[c + d*x]^7*(1765*a^2 + 680*b^2 + 1536*a*b*Sin[c +
d*x]) + 5376*a*b*Csc[c + d*x]^8*Sin[4*(c + d*x)] + 2304*a*b*Csc[c + d*x]^8*Sin[6*(c + d*x)] + 384*a*b*Csc[c +
d*x]^8*Sin[8*(c + d*x)])/d

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.36

method	result	size
parallelrisch	\(\frac {\left (-1720320 a^{2}-13762560 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12355 \left (a^{2} \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )+\frac {179 \cos \left (3 d x +3 c \right )}{353}+\frac {397 \cos \left (5 d x +5 c \right )}{1765}+\frac {3 \cos \left (7 d x +7 c \right )}{353}\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {1536 b \left (\cos \left (d x +c \right )+\frac {3 \cos \left (3 d x +3 c \right )}{5}+\frac {\cos \left (5 d x +5 c \right )}{5}+\frac {\cos \left (7 d x +7 c \right )}{35}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{353}+\frac {2304 b^{2} \left (\cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{18}+\frac {11 \cos \left (5 d x +5 c \right )}{30}\right )}{353}\right ) \left (\csc ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{44040192 d}\)	\(216\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )-\frac {2 a b \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+b^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(254\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )-\frac {2 a b \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+b^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(254\)
risch	\(-\frac {-16128 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}+105 a^{2} {\mathrm e}^{15 i \left (d x +c \right )}-1848 b^{2} {\mathrm e}^{15 i \left (d x +c \right )}+5376 i a b \,{\mathrm e}^{12 i \left (d x +c \right )}+2779 a^{2} {\mathrm e}^{13 i \left (d x +c \right )}+3416 b^{2} {\mathrm e}^{13 i \left (d x +c \right )}+768 i a b +6265 a^{2} {\mathrm e}^{11 i \left (d x +c \right )}-6328 b^{2} {\mathrm e}^{11 i \left (d x +c \right )}-26880 i a b \,{\mathrm e}^{10 i \left (d x +c \right )}+12355 a^{2} {\mathrm e}^{9 i \left (d x +c \right )}+4760 b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-768 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+12355 a^{2} {\mathrm e}^{7 i \left (d x +c \right )}+4760 b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-5376 i a b \,{\mathrm e}^{14 i \left (d x +c \right )}+6265 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-6328 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+16128 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+2779 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+3416 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+26880 i a b \,{\mathrm e}^{8 i \left (d x +c \right )}+105 a^{2} {\mathrm e}^{i \left (d x +c \right )}-1848 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{1344 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}-\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{16 d}+\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{16 d}\)	\(428\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^9*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/44040192*((-1720320*a^2-13762560*b^2)*ln(tan(1/2*d*x+1/2*c))-12355*(a^2*sec(1/2*d*x+1/2*c)^2*(cos(d*x+c)+179
/353*cos(3*d*x+3*c)+397/1765*cos(5*d*x+5*c)+3/353*cos(7*d*x+7*c))*csc(1/2*d*x+1/2*c)^2+1536/353*b*(cos(d*x+c)+
3/5*cos(3*d*x+3*c)+1/5*cos(5*d*x+5*c)+1/35*cos(7*d*x+7*c))*sec(1/2*d*x+1/2*c)*a*csc(1/2*d*x+1/2*c)+2304/353*b^
2*(cos(d*x+c)+1/18*cos(3*d*x+3*c)+11/30*cos(5*d*x+5*c)))*csc(1/2*d*x+1/2*c)^6*sec(1/2*d*x+1/2*c)^6)/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (147) = 294\).

Time = 0.35 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.13 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {1536 \, a b \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) + 42 \, {\left (5 \, a^{2} - 88 \, b^{2}\right )} \cos \left (d x + c\right )^{7} + 1022 \, {\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 770 \, {\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 210 \, {\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right ) - 105 \, {\left ({\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{8} - 4 \, {\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 8 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left ({\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{8} - 4 \, {\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (a^{2} + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 8 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{5376 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/5376*(1536*a*b*cos(d*x + c)^7*sin(d*x + c) + 42*(5*a^2 - 88*b^2)*cos(d*x + c)^7 + 1022*(a^2 + 8*b^2)*cos(d*
x + c)^5 - 770*(a^2 + 8*b^2)*cos(d*x + c)^3 + 210*(a^2 + 8*b^2)*cos(d*x + c) - 105*((a^2 + 8*b^2)*cos(d*x + c)
^8 - 4*(a^2 + 8*b^2)*cos(d*x + c)^6 + 6*(a^2 + 8*b^2)*cos(d*x + c)^4 - 4*(a^2 + 8*b^2)*cos(d*x + c)^2 + a^2 +
8*b^2)*log(1/2*cos(d*x + c) + 1/2) + 105*((a^2 + 8*b^2)*cos(d*x + c)^8 - 4*(a^2 + 8*b^2)*cos(d*x + c)^6 + 6*(a
^2 + 8*b^2)*cos(d*x + c)^4 - 4*(a^2 + 8*b^2)*cos(d*x + c)^2 + a^2 + 8*b^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*co
s(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*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 = 220, normalized size of antiderivative = 1.38 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {7 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 56 \, b^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {1536 \, a b}{\tan \left (d x + c\right )^{7}}}{5376 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/5376*(7*a^2*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^
8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x +
c) - 1)) - 56*b^2*(2*(33*cos(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)) + 1536*a*b/tan(d*x + c)^7)/
d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (147) = 294\).

Time = 0.44 (sec) , antiderivative size = 402, normalized size of antiderivative = 2.53 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 96 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 112 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 112 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 672 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 168 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1008 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2016 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5040 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3360 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1680 \, {\left (a^{2} + 8 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {4566 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 36528 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 3360 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 5040 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2016 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1008 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 112 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 96 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{43008 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/43008*(21*a^2*tan(1/2*d*x + 1/2*c)^8 + 96*a*b*tan(1/2*d*x + 1/2*c)^7 - 112*a^2*tan(1/2*d*x + 1/2*c)^6 + 112*
b^2*tan(1/2*d*x + 1/2*c)^6 - 672*a*b*tan(1/2*d*x + 1/2*c)^5 + 168*a^2*tan(1/2*d*x + 1/2*c)^4 - 1008*b^2*tan(1/
2*d*x + 1/2*c)^4 + 2016*a*b*tan(1/2*d*x + 1/2*c)^3 + 336*a^2*tan(1/2*d*x + 1/2*c)^2 + 5040*b^2*tan(1/2*d*x + 1
/2*c)^2 - 3360*a*b*tan(1/2*d*x + 1/2*c) - 1680*(a^2 + 8*b^2)*log(abs(tan(1/2*d*x + 1/2*c))) + (4566*a^2*tan(1/
2*d*x + 1/2*c)^8 + 36528*b^2*tan(1/2*d*x + 1/2*c)^8 + 3360*a*b*tan(1/2*d*x + 1/2*c)^7 - 336*a^2*tan(1/2*d*x +
1/2*c)^6 - 5040*b^2*tan(1/2*d*x + 1/2*c)^6 - 2016*a*b*tan(1/2*d*x + 1/2*c)^5 - 168*a^2*tan(1/2*d*x + 1/2*c)^4
+ 1008*b^2*tan(1/2*d*x + 1/2*c)^4 + 672*a*b*tan(1/2*d*x + 1/2*c)^3 + 112*a^2*tan(1/2*d*x + 1/2*c)^2 - 112*b^2*
tan(1/2*d*x + 1/2*c)^2 - 96*a*b*tan(1/2*d*x + 1/2*c) - 21*a^2)/tan(1/2*d*x + 1/2*c)^8)/d

Mupad [B] (verification not implemented)

Time = 12.64 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.16 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {5\,a^2}{128}+\frac {5\,b^2}{16}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (2\,a^2+30\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {2\,a^2}{3}-\frac {2\,b^2}{3}\right )+\frac {a^2}{8}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2-6\,b^2\right )-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-20\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{7}\right )}{256\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{128}+\frac {15\,b^2}{128}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{256}-\frac {3\,b^2}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^2}{384}-\frac {b^2}{384}\right )}{d}+\frac {3\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{64\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{448\,d}-\frac {5\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,d} \]

[In]

int((cos(c + d*x)^6*(a + b*sin(c + d*x))^2)/sin(c + d*x)^9,x)

[Out]

(a^2*tan(c/2 + (d*x)/2)^8)/(2048*d) - (log(tan(c/2 + (d*x)/2))*((5*a^2)/128 + (5*b^2)/16))/d - (cot(c/2 + (d*x
)/2)^8*(tan(c/2 + (d*x)/2)^6*(2*a^2 + 30*b^2) - tan(c/2 + (d*x)/2)^2*((2*a^2)/3 - (2*b^2)/3) + a^2/8 + tan(c/2
+ (d*x)/2)^4*(a^2 - 6*b^2) - 4*a*b*tan(c/2 + (d*x)/2)^3 + 12*a*b*tan(c/2 + (d*x)/2)^5 - 20*a*b*tan(c/2 + (d*x
)/2)^7 + (4*a*b*tan(c/2 + (d*x)/2))/7))/(256*d) + (tan(c/2 + (d*x)/2)^2*(a^2/128 + (15*b^2)/128))/d + (tan(c/2
+ (d*x)/2)^4*(a^2/256 - (3*b^2)/128))/d - (tan(c/2 + (d*x)/2)^6*(a^2/384 - b^2/384))/d + (3*a*b*tan(c/2 + (d*
x)/2)^3)/(64*d) - (a*b*tan(c/2 + (d*x)/2)^5)/(64*d) + (a*b*tan(c/2 + (d*x)/2)^7)/(448*d) - (5*a*b*tan(c/2 + (d
*x)/2))/(64*d)