3.1126 \(\int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=334 \[ -\frac {b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {3 a \left (a^2+8 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d} \]

[Out]

-3/128*a*(a^2+8*b^2)*arctanh(cos(d*x+c))/d-1/35*b*(6*a^2+7*b^2)*cot(d*x+c)/d-3/128*a*(a^2+8*b^2)*cot(d*x+c)*cs
c(d*x+c)/d-1/280*b*(24*a^4-25*a^2*b^2+4*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^2/d-1/2240*(35*a^4-148*a^2*b^2+24*b^4)*
cot(d*x+c)*csc(d*x+c)^3/a/d+3/560*b*(23*a^2-4*b^2)*cot(d*x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^2/a^2/d+1/112*(21*
a^2-4*b^2)*cot(d*x+c)*csc(d*x+c)^5*(a+b*sin(d*x+c))^3/a^2/d+1/14*b*cot(d*x+c)*csc(d*x+c)^6*(a+b*sin(d*x+c))^4/
a^2/d-1/8*cot(d*x+c)*csc(d*x+c)^7*(a+b*sin(d*x+c))^4/a/d

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Rubi [A]  time = 0.91, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2893, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8} \[ -\frac {b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {3 a \left (a^2+8 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {\left (-148 a^2 b^2+35 a^4+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}-\frac {b \left (-25 a^2 b^2+24 a^4+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]

[Out]

(-3*a*(a^2 + 8*b^2)*ArcTanh[Cos[c + d*x]])/(128*d) - (b*(6*a^2 + 7*b^2)*Cot[c + d*x])/(35*d) - (3*a*(a^2 + 8*b
^2)*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (b*(24*a^4 - 25*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(280*a^
2*d) - ((35*a^4 - 148*a^2*b^2 + 24*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*a*d) + (3*b*(23*a^2 - 4*b^2)*Cot[c
+ d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(560*a^2*d) + ((21*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x]^5*(a +
b*Sin[c + d*x])^3)/(112*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^4)/(14*a^2*d) - (Cot[c +
d*x]*Csc[c + d*x]^7*(a + b*Sin[c + d*x])^4)/(8*a*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2893

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1))/(a*d*f*(n + 1)), x] +
(-Dist[1/(a^2*d^2*(n + 1)*(n + 2)), Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) -
b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e +
f*x]^2, x], x], x] - Simp[(b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 2))/
(a^2*d^2*f*(n + 1)*(n + 2)), x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || Intege
rsQ[2*m, 2*n]) &&  !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])

Rule 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 3031

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

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]

Rubi steps

\begin {align*} \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}-\frac {\int \csc ^7(c+d x) (a+b \sin (c+d x))^3 \left (3 \left (21 a^2-4 b^2\right )+3 a b \sin (c+d x)-8 \left (7 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{56 a^2}\\ &=\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}-\frac {\int \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (9 b \left (23 a^2-4 b^2\right )-3 a \left (7 a^2-2 b^2\right ) \sin (c+d x)-6 b \left (35 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{336 a^2}\\ &=\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}-\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x)) \left (-3 \left (35 a^4-148 a^2 b^2+24 b^4\right )-3 a b \left (109 a^2-2 b^2\right ) \sin (c+d x)-12 b^2 \left (53 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{1680 a^2}\\ &=-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {\int \csc ^4(c+d x) \left (72 b \left (24 a^4-25 a^2 b^2+4 b^4\right )+315 a^3 \left (a^2+8 b^2\right ) \sin (c+d x)+48 b^3 \left (53 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6720 a^2}\\ &=-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {\int \csc ^3(c+d x) \left (945 a^3 \left (a^2+8 b^2\right )+576 a^2 b \left (6 a^2+7 b^2\right ) \sin (c+d x)\right ) \, dx}{20160 a^2}\\ &=-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {1}{35} \left (b \left (6 a^2+7 b^2\right )\right ) \int \csc ^2(c+d x) \, dx+\frac {1}{64} \left (3 a \left (a^2+8 b^2\right )\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {1}{128} \left (3 a \left (a^2+8 b^2\right )\right ) \int \csc (c+d x) \, dx-\frac {\left (b \left (6 a^2+7 b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{35 d}\\ &=-\frac {3 a \left (a^2+8 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\\ \end {align*}

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Mathematica [A]  time = 1.60, size = 268, normalized size = 0.80 \[ -\frac {-6720 a \left (a^2+8 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6720 a \left (a^2+8 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^8(c+d x) \left (35 \left (333 a^3+104 a b^2\right ) \cos (3 (c+d x))+805 a^3 \cos (5 (c+d x))-105 a^3 \cos (7 (c+d x))+35 a \left (671 a^2+248 b^2\right ) \cos (c+d x)+21504 a^2 b \sin (2 (c+d x))+16128 a^2 b \sin (4 (c+d x))+3072 a^2 b \sin (6 (c+d x))-384 a^2 b \sin (8 (c+d x))-11480 a b^2 \cos (5 (c+d x))-840 a b^2 \cos (7 (c+d x))+2688 b^3 \sin (2 (c+d x))+896 b^3 \sin (4 (c+d x))-896 b^3 \sin (6 (c+d x))-448 b^3 \sin (8 (c+d x))\right )}{286720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]

[Out]

-1/286720*(6720*a*(a^2 + 8*b^2)*Log[Cos[(c + d*x)/2]] - 6720*a*(a^2 + 8*b^2)*Log[Sin[(c + d*x)/2]] + Csc[c + d
*x]^8*(35*a*(671*a^2 + 248*b^2)*Cos[c + d*x] + 35*(333*a^3 + 104*a*b^2)*Cos[3*(c + d*x)] + 805*a^3*Cos[5*(c +
d*x)] - 11480*a*b^2*Cos[5*(c + d*x)] - 105*a^3*Cos[7*(c + d*x)] - 840*a*b^2*Cos[7*(c + d*x)] + 21504*a^2*b*Sin
[2*(c + d*x)] + 2688*b^3*Sin[2*(c + d*x)] + 16128*a^2*b*Sin[4*(c + d*x)] + 896*b^3*Sin[4*(c + d*x)] + 3072*a^2
*b*Sin[6*(c + d*x)] - 896*b^3*Sin[6*(c + d*x)] - 384*a^2*b*Sin[8*(c + d*x)] - 448*b^3*Sin[8*(c + d*x)]))/d

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fricas [A]  time = 0.68, size = 384, normalized size = 1.15 \[ \frac {210 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} - 70 \, {\left (11 \, a^{3} - 40 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 770 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 210 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right ) - 105 \, {\left ({\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{8} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 8 \, a b^{2} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left ({\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{8} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 8 \, a b^{2} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 256 \, {\left ({\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{7} - 7 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{8960 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/8960*(210*(a^3 + 8*a*b^2)*cos(d*x + c)^7 - 70*(11*a^3 - 40*a*b^2)*cos(d*x + c)^5 - 770*(a^3 + 8*a*b^2)*cos(d
*x + c)^3 + 210*(a^3 + 8*a*b^2)*cos(d*x + c) - 105*((a^3 + 8*a*b^2)*cos(d*x + c)^8 - 4*(a^3 + 8*a*b^2)*cos(d*x
+ c)^6 + 6*(a^3 + 8*a*b^2)*cos(d*x + c)^4 + a^3 + 8*a*b^2 - 4*(a^3 + 8*a*b^2)*cos(d*x + c)^2)*log(1/2*cos(d*x
+ c) + 1/2) + 105*((a^3 + 8*a*b^2)*cos(d*x + c)^8 - 4*(a^3 + 8*a*b^2)*cos(d*x + c)^6 + 6*(a^3 + 8*a*b^2)*cos(
d*x + c)^4 + a^3 + 8*a*b^2 - 4*(a^3 + 8*a*b^2)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) + 256*((6*a^2*b +
7*b^3)*cos(d*x + c)^7 - 7*(3*a^2*b + b^3)*cos(d*x + c)^5)*sin(d*x + c))/(d*cos(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)

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giac [A]  time = 0.39, size = 457, normalized size = 1.37 \[ \frac {35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 240 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 560 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 336 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 448 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1680 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5040 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4480 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1680 \, {\left (a^{3} + 8 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {4566 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 36528 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 5040 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4480 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1680 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 448 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 35 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{71680 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/71680*(35*a^3*tan(1/2*d*x + 1/2*c)^8 + 240*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 560*a*b^2*tan(1/2*d*x + 1/2*c)^6 -
336*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 448*b^3*tan(1/2*d*x + 1/2*c)^5 - 280*a^3*tan(1/2*d*x + 1/2*c)^4 - 1680*a*b
^2*tan(1/2*d*x + 1/2*c)^4 - 1680*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 2240*b^3*tan(1/2*d*x + 1/2*c)^3 - 1680*a*b^2*t
an(1/2*d*x + 1/2*c)^2 + 5040*a^2*b*tan(1/2*d*x + 1/2*c) + 4480*b^3*tan(1/2*d*x + 1/2*c) + 1680*(a^3 + 8*a*b^2)
*log(abs(tan(1/2*d*x + 1/2*c))) - (4566*a^3*tan(1/2*d*x + 1/2*c)^8 + 36528*a*b^2*tan(1/2*d*x + 1/2*c)^8 + 5040
*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 4480*b^3*tan(1/2*d*x + 1/2*c)^7 - 1680*a*b^2*tan(1/2*d*x + 1/2*c)^6 - 1680*a^2
*b*tan(1/2*d*x + 1/2*c)^5 - 2240*b^3*tan(1/2*d*x + 1/2*c)^5 - 280*a^3*tan(1/2*d*x + 1/2*c)^4 - 1680*a*b^2*tan(
1/2*d*x + 1/2*c)^4 - 336*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 448*b^3*tan(1/2*d*x + 1/2*c)^3 + 560*a*b^2*tan(1/2*d*x
+ 1/2*c)^2 + 240*a^2*b*tan(1/2*d*x + 1/2*c) + 35*a^3)/tan(1/2*d*x + 1/2*c)^8)/d

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maple [A]  time = 0.50, size = 358, normalized size = 1.07 \[ -\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{6}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{64 d \sin \left (d x +c \right )^{4}}+\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{2}}+\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{128 d}+\frac {3 a^{3} \cos \left (d x +c \right )}{128 d}+\frac {3 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d}-\frac {3 a^{2} b \left (\cos ^{5}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}}-\frac {6 a^{2} b \left (\cos ^{5}\left (d x +c \right )\right )}{35 d \sin \left (d x +c \right )^{5}}-\frac {a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{6}}-\frac {a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{4}}+\frac {a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{2}}+\frac {a \,b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{16 d}+\frac {3 a \,b^{2} \cos \left (d x +c \right )}{16 d}+\frac {3 a \,b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {b^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^9*(a+b*sin(d*x+c))^3,x)

[Out]

-1/8/d*a^3/sin(d*x+c)^8*cos(d*x+c)^5-1/16/d*a^3/sin(d*x+c)^6*cos(d*x+c)^5-1/64/d*a^3/sin(d*x+c)^4*cos(d*x+c)^5
+1/128/d*a^3/sin(d*x+c)^2*cos(d*x+c)^5+1/128*a^3*cos(d*x+c)^3/d+3/128*a^3*cos(d*x+c)/d+3/128/d*a^3*ln(csc(d*x+
c)-cot(d*x+c))-3/7/d*a^2*b/sin(d*x+c)^7*cos(d*x+c)^5-6/35/d*a^2*b/sin(d*x+c)^5*cos(d*x+c)^5-1/2/d*a*b^2/sin(d*
x+c)^6*cos(d*x+c)^5-1/8/d*a*b^2/sin(d*x+c)^4*cos(d*x+c)^5+1/16/d*a*b^2/sin(d*x+c)^2*cos(d*x+c)^5+1/16*a*b^2*co
s(d*x+c)^3/d+3/16*a*b^2*cos(d*x+c)/d+3/16/d*a*b^2*ln(csc(d*x+c)-cot(d*x+c))-1/5/d*b^3/sin(d*x+c)^5*cos(d*x+c)^
5

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maxima [A]  time = 0.33, size = 248, normalized size = 0.74 \[ \frac {35 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 280 \, a b^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1792 \, b^{3}}{\tan \left (d x + c\right )^{5}} - \frac {768 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2} b}{\tan \left (d x + c\right )^{7}}}{8960 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/8960*(35*a^3*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(d*x + c)^3 + 3*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) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) -
1)) + 280*a*b^2*(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)) - 1792*b^3/tan(d*x + c)^5 - 768*(
7*tan(d*x + c)^2 + 5)*a^2*b/tan(d*x + c)^7)/d

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mupad [B]  time = 9.95, size = 381, normalized size = 1.14 \[ \frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3}{256}+\frac {3\,a\,b^2}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {3\,a^2\,b}{128}+\frac {b^3}{32}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,a^2\,b}{640}-\frac {b^3}{160}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^3}{128}+\frac {3\,a\,b^2}{16}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^3+6\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (6\,a^2\,b+8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {6\,a^2\,b}{5}-\frac {8\,b^3}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (18\,a^2\,b+16\,b^3\right )-\frac {a^3}{8}-\frac {6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{7}-2\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{256\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {9\,a^2\,b}{128}+\frac {b^3}{16}\right )}{d}-\frac {3\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^4*(a + b*sin(c + d*x))^3)/sin(c + d*x)^9,x)

[Out]

(a^3*tan(c/2 + (d*x)/2)^8)/(2048*d) - (tan(c/2 + (d*x)/2)^4*((3*a*b^2)/128 + a^3/256))/d - (tan(c/2 + (d*x)/2)
^3*((3*a^2*b)/128 + b^3/32))/d - (tan(c/2 + (d*x)/2)^5*((3*a^2*b)/640 - b^3/160))/d + (log(tan(c/2 + (d*x)/2))
*((3*a*b^2)/16 + (3*a^3)/128))/d + (tan(c/2 + (d*x)/2)^4*(6*a*b^2 + a^3) + tan(c/2 + (d*x)/2)^5*(6*a^2*b + 8*b
^3) + tan(c/2 + (d*x)/2)^3*((6*a^2*b)/5 - (8*b^3)/5) - tan(c/2 + (d*x)/2)^7*(18*a^2*b + 16*b^3) - a^3/8 - (6*a
^2*b*tan(c/2 + (d*x)/2))/7 - 2*a*b^2*tan(c/2 + (d*x)/2)^2 + 6*a*b^2*tan(c/2 + (d*x)/2)^6)/(256*d*tan(c/2 + (d*
x)/2)^8) + (tan(c/2 + (d*x)/2)*((9*a^2*b)/128 + b^3/16))/d - (3*a*b^2*tan(c/2 + (d*x)/2)^2)/(128*d) + (a*b^2*t
an(c/2 + (d*x)/2)^6)/(128*d) + (3*a^2*b*tan(c/2 + (d*x)/2)^7)/(896*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**9*(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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