3.1115 \(\int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=261 \[ -\frac {\left (2 a^2+7 b^2\right ) \cot (c+d x)}{35 d}+\frac {b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac {2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}-\frac {a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d} \]

[Out]

-1/8*a*b*arctanh(cos(d*x+c))/d-1/35*(2*a^2+7*b^2)*cot(d*x+c)/d-1/8*a*b*cot(d*x+c)*csc(d*x+c)/d-1/105*(3*a^4-18
*a^2*b^2+4*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^2/d+1/420*b*(53*a^2-12*b^2)*cot(d*x+c)*csc(d*x+c)^3/a/d+2/35*(4*a^2-
b^2)*cot(d*x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^2/a^2/d+2/21*b*cot(d*x+c)*csc(d*x+c)^5*(a+b*sin(d*x+c))^3/a^2/d-
1/7*cot(d*x+c)*csc(d*x+c)^6*(a+b*sin(d*x+c))^3/a/d

________________________________________________________________________________________

Rubi [A]  time = 0.63, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {\left (2 a^2+7 b^2\right ) \cot (c+d x)}{35 d}+\frac {b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}-\frac {\left (-18 a^2 b^2+3 a^4+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac {2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*b*ArcTanh[Cos[c + d*x]])/(8*d) - ((2*a^2 + 7*b^2)*Cot[c + d*x])/(35*d) - (a*b*Cot[c + d*x]*Csc[c + d*x])/(
8*d) - ((3*a^4 - 18*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(105*a^2*d) + (b*(53*a^2 - 12*b^2)*Cot[c + d
*x]*Csc[c + d*x]^3)/(420*a*d) + (2*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(35*a^2*d
) + (2*b*Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(21*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + b*
Sin[c + d*x])^3)/(7*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 ^4(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}-\frac {\int \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (12 \left (4 a^2-b^2\right )+2 a b \sin (c+d x)-2 \left (21 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{42 a^2}\\ &=\frac {2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}-\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x)) \left (2 b \left (53 a^2-12 b^2\right )-2 a \left (9 a^2-b^2\right ) \sin (c+d x)-2 b \left (57 a^2-8 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{210 a^2}\\ &=\frac {b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac {2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}+\frac {\int \csc ^4(c+d x) \left (24 \left (3 a^4-18 a^2 b^2+4 b^4\right )+210 a^3 b \sin (c+d x)+8 b^2 \left (57 a^2-8 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{840 a^2}\\ &=-\frac {\left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac {b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac {2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}+\frac {\int \csc ^3(c+d x) \left (630 a^3 b+72 a^2 \left (2 a^2+7 b^2\right ) \sin (c+d x)\right ) \, dx}{2520 a^2}\\ &=-\frac {\left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac {b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac {2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}+\frac {1}{4} (a b) \int \csc ^3(c+d x) \, dx+\frac {1}{35} \left (2 a^2+7 b^2\right ) \int \csc ^2(c+d x) \, dx\\ &=-\frac {a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac {\left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac {b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac {2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}+\frac {1}{8} (a b) \int \csc (c+d x) \, dx-\frac {\left (2 a^2+7 b^2\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{35 d}\\ &=-\frac {a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {\left (2 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac {\left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac {b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac {2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.33, size = 322, normalized size = 1.23 \[ -\frac {\csc ^7(c+d x) \left (840 \left (6 a^2+b^2\right ) \cos (c+d x)+168 \left (14 a^2-b^2\right ) \cos (3 (c+d x))+336 a^2 \cos (5 (c+d x))-48 a^2 \cos (7 (c+d x))+2170 a b \sin (2 (c+d x))+3080 a b \sin (4 (c+d x))+210 a b \sin (6 (c+d x))-3675 a b \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2205 a b \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-735 a b \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+105 a b \sin (7 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3675 a b \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2205 a b \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+735 a b \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-105 a b \sin (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-504 b^2 \cos (5 (c+d x))-168 b^2 \cos (7 (c+d x))\right )}{53760 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/53760*(Csc[c + d*x]^7*(840*(6*a^2 + b^2)*Cos[c + d*x] + 168*(14*a^2 - b^2)*Cos[3*(c + d*x)] + 336*a^2*Cos[5
*(c + d*x)] - 504*b^2*Cos[5*(c + d*x)] - 48*a^2*Cos[7*(c + d*x)] - 168*b^2*Cos[7*(c + d*x)] + 3675*a*b*Log[Cos
[(c + d*x)/2]]*Sin[c + d*x] - 3675*a*b*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 2170*a*b*Sin[2*(c + d*x)] - 2205*a
*b*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 2205*a*b*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 3080*a*b*Sin[4*(
c + d*x)] + 735*a*b*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 735*a*b*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] +
210*a*b*Sin[6*(c + d*x)] - 105*a*b*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 105*a*b*Log[Sin[(c + d*x)/2]]*Sin[
7*(c + d*x)]))/d

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 248, normalized size = 0.95 \[ -\frac {48 \, {\left (2 \, a^{2} + 7 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 336 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (a b \cos \left (d x + c\right )^{6} - 3 \, a b \cos \left (d x + c\right )^{4} + 3 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 105 \, {\left (a b \cos \left (d x + c\right )^{6} - 3 \, a b \cos \left (d x + c\right )^{4} + 3 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 70 \, {\left (3 \, a b \cos \left (d x + c\right )^{5} + 8 \, a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/1680*(48*(2*a^2 + 7*b^2)*cos(d*x + c)^7 - 336*(a^2 + b^2)*cos(d*x + c)^5 + 105*(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) - 105*(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) - 70*(3*a*b*c
os(d*x + c)^5 + 8*a*b*cos(d*x + c)^3 - 3*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))

________________________________________________________________________________________

giac [A]  time = 0.29, size = 347, normalized size = 1.33 \[ \frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 70 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 84 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 210 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 420 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 210 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1680 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 840 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {4356 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 840 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 210 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 420 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 210 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 84 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 70 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*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 - 21*a^2*tan(1/2*d*x + 1/2*c)^5 + 84*b^
2*tan(1/2*d*x + 1/2*c)^5 - 210*a*b*tan(1/2*d*x + 1/2*c)^4 - 105*a^2*tan(1/2*d*x + 1/2*c)^3 - 420*b^2*tan(1/2*d
*x + 1/2*c)^3 - 210*a*b*tan(1/2*d*x + 1/2*c)^2 + 1680*a*b*log(abs(tan(1/2*d*x + 1/2*c))) + 315*a^2*tan(1/2*d*x
+ 1/2*c) + 840*b^2*tan(1/2*d*x + 1/2*c) - (4356*a*b*tan(1/2*d*x + 1/2*c)^7 + 315*a^2*tan(1/2*d*x + 1/2*c)^6 +
840*b^2*tan(1/2*d*x + 1/2*c)^6 - 210*a*b*tan(1/2*d*x + 1/2*c)^5 - 105*a^2*tan(1/2*d*x + 1/2*c)^4 - 420*b^2*ta
n(1/2*d*x + 1/2*c)^4 - 210*a*b*tan(1/2*d*x + 1/2*c)^3 - 21*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

________________________________________________________________________________________

maple [A]  time = 0.47, size = 194, normalized size = 0.74 \[ -\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}}-\frac {2 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{35 d \sin \left (d x +c \right )^{5}}-\frac {a b \left (\cos ^{5}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{6}}-\frac {a b \left (\cos ^{5}\left (d x +c \right )\right )}{12 d \sin \left (d x +c \right )^{4}}+\frac {a b \left (\cos ^{5}\left (d x +c \right )\right )}{24 d \sin \left (d x +c \right )^{2}}+\frac {a b \left (\cos ^{3}\left (d x +c \right )\right )}{24 d}+\frac {a b \cos \left (d x +c \right )}{8 d}+\frac {a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^8*(a+b*sin(d*x+c))^2,x)

[Out]

-1/7/d*a^2/sin(d*x+c)^7*cos(d*x+c)^5-2/35/d*a^2/sin(d*x+c)^5*cos(d*x+c)^5-1/3/d*a*b/sin(d*x+c)^6*cos(d*x+c)^5-
1/12/d*a*b/sin(d*x+c)^4*cos(d*x+c)^5+1/24/d*a*b/sin(d*x+c)^2*cos(d*x+c)^5+1/24*a*b*cos(d*x+c)^3/d+1/8*a*b*cos(
d*x+c)/d+1/8/d*a*b*ln(csc(d*x+c)-cot(d*x+c))-1/5/d*b^2/sin(d*x+c)^5*cos(d*x+c)^5

________________________________________________________________________________________

maxima [A]  time = 0.33, size = 134, normalized size = 0.51 \[ \frac {35 \, a b {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {336 \, b^{2}}{\tan \left (d x + c\right )^{5}} - \frac {48 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2}}{\tan \left (d x + c\right )^{7}}}{1680 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^8*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/1680*(35*a*b*(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)) - 336*b^2/tan(d*x + c)^5 - 48*(7*ta
n(d*x + c)^2 + 5)*a^2/tan(d*x + c)^7)/d

________________________________________________________________________________________

mupad [B]  time = 9.92, size = 302, normalized size = 1.16 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{5}-\frac {4\,b^2}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (3\,a^2+8\,b^2\right )-\frac {a^2}{7}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2+4\,b^2\right )+2\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {2\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}\right )}{128\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^2}{128}+\frac {b^2}{16}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^2}{128}+\frac {b^2}{32}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {a^2}{640}-\frac {b^2}{160}\right )}{d}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {a\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^4*(a + b*sin(c + d*x))^2)/sin(c + d*x)^8,x)

[Out]

(a^2*tan(c/2 + (d*x)/2)^7)/(896*d) + (cot(c/2 + (d*x)/2)^7*(tan(c/2 + (d*x)/2)^2*(a^2/5 - (4*b^2)/5) - tan(c/2
+ (d*x)/2)^6*(3*a^2 + 8*b^2) - a^2/7 + tan(c/2 + (d*x)/2)^4*(a^2 + 4*b^2) + 2*a*b*tan(c/2 + (d*x)/2)^3 + 2*a*
b*tan(c/2 + (d*x)/2)^5 - (2*a*b*tan(c/2 + (d*x)/2))/3))/(128*d) + (tan(c/2 + (d*x)/2)*((3*a^2)/128 + b^2/16))/
d - (tan(c/2 + (d*x)/2)^3*(a^2/128 + b^2/32))/d - (tan(c/2 + (d*x)/2)^5*(a^2/640 - b^2/160))/d - (a*b*tan(c/2
+ (d*x)/2)^2)/(64*d) - (a*b*tan(c/2 + (d*x)/2)^4)/(64*d) + (a*b*tan(c/2 + (d*x)/2)^6)/(192*d) + (a*b*log(tan(c
/2 + (d*x)/2)))/(8*d)

________________________________________________________________________________________

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)**8*(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

________________________________________________________________________________________