3.1124 \(\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=275 \[ -\frac {a \left (a^2+18 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{60 a^2 d}-\frac {\left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}+b^3 x \]

[Out]

b^3*x-1/16*a*(a^2+18*b^2)*arctanh(cos(d*x+c))/d-1/60*b*(36*a^4-43*a^2*b^2+2*b^4)*cot(d*x+c)/a^2/d-1/240*(15*a^
4-84*a^2*b^2+4*b^4)*cot(d*x+c)*csc(d*x+c)/a/d+1/120*b*(39*a^2-2*b^2)*cot(d*x+c)*csc(d*x+c)^2*(a+b*sin(d*x+c))^
2/a^2/d+1/120*(35*a^2-2*b^2)*cot(d*x+c)*csc(d*x+c)^3*(a+b*sin(d*x+c))^3/a^2/d+1/15*b*cot(d*x+c)*csc(d*x+c)^4*(
a+b*sin(d*x+c))^4/a^2/d-1/6*cot(d*x+c)*csc(d*x+c)^5*(a+b*sin(d*x+c))^4/a/d

________________________________________________________________________________________

Rubi [A]  time = 0.76, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2893, 3047, 3031, 3021, 2735, 3770} \[ -\frac {b \left (-43 a^2 b^2+36 a^4+2 b^4\right ) \cot (c+d x)}{60 a^2 d}-\frac {a \left (a^2+18 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {\left (-84 a^2 b^2+15 a^4+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a d}+\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}+b^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

b^3*x - (a*(a^2 + 18*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) - (b*(36*a^4 - 43*a^2*b^2 + 2*b^4)*Cot[c + d*x])/(60*a
^2*d) - ((15*a^4 - 84*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x])/(240*a*d) + (b*(39*a^2 - 2*b^2)*Cot[c + d*x]
*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(120*a^2*d) + ((35*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin
[c + d*x])^3)/(120*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(15*a^2*d) - (Cot[c + d*x]*
Csc[c + d*x]^5*(a + b*Sin[c + d*x])^4)/(6*a*d)

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

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 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 ^3(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}-\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x))^3 \left (35 a^2-2 b^2+3 a b \sin (c+d x)-30 a^2 \sin ^2(c+d x)\right ) \, dx}{30 a^2}\\ &=\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}-\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x))^2 \left (3 b \left (39 a^2-2 b^2\right )-3 a \left (5 a^2-2 b^2\right ) \sin (c+d x)-120 a^2 b \sin ^2(c+d x)\right ) \, dx}{120 a^2}\\ &=\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}-\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x)) \left (-3 \left (15 a^4-84 a^2 b^2+4 b^4\right )-3 a b \left (57 a^2-2 b^2\right ) \sin (c+d x)-360 a^2 b^2 \sin ^2(c+d x)\right ) \, dx}{360 a^2}\\ &=-\frac {\left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a d}+\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}+\frac {\int \csc ^2(c+d x) \left (12 b \left (36 a^4-43 a^2 b^2+2 b^4\right )+45 a^3 \left (a^2+18 b^2\right ) \sin (c+d x)+720 a^2 b^3 \sin ^2(c+d x)\right ) \, dx}{720 a^2}\\ &=-\frac {b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{60 a^2 d}-\frac {\left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a d}+\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}+\frac {\int \csc (c+d x) \left (45 a^3 \left (a^2+18 b^2\right )+720 a^2 b^3 \sin (c+d x)\right ) \, dx}{720 a^2}\\ &=b^3 x-\frac {b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{60 a^2 d}-\frac {\left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a d}+\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}+\frac {1}{16} \left (a \left (a^2+18 b^2\right )\right ) \int \csc (c+d x) \, dx\\ &=b^3 x-\frac {a \left (a^2+18 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{60 a^2 d}-\frac {\left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a d}+\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.87, size = 408, normalized size = 1.48 \[ \frac {-30 \left (a^3-30 a b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )+5 a^3 \sec ^6\left (\frac {1}{2} (c+d x)\right )-30 a^3 \sec ^4\left (\frac {1}{2} (c+d x)\right )+30 a^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+120 a^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-120 a^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-64 \left (9 a^2 b-20 b^3\right ) \cot \left (\frac {1}{2} (c+d x)\right )+576 a^2 b \tan \left (\frac {1}{2} (c+d x)\right )-a^2 \csc ^6\left (\frac {1}{2} (c+d x)\right ) (5 a+18 b \sin (c+d x))-2016 a^2 b \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+36 a^2 b \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )+2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (15 \left (a^3-3 a b^2\right )+b \left (63 a^2-20 b^2\right ) \sin (c+d x)\right )+90 a b^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )-900 a b^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )+2160 a b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2160 a b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-1280 b^3 \tan \left (\frac {1}{2} (c+d x)\right )+640 b^3 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+1920 b^3 c+1920 b^3 d x}{1920 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1920*b^3*c + 1920*b^3*d*x - 64*(9*a^2*b - 20*b^3)*Cot[(c + d*x)/2] - 30*(a^3 - 30*a*b^2)*Csc[(c + d*x)/2]^2 -
120*a^3*Log[Cos[(c + d*x)/2]] - 2160*a*b^2*Log[Cos[(c + d*x)/2]] + 120*a^3*Log[Sin[(c + d*x)/2]] + 2160*a*b^2
*Log[Sin[(c + d*x)/2]] + 30*a^3*Sec[(c + d*x)/2]^2 - 900*a*b^2*Sec[(c + d*x)/2]^2 - 30*a^3*Sec[(c + d*x)/2]^4
+ 90*a*b^2*Sec[(c + d*x)/2]^4 + 5*a^3*Sec[(c + d*x)/2]^6 - 2016*a^2*b*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 640*
b^3*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - a^2*Csc[(c + d*x)/2]^6*(5*a + 18*b*Sin[c + d*x]) + 2*Csc[(c + d*x)/2]^
4*(15*(a^3 - 3*a*b^2) + b*(63*a^2 - 20*b^2)*Sin[c + d*x]) + 576*a^2*b*Tan[(c + d*x)/2] - 1280*b^3*Tan[(c + d*x
)/2] + 36*a^2*b*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(1920*d)

________________________________________________________________________________________

fricas [A]  time = 0.85, size = 373, normalized size = 1.36 \[ \frac {480 \, b^{3} d x \cos \left (d x + c\right )^{6} - 1440 \, b^{3} d x \cos \left (d x + c\right )^{4} + 1440 \, b^{3} d x \cos \left (d x + c\right )^{2} + 30 \, {\left (a^{3} - 30 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 480 \, b^{3} d x + 80 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right ) - 15 \, {\left ({\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 18 \, a b^{2} + 3 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left ({\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 18 \, a b^{2} + 3 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 32 \, {\left (35 \, b^{3} \cos \left (d x + c\right )^{3} + {\left (9 \, a^{2} b - 20 \, b^{3}\right )} \cos \left (d x + c\right )^{5} - 15 \, b^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7*(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/480*(480*b^3*d*x*cos(d*x + c)^6 - 1440*b^3*d*x*cos(d*x + c)^4 + 1440*b^3*d*x*cos(d*x + c)^2 + 30*(a^3 - 30*a
*b^2)*cos(d*x + c)^5 - 480*b^3*d*x + 80*(a^3 + 18*a*b^2)*cos(d*x + c)^3 - 30*(a^3 + 18*a*b^2)*cos(d*x + c) - 1
5*((a^3 + 18*a*b^2)*cos(d*x + c)^6 - 3*(a^3 + 18*a*b^2)*cos(d*x + c)^4 - a^3 - 18*a*b^2 + 3*(a^3 + 18*a*b^2)*c
os(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + 15*((a^3 + 18*a*b^2)*cos(d*x + c)^6 - 3*(a^3 + 18*a*b^2)*cos(d*x
+ c)^4 - a^3 - 18*a*b^2 + 3*(a^3 + 18*a*b^2)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) + 32*(35*b^3*cos(d*x
+ c)^3 + (9*a^2*b - 20*b^3)*cos(d*x + c)^5 - 15*b^3*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)

________________________________________________________________________________________

giac [A]  time = 0.36, size = 399, normalized size = 1.45 \[ \frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 720 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1920 \, {\left (d x + c\right )} b^{3} + 360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1200 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, {\left (a^{3} + 18 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {294 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 5292 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1200 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 720 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 90 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7*(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/1920*(5*a^3*tan(1/2*d*x + 1/2*c)^6 + 36*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 15*a^3*tan(1/2*d*x + 1/2*c)^4 + 90*a*
b^2*tan(1/2*d*x + 1/2*c)^4 - 180*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 80*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*a^3*tan(1/2
*d*x + 1/2*c)^2 - 720*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 1920*(d*x + c)*b^3 + 360*a^2*b*tan(1/2*d*x + 1/2*c) - 120
0*b^3*tan(1/2*d*x + 1/2*c) + 120*(a^3 + 18*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c))) - (294*a^3*tan(1/2*d*x + 1/2*
c)^6 + 5292*a*b^2*tan(1/2*d*x + 1/2*c)^6 + 360*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 1200*b^3*tan(1/2*d*x + 1/2*c)^5
- 15*a^3*tan(1/2*d*x + 1/2*c)^4 - 720*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 180*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 80*b^3
*tan(1/2*d*x + 1/2*c)^3 - 15*a^3*tan(1/2*d*x + 1/2*c)^2 + 90*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 36*a^2*b*tan(1/2*d
*x + 1/2*c) + 5*a^3)/tan(1/2*d*x + 1/2*c)^6)/d

________________________________________________________________________________________

maple [A]  time = 0.51, size = 302, normalized size = 1.10 \[ -\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{6 d \sin \left (d x +c \right )^{6}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{24 d \sin \left (d x +c \right )^{4}}+\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{2}}+\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{48 d}+\frac {a^{3} \cos \left (d x +c \right )}{16 d}+\frac {a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {3 a^{2} b \left (\cos ^{5}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}}-\frac {3 a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}+\frac {3 a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}+\frac {3 a \,b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{8 d}+\frac {9 a \,b^{2} \cos \left (d x +c \right )}{8 d}+\frac {9 a \,b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {b^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {\cot \left (d x +c \right ) b^{3}}{d}+b^{3} x +\frac {b^{3} c}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^7*(a+b*sin(d*x+c))^3,x)

[Out]

-1/6/d*a^3/sin(d*x+c)^6*cos(d*x+c)^5-1/24/d*a^3/sin(d*x+c)^4*cos(d*x+c)^5+1/48/d*a^3/sin(d*x+c)^2*cos(d*x+c)^5
+1/48*a^3*cos(d*x+c)^3/d+1/16*a^3*cos(d*x+c)/d+1/16/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-3/5/d*a^2*b/sin(d*x+c)^5*c
os(d*x+c)^5-3/4/d*a*b^2/sin(d*x+c)^4*cos(d*x+c)^5+3/8/d*a*b^2/sin(d*x+c)^2*cos(d*x+c)^5+3/8*a*b^2*cos(d*x+c)^3
/d+9/8*a*b^2*cos(d*x+c)/d+9/8/d*a*b^2*ln(csc(d*x+c)-cot(d*x+c))-1/3/d*b^3*cot(d*x+c)^3+1/d*cot(d*x+c)*b^3+b^3*
x+1/d*b^3*c

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 217, normalized size = 0.79 \[ \frac {160 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} b^{3} + 5 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 90 \, a b^{2} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {288 \, a^{2} b}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7*(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/480*(160*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*b^3 + 5*a^3*(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)) - 90*a*b^2*(2*(5*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^
2 + 1) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) - 288*a^2*b/tan(d*x + c)^5)/d

________________________________________________________________________________________

mupad [B]  time = 9.93, size = 446, normalized size = 1.62 \[ \frac {2\,b^3\,\mathrm {atan}\left (\frac {4\,b^6}{\frac {a^3\,b^3}{4}+\frac {9\,a\,b^5}{2}-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6}+\frac {9\,a\,b^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {a^3\,b^3}{4}+\frac {9\,a\,b^5}{2}-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6\right )}+\frac {a^3\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {a^3\,b^3}{4}+\frac {9\,a\,b^5}{2}-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6\right )}\right )}{d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3}{128}+\frac {3\,a\,b^2}{8}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {3\,a\,b^2}{64}-\frac {a^3}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {3\,a^2\,b}{32}-\frac {b^3}{24}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (3\,a\,b^2-\frac {a^3}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3}{2}+24\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (6\,a^2\,b-\frac {8\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (12\,a^2\,b-40\,b^3\right )+\frac {a^3}{6}+\frac {6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}}{64\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^2\,b}{16}-\frac {5\,b^3}{8}\right )}{d}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2+18\,b^2\right )}{16\,d}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d} \]

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)^7,x)

[Out]

(2*b^3*atan((4*b^6)/((9*a*b^5)/2 + (a^3*b^3)/4 - 4*b^6*tan(c/2 + (d*x)/2)) + (9*a*b^5*tan(c/2 + (d*x)/2))/(2*(
(9*a*b^5)/2 + (a^3*b^3)/4 - 4*b^6*tan(c/2 + (d*x)/2))) + (a^3*b^3*tan(c/2 + (d*x)/2))/(4*((9*a*b^5)/2 + (a^3*b
^3)/4 - 4*b^6*tan(c/2 + (d*x)/2)))))/d + (a^3*tan(c/2 + (d*x)/2)^6)/(384*d) - (tan(c/2 + (d*x)/2)^2*((3*a*b^2)
/8 + a^3/128))/d + (tan(c/2 + (d*x)/2)^4*((3*a*b^2)/64 - a^3/128))/d - (tan(c/2 + (d*x)/2)^3*((3*a^2*b)/32 - b
^3/24))/d - (tan(c/2 + (d*x)/2)^2*(3*a*b^2 - a^3/2) - tan(c/2 + (d*x)/2)^4*(24*a*b^2 + a^3/2) - tan(c/2 + (d*x
)/2)^3*(6*a^2*b - (8*b^3)/3) + tan(c/2 + (d*x)/2)^5*(12*a^2*b - 40*b^3) + a^3/6 + (6*a^2*b*tan(c/2 + (d*x)/2))
/5)/(64*d*tan(c/2 + (d*x)/2)^6) + (tan(c/2 + (d*x)/2)*((3*a^2*b)/16 - (5*b^3)/8))/d + (a*log(tan(c/2 + (d*x)/2
))*(a^2 + 18*b^2))/(16*d) + (3*a^2*b*tan(c/2 + (d*x)/2)^5)/(160*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)**7*(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

________________________________________________________________________________________