3.1123 \(\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=227 \[ -\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}-\frac {3 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}+3 a b^2 x-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d} \]

[Out]

3*a*b^2*x-3/8*b*(3*a^2-4*b^2)*arctanh(cos(d*x+c))/d-1/40*b^3*(83*a^2+2*b^2)*cos(d*x+c)/a^2/d-1/20*a*(4*a^2-29*
b^2)*cot(d*x+c)/d+27/40*b*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^2/d+2/5*cot(d*x+c)*csc(d*x+c)^2*(a+b*sin(d*x+
c))^3/d+1/20*b*cot(d*x+c)*csc(d*x+c)^3*(a+b*sin(d*x+c))^4/a^2/d-1/5*cot(d*x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^4
/a/d

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Rubi [A]  time = 0.71, antiderivative size = 227, 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, 3023, 2735, 3770} \[ -\frac {b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}-\frac {3 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}+3 a b^2 x-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]

[Out]

3*a*b^2*x - (3*b*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) - (b^3*(83*a^2 + 2*b^2)*Cos[c + d*x])/(40*a^2*d)
- (a*(4*a^2 - 29*b^2)*Cot[c + d*x])/(20*d) + (27*b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^2)/(40*d) +
(2*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3)/(5*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c +
d*x])^4)/(20*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(5*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 3023

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

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 ^2(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}-\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x))^3 \left (24 a^2+3 a b \sin (c+d x)-\left (20 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{20 a^2}\\ &=\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}-\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x))^2 \left (81 a^2 b-6 a \left (2 a^2-b^2\right ) \sin (c+d x)-3 b \left (28 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{60 a^2}\\ &=\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}-\frac {\int \csc ^2(c+d x) (a+b \sin (c+d x)) \left (-6 a^2 \left (4 a^2-29 b^2\right )-3 a b \left (37 a^2-2 b^2\right ) \sin (c+d x)-3 b^2 \left (83 a^2+2 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2}\\ &=-\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac {\int \csc (c+d x) \left (45 a^2 b \left (3 a^2-4 b^2\right )+360 a^3 b^2 \sin (c+d x)+3 b^3 \left (83 a^2+2 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2}\\ &=-\frac {b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac {\int \csc (c+d x) \left (45 a^2 b \left (3 a^2-4 b^2\right )+360 a^3 b^2 \sin (c+d x)\right ) \, dx}{120 a^2}\\ &=3 a b^2 x-\frac {b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac {1}{8} \left (3 b \left (3 a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx\\ &=3 a b^2 x-\frac {3 b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}\\ \end {align*}

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Mathematica [A]  time = 1.30, size = 405, normalized size = 1.78 \[ \frac {-32 \left (a^3-20 a b^2\right ) \cot \left (\frac {1}{2} (c+d x)\right )+32 a^3 \tan \left (\frac {1}{2} (c+d x)\right )-a^3 \sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )+64 a^3 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \csc ^5(c+d x)+7 a^3 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )-112 a^3 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)-15 a^2 b \csc ^4\left (\frac {1}{2} (c+d x)\right )+150 a^2 b \csc ^2\left (\frac {1}{2} (c+d x)\right )+15 a^2 b \sec ^4\left (\frac {1}{2} (c+d x)\right )-150 a^2 b \sec ^2\left (\frac {1}{2} (c+d x)\right )+360 a^2 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-360 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-640 a b^2 \tan \left (\frac {1}{2} (c+d x)\right )-20 a b^2 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+320 a b^2 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+960 a b^2 c+960 a b^2 d x-320 b^3 \cos (c+d x)-40 b^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )+40 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )-480 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{320 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]

[Out]

(960*a*b^2*c + 960*a*b^2*d*x - 320*b^3*Cos[c + d*x] - 32*(a^3 - 20*a*b^2)*Cot[(c + d*x)/2] + 150*a^2*b*Csc[(c
+ d*x)/2]^2 - 40*b^3*Csc[(c + d*x)/2]^2 - 15*a^2*b*Csc[(c + d*x)/2]^4 - 360*a^2*b*Log[Cos[(c + d*x)/2]] + 480*
b^3*Log[Cos[(c + d*x)/2]] + 360*a^2*b*Log[Sin[(c + d*x)/2]] - 480*b^3*Log[Sin[(c + d*x)/2]] - 150*a^2*b*Sec[(c
+ d*x)/2]^2 + 40*b^3*Sec[(c + d*x)/2]^2 + 15*a^2*b*Sec[(c + d*x)/2]^4 - 112*a^3*Csc[c + d*x]^3*Sin[(c + d*x)/
2]^4 + 320*a*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 64*a^3*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 + 7*a^3*Csc[(c +
d*x)/2]^4*Sin[c + d*x] - 20*a*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - a^3*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 32*
a^3*Tan[(c + d*x)/2] - 640*a*b^2*Tan[(c + d*x)/2])/(320*d)

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fricas [A]  time = 0.91, size = 334, normalized size = 1.47 \[ -\frac {560 \, a b^{2} \cos \left (d x + c\right )^{3} + 16 \, {\left (a^{3} - 20 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 240 \, a b^{2} \cos \left (d x + c\right ) + 15 \, {\left ({\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b - 4 \, b^{3} - 2 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left ({\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b - 4 \, b^{3} - 2 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 10 \, {\left (24 \, a b^{2} d x \cos \left (d x + c\right )^{4} - 8 \, b^{3} \cos \left (d x + c\right )^{5} - 48 \, a b^{2} d x \cos \left (d x + c\right )^{2} + 24 \, a b^{2} d x - 5 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/80*(560*a*b^2*cos(d*x + c)^3 + 16*(a^3 - 20*a*b^2)*cos(d*x + c)^5 - 240*a*b^2*cos(d*x + c) + 15*((3*a^2*b -
4*b^3)*cos(d*x + c)^4 + 3*a^2*b - 4*b^3 - 2*(3*a^2*b - 4*b^3)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin
(d*x + c) - 15*((3*a^2*b - 4*b^3)*cos(d*x + c)^4 + 3*a^2*b - 4*b^3 - 2*(3*a^2*b - 4*b^3)*cos(d*x + c)^2)*log(-
1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 10*(24*a*b^2*d*x*cos(d*x + c)^4 - 8*b^3*cos(d*x + c)^5 - 48*a*b^2*d*x*c
os(d*x + c)^2 + 24*a*b^2*d*x - 5*(3*a^2*b - 4*b^3)*cos(d*x + c)^3 + 3*(3*a^2*b - 4*b^3)*cos(d*x + c))*sin(d*x
+ c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))

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giac [A]  time = 0.35, size = 356, normalized size = 1.57 \[ \frac {2 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 960 \, {\left (d x + c\right )} a b^{2} + 20 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 600 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {640 \, b^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 120 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {822 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1096 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 600 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/320*(2*a^3*tan(1/2*d*x + 1/2*c)^5 + 15*a^2*b*tan(1/2*d*x + 1/2*c)^4 - 10*a^3*tan(1/2*d*x + 1/2*c)^3 + 40*a*b
^2*tan(1/2*d*x + 1/2*c)^3 - 120*a^2*b*tan(1/2*d*x + 1/2*c)^2 + 40*b^3*tan(1/2*d*x + 1/2*c)^2 + 960*(d*x + c)*a
*b^2 + 20*a^3*tan(1/2*d*x + 1/2*c) - 600*a*b^2*tan(1/2*d*x + 1/2*c) - 640*b^3/(tan(1/2*d*x + 1/2*c)^2 + 1) + 1
20*(3*a^2*b - 4*b^3)*log(abs(tan(1/2*d*x + 1/2*c))) - (822*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 1096*b^3*tan(1/2*d*x
+ 1/2*c)^5 + 20*a^3*tan(1/2*d*x + 1/2*c)^4 - 600*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 120*a^2*b*tan(1/2*d*x + 1/2*c
)^3 + 40*b^3*tan(1/2*d*x + 1/2*c)^3 - 10*a^3*tan(1/2*d*x + 1/2*c)^2 + 40*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 15*a^2
*b*tan(1/2*d*x + 1/2*c) + 2*a^3)/tan(1/2*d*x + 1/2*c)^5)/d

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maple [A]  time = 0.51, size = 260, normalized size = 1.15 \[ -\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}}-\frac {3 a^{2} b \left (\cos ^{5}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}+\frac {3 a^{2} b \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}+\frac {3 a^{2} b \left (\cos ^{3}\left (d x +c \right )\right )}{8 d}+\frac {9 a^{2} b \cos \left (d x +c \right )}{8 d}+\frac {9 a^{2} b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {a \,b^{2} \left (\cot ^{3}\left (d x +c \right )\right )}{d}+3 a \,b^{2} x +\frac {3 a \,b^{2} \cot \left (d x +c \right )}{d}+\frac {3 a \,b^{2} c}{d}-\frac {b^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {b^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{2 d}-\frac {3 b^{3} \cos \left (d x +c \right )}{2 d}-\frac {3 b^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^6*(a+b*sin(d*x+c))^3,x)

[Out]

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

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maxima [A]  time = 0.49, size = 182, normalized size = 0.80 \[ \frac {80 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a b^{2} - 15 \, a^{2} b {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 20 \, b^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {16 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{80 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/80*(80*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a*b^2 - 15*a^2*b*(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)) + 20*b^3
*(2*cos(d*x + c)/(cos(d*x + c)^2 - 1) - 4*cos(d*x + c) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) -
16*a^3/tan(d*x + c)^5)/d

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mupad [B]  time = 12.28, size = 1007, normalized size = 4.44 \[ \text {result too large to display} \]

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)^6,x)

[Out]

-(2*a^3*cos(c/2 + (d*x)/2)^12 - 2*a^3*sin(c/2 + (d*x)/2)^12 + 8*a^3*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10
- 10*a^3*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 + 10*a^3*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 - 8*a^3
*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 - 40*b^3*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^9 - 40*b^3*cos(c/
2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 + 680*b^3*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 + 40*b^3*cos(c/2 + (d*
x)/2)^9*sin(c/2 + (d*x)/2)^3 + 480*b^3*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^5*sin(c/2
+ (d*x)/2)^7 + 480*b^3*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 -
15*a^2*b*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^11 + 15*a^2*b*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2) - 40*a*
b^2*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 560*a*b^2*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 - 560*a*b
^2*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 + 40*a*b^2*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 105*a^2*b
*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^9 + 120*a^2*b*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 - 120*a^2*b*c
os(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 - 105*a^2*b*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 + 1920*a*b^2*at
an((3*a^2*sin(c/2 + (d*x)/2) - 4*b^2*sin(c/2 + (d*x)/2) + 8*a*b*cos(c/2 + (d*x)/2))/(4*b^2*cos(c/2 + (d*x)/2)
- 3*a^2*cos(c/2 + (d*x)/2) + 8*a*b*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 + 1920*a*b^2
*atan((3*a^2*sin(c/2 + (d*x)/2) - 4*b^2*sin(c/2 + (d*x)/2) + 8*a*b*cos(c/2 + (d*x)/2))/(4*b^2*cos(c/2 + (d*x)/
2) - 3*a^2*cos(c/2 + (d*x)/2) + 8*a*b*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 - 360*a^2
*b*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 - 360*a^2*b*log(sin(c/
2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5)/(320*d*cos(c/2 + (d*x)/2)^5*sin(c/
2 + (d*x)/2)^5*(cos(c/2 + (d*x)/2)^2 + sin(c/2 + (d*x)/2)^2))

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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)**6*(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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