3.1118 \(\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=250 \[ -\frac{a \left (-43 a^2 b^2+2 a^4+36 b^4\right ) \cos (c+d x)}{60 b^2 d}-\frac{\left (2 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}-\frac{a \left (2 a^2-39 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}-\frac{\left (-84 a^2 b^2+4 a^4+15 b^4\right ) \sin (c+d x) \cos (c+d x)}{240 b d}+\frac{1}{16} b x \left (18 a^2+b^2\right )-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{6 b d} \]

[Out]

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

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Rubi [A]  time = 0.659357, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2895, 3049, 3033, 3023, 2735, 3770} \[ -\frac{a \left (-43 a^2 b^2+2 a^4+36 b^4\right ) \cos (c+d x)}{60 b^2 d}-\frac{\left (2 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}-\frac{a \left (2 a^2-39 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}-\frac{\left (-84 a^2 b^2+4 a^4+15 b^4\right ) \sin (c+d x) \cos (c+d x)}{240 b d}+\frac{1}{16} b x \left (18 a^2+b^2\right )-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{6 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2895

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(a*(n + 3)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(b^2*d*f*(m
+ n + 3)*(m + n + 4)), x] + (-Dist[1/(b^2*(m + n + 3)*(m + n + 4)), Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x
])^m*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*
(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x], x] - Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e
+ f*x])^(m + 1))/(b*d^2*f*(m + n + 4)), x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[
m, 0] || IntegersQ[2*m, 2*n]) &&  !m < -1 &&  !LtQ[n, -1] && NeQ[m + n + 3, 0] && NeQ[m + n + 4, 0]

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((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*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3033

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[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b
*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*
(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e +
f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

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

Mathematica [A]  time = 0.51171, size = 191, normalized size = 0.76 \[ \frac{120 a \left (10 a^2-3 b^2\right ) \cos (c+d x)+20 \left (4 a^3-9 a b^2\right ) \cos (3 (c+d x))+720 a^2 b \sin (2 (c+d x))+90 a^2 b \sin (4 (c+d x))+1080 a^2 b c+1080 a^2 b d x+960 a^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-960 a^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-36 a b^2 \cos (5 (c+d x))+15 b^3 \sin (2 (c+d x))-15 b^3 \sin (4 (c+d x))-5 b^3 \sin (6 (c+d x))+60 b^3 c+60 b^3 d x}{960 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1080*a^2*b*c + 60*b^3*c + 1080*a^2*b*d*x + 60*b^3*d*x + 120*a*(10*a^2 - 3*b^2)*Cos[c + d*x] + 20*(4*a^3 - 9*a
*b^2)*Cos[3*(c + d*x)] - 36*a*b^2*Cos[5*(c + d*x)] - 960*a^3*Log[Cos[(c + d*x)/2]] + 960*a^3*Log[Sin[(c + d*x)
/2]] + 720*a^2*b*Sin[2*(c + d*x)] + 15*b^3*Sin[2*(c + d*x)] + 90*a^2*b*Sin[4*(c + d*x)] - 15*b^3*Sin[4*(c + d*
x)] - 5*b^3*Sin[6*(c + d*x)])/(960*d)

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Maple [A]  time = 0.098, size = 211, normalized size = 0.8 \begin{align*}{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{3}\cos \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,{a}^{2}b\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{9\,{a}^{2}b\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{9\,{a}^{2}bx}{8}}+{\frac{9\,{a}^{2}bc}{8\,d}}-{\frac{3\,a \left ( \cos \left ( dx+c \right ) \right ) ^{5}{b}^{2}}{5\,d}}-{\frac{{b}^{3}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6\,d}}+{\frac{{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{24\,d}}+{\frac{{b}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{16\,d}}+{\frac{{b}^{3}x}{16}}+{\frac{{b}^{3}c}{16\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.159, size = 185, normalized size = 0.74 \begin{align*} -\frac{576 \, a b^{2} \cos \left (d x + c\right )^{5} - 160 \,{\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \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 )} a^{3} - 90 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} b - 5 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{3}}{960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*(576*a*b^2*cos(d*x + c)^5 - 160*(2*cos(d*x + c)^3 + 6*cos(d*x + c) - 3*log(cos(d*x + c) + 1) + 3*log(co
s(d*x + c) - 1))*a^3 - 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^2*b - 5*(4*sin(2*d*x + 2*c
)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*b^3)/d

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Fricas [A]  time = 1.96491, size = 401, normalized size = 1.6 \begin{align*} -\frac{144 \, a b^{2} \cos \left (d x + c\right )^{5} - 80 \, a^{3} \cos \left (d x + c\right )^{3} - 240 \, a^{3} \cos \left (d x + c\right ) + 120 \, a^{3} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 120 \, a^{3} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 15 \,{\left (18 \, a^{2} b + b^{3}\right )} d x + 5 \,{\left (8 \, b^{3} \cos \left (d x + c\right )^{5} - 2 \,{\left (18 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (18 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/240*(144*a*b^2*cos(d*x + c)^5 - 80*a^3*cos(d*x + c)^3 - 240*a^3*cos(d*x + c) + 120*a^3*log(1/2*cos(d*x + c)
+ 1/2) - 120*a^3*log(-1/2*cos(d*x + c) + 1/2) - 15*(18*a^2*b + b^3)*d*x + 5*(8*b^3*cos(d*x + c)^5 - 2*(18*a^2
*b + b^3)*cos(d*x + c)^3 - 3*(18*a^2*b + b^3)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.30763, size = 576, normalized size = 2.3 \begin{align*} \frac{240 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 15 \,{\left (18 \, a^{2} b + b^{3}\right )}{\left (d x + c\right )} - \frac{2 \,{\left (450 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 15 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 480 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 720 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 630 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 235 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1920 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 720 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 180 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 390 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 3200 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1440 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 180 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 390 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2880 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1440 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 630 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 235 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1440 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 144 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 450 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 320 \, a^{3} + 144 \, a b^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(240*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 15*(18*a^2*b + b^3)*(d*x + c) - 2*(450*a^2*b*tan(1/2*d*x + 1/2
*c)^11 - 15*b^3*tan(1/2*d*x + 1/2*c)^11 - 480*a^3*tan(1/2*d*x + 1/2*c)^10 + 720*a*b^2*tan(1/2*d*x + 1/2*c)^10
+ 630*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 235*b^3*tan(1/2*d*x + 1/2*c)^9 - 1920*a^3*tan(1/2*d*x + 1/2*c)^8 + 720*a*
b^2*tan(1/2*d*x + 1/2*c)^8 + 180*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 390*b^3*tan(1/2*d*x + 1/2*c)^7 - 3200*a^3*tan(
1/2*d*x + 1/2*c)^6 + 1440*a*b^2*tan(1/2*d*x + 1/2*c)^6 - 180*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 390*b^3*tan(1/2*d*
x + 1/2*c)^5 - 2880*a^3*tan(1/2*d*x + 1/2*c)^4 + 1440*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 630*a^2*b*tan(1/2*d*x + 1
/2*c)^3 - 235*b^3*tan(1/2*d*x + 1/2*c)^3 - 1440*a^3*tan(1/2*d*x + 1/2*c)^2 + 144*a*b^2*tan(1/2*d*x + 1/2*c)^2
- 450*a^2*b*tan(1/2*d*x + 1/2*c) + 15*b^3*tan(1/2*d*x + 1/2*c) - 320*a^3 + 144*a*b^2)/(tan(1/2*d*x + 1/2*c)^2
+ 1)^6)/d