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

[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)

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Rubi [A]  time = 0.755893, antiderivative size = 275, normalized size of antiderivative = 1., 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 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 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 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 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 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 \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 = 1.77393, size = 408, normalized size = 1.48 \[ \frac{-64 \left (9 a^2 b-20 b^3\right ) \cot \left (\frac{1}{2} (c+d x)\right )-30 \left (a^3-30 a b^2\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )+2 \csc ^4\left (\frac{1}{2} (c+d x)\right ) \left (b \left (63 a^2-20 b^2\right ) \sin (c+d x)+15 \left (a^3-3 a b^2\right )\right )+576 a^2 b \tan \left (\frac{1}{2} (c+d x)\right )-2016 a^2 b \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-a^2 \csc ^6\left (\frac{1}{2} (c+d x)\right ) (5 a+18 b \sin (c+d x))+36 a^2 b \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )+5 a^3 \sec ^6\left (\frac{1}{2} (c+d x)\right )-30 a^3 \sec ^4\left (\frac{1}{2} (c+d x)\right )+30 a^3 \sec ^2\left (\frac{1}{2} (c+d x)\right )+120 a^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-120 a^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+90 a b^2 \sec ^4\left (\frac{1}{2} (c+d x)\right )-900 a b^2 \sec ^2\left (\frac{1}{2} (c+d x)\right )+2160 a b^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2160 a b^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-1280 b^3 \tan \left (\frac{1}{2} (c+d x)\right )+640 b^3 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)+1920 b^3 c+1920 b^3 d x}{1920 d} \]

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)

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Maple [A]  time = 0.112, size = 302, normalized size = 1.1 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{48\,d}}+{\frac{{a}^{3}\cos \left ( dx+c \right ) }{16\,d}}+{\frac{{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{3\,{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3\,a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{9\,a{b}^{2}\cos \left ( dx+c \right ) }{8\,d}}+{\frac{9\,a{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{b}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{\cot \left ( dx+c \right ){b}^{3}}{d}}+{b}^{3}x+{\frac{{b}^{3}c}{d}} \end{align*}

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

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Maxima [A]  time = 1.64595, size = 293, normalized size = 1.07 \begin{align*} \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} \end{align*}

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

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Fricas [A]  time = 1.90801, size = 941, normalized size = 3.42 \begin{align*} \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 )}} \end{align*}

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)

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

[Out]

Timed out

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Giac [A]  time = 1.45618, size = 539, normalized size = 1.96 \begin{align*} \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} \end{align*}

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