3.1077 \(\int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=212 \[ \frac{b \left (6 a^2+5 b^2\right ) \cot (c+d x)}{15 d}+\frac{a \left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{120 d}+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac{a \left (a^2+6 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}-\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d} \]

[Out]

(a*(a^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) + (b*(6*a^2 + 5*b^2)*Cot[c + d*x])/(15*d) + (a*(a^2 + 6*b^2)*Co
t[c + d*x]*Csc[c + d*x])/(16*d) + (b*(3*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) + (a*(5*a^2 - 6*b^2)*Co
t[c + d*x]*Csc[c + d*x]^3)/(120*d) - (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(10*d) - (Cot[c +
d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(6*d)

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Rubi [A]  time = 0.598271, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {2889, 3048, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{b \left (6 a^2+5 b^2\right ) \cot (c+d x)}{15 d}+\frac{a \left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{120 d}+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac{a \left (a^2+6 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}-\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]

[Out]

(a*(a^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) + (b*(6*a^2 + 5*b^2)*Cot[c + d*x])/(15*d) + (a*(a^2 + 6*b^2)*Co
t[c + d*x]*Csc[c + d*x])/(16*d) + (b*(3*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) + (a*(5*a^2 - 6*b^2)*Co
t[c + d*x]*Csc[c + d*x]^3)/(120*d) - (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(10*d) - (Cot[c +
d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(6*d)

Rule 2889

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

Rule 3048

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n +
2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(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, 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 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 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 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]

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx &=\int \csc ^7(c+d x) (a+b \sin (c+d x))^3 \left (1-\sin ^2(c+d x)\right ) \, dx\\ &=-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}+\frac{1}{6} \int \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (3 b-a \sin (c+d x)-4 b \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}+\frac{1}{30} \int \csc ^5(c+d x) (a+b \sin (c+d x)) \left (-5 a^2+6 b^2-13 a b \sin (c+d x)-14 b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{120 d}-\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}-\frac{1}{120} \int \csc ^4(c+d x) \left (24 b \left (3 a^2-b^2\right )+15 a \left (a^2+6 b^2\right ) \sin (c+d x)+56 b^3 \sin ^2(c+d x)\right ) \, dx\\ &=\frac{b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac{a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{120 d}-\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}-\frac{1}{360} \int \csc ^3(c+d x) \left (45 a \left (a^2+6 b^2\right )+24 b \left (6 a^2+5 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac{b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac{a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{120 d}-\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}-\frac{1}{15} \left (b \left (6 a^2+5 b^2\right )\right ) \int \csc ^2(c+d x) \, dx-\frac{1}{8} \left (a \left (a^2+6 b^2\right )\right ) \int \csc ^3(c+d x) \, dx\\ &=\frac{a \left (a^2+6 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac{a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{120 d}-\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}-\frac{1}{16} \left (a \left (a^2+6 b^2\right )\right ) \int \csc (c+d x) \, dx+\frac{\left (b \left (6 a^2+5 b^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{15 d}\\ &=\frac{a \left (a^2+6 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{b \left (6 a^2+5 b^2\right ) \cot (c+d x)}{15 d}+\frac{a \left (a^2+6 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac{a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{120 d}-\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\\ \end{align*}

Mathematica [A]  time = 2.02905, size = 369, normalized size = 1.74 \[ -\frac{-64 \left (6 a^2 b+5 b^3\right ) \cot \left (\frac{1}{2} (c+d x)\right )-30 \left (a^3+6 a b^2\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )+2 b \csc ^4\left (\frac{1}{2} (c+d x)\right ) \left (\left (20 b^2-3 a^2\right ) \sin (c+d x)+45 a b\right )+384 a^2 b \tan \left (\frac{1}{2} (c+d x)\right )+96 a^2 b \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)+a^2 \csc ^6\left (\frac{1}{2} (c+d x)\right ) (5 a+18 b \sin (c+d x))-36 a^2 b \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )-5 a^3 \sec ^6\left (\frac{1}{2} (c+d x)\right )+30 a^3 \sec ^2\left (\frac{1}{2} (c+d x)\right )+120 a^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-120 a^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-90 a b^2 \sec ^4\left (\frac{1}{2} (c+d x)\right )+180 a b^2 \sec ^2\left (\frac{1}{2} (c+d x)\right )+720 a b^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-720 a b^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+320 b^3 \tan \left (\frac{1}{2} (c+d x)\right )-640 b^3 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)}{1920 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]

[Out]

-(-64*(6*a^2*b + 5*b^3)*Cot[(c + d*x)/2] - 30*(a^3 + 6*a*b^2)*Csc[(c + d*x)/2]^2 - 120*a^3*Log[Cos[(c + d*x)/2
]] - 720*a*b^2*Log[Cos[(c + d*x)/2]] + 120*a^3*Log[Sin[(c + d*x)/2]] + 720*a*b^2*Log[Sin[(c + d*x)/2]] + 30*a^
3*Sec[(c + d*x)/2]^2 + 180*a*b^2*Sec[(c + d*x)/2]^2 - 90*a*b^2*Sec[(c + d*x)/2]^4 - 5*a^3*Sec[(c + d*x)/2]^6 +
96*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*b*Csc[(c + d*x)/2]^4*(45*a*b + (-3*a^2 + 20*b^2)*Sin[c + d*x]) + 384*a^2*b*Ta
n[(c + d*x)/2] + 320*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 = 0.106, size = 276, normalized size = 1.3 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\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 ) ^{3}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{2\,{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{3\,a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,a{b}^{2}\cos \left ( dx+c \right ) }{8\,d}}-{\frac{3\,a{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.12276, size = 273, normalized size = 1.29 \begin{align*} -\frac{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 (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{160 \, b^{3}}{\tan \left (d x + c\right )^{3}} + \frac{96 \,{\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} 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)^2*csc(d*x+c)^7*(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/480*(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*(cos(d*x + c)^3 + cos(
d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) + 160*b^3/t
an(d*x + c)^3 + 96*(5*tan(d*x + c)^2 + 3)*a^2*b/tan(d*x + c)^5)/d

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Fricas [A]  time = 1.50196, size = 755, normalized size = 3.56 \begin{align*} \frac{80 \, a^{3} \cos \left (d x + c\right )^{3} - 30 \,{\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 30 \,{\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right ) + 15 \,{\left ({\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 6 \, a b^{2} + 3 \,{\left (a^{3} + 6 \, 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} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 6 \, a b^{2} + 3 \,{\left (a^{3} + 6 \, 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 ({\left (6 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{5} - 5 \,{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3}\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)^2*csc(d*x+c)^7*(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/480*(80*a^3*cos(d*x + c)^3 - 30*(a^3 + 6*a*b^2)*cos(d*x + c)^5 + 30*(a^3 + 6*a*b^2)*cos(d*x + c) + 15*((a^3
+ 6*a*b^2)*cos(d*x + c)^6 - 3*(a^3 + 6*a*b^2)*cos(d*x + c)^4 - a^3 - 6*a*b^2 + 3*(a^3 + 6*a*b^2)*cos(d*x + c)^
2)*log(1/2*cos(d*x + c) + 1/2) - 15*((a^3 + 6*a*b^2)*cos(d*x + c)^6 - 3*(a^3 + 6*a*b^2)*cos(d*x + c)^4 - a^3 -
6*a*b^2 + 3*(a^3 + 6*a*b^2)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 32*((6*a^2*b + 5*b^3)*cos(d*x + c)
^5 - 5*(3*a^2*b + b^3)*cos(d*x + c)^3)*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)**2*csc(d*x+c)**7*(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.32878, size = 478, normalized size = 2.25 \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} + 60 \, 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} - 360 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 240 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 120 \,{\left (a^{3} + 6 \, 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} + 1764 \, 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} + 240 \, 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} - 60 \, 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)^2*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 + 60*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 - 360*a^2*b*tan(1/2*d*x + 1/2*c) - 240*b^3*tan(1/2*d*x + 1/2*c) - 120*(a^3 + 6*a*b^2)*log(abs(t
an(1/2*d*x + 1/2*c))) + (294*a^3*tan(1/2*d*x + 1/2*c)^6 + 1764*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 + 240*b^3*tan(1/2*d*x + 1/2*c)^5 + 15*a^3*tan(1/2*d*x + 1/2*c)^4 - 60*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