3.1254 \(\int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=210 \[ \frac{\left (3 a^2+10 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{256 d}+\frac{\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac{\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}+\frac{\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}-\frac{a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}-\frac{2 a b \cot ^9(c+d x)}{9 d}-\frac{2 a b \cot ^7(c+d x)}{7 d} \]

[Out]

((3*a^2 + 10*b^2)*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a*b*Cot[c + d*x]^7)/(7*d) - (2*a*b*Cot[c + d*x]^9)/(9*d)
+ ((3*a^2 + 10*b^2)*Cot[c + d*x]*Csc[c + d*x])/(256*d) + ((3*a^2 - 118*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(384
*d) - ((93*a^2 - 170*b^2)*Cot[c + d*x]*Csc[c + d*x]^5)/(480*d) + ((21*a^2 - 10*b^2)*Cot[c + d*x]*Csc[c + d*x]^
7)/(80*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^9)/(10*d)

________________________________________________________________________________________

Rubi [A]  time = 0.344888, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {2911, 2607, 14, 4366, 455, 1814, 1157, 385, 199, 206} \[ \frac{\left (3 a^2+10 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{256 d}+\frac{\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac{\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}+\frac{\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}-\frac{a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}-\frac{2 a b \cot ^9(c+d x)}{9 d}-\frac{2 a b \cot ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]

[Out]

((3*a^2 + 10*b^2)*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a*b*Cot[c + d*x]^7)/(7*d) - (2*a*b*Cot[c + d*x]^9)/(9*d)
+ ((3*a^2 + 10*b^2)*Cot[c + d*x]*Csc[c + d*x])/(256*d) + ((3*a^2 - 118*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(384
*d) - ((93*a^2 - 170*b^2)*Cot[c + d*x]*Csc[c + d*x]^5)/(480*d) + ((21*a^2 - 10*b^2)*Cot[c + d*x]*Csc[c + d*x]^
7)/(80*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^9)/(10*d)

Rule 2911

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^2, x_Symbol] :> Dist[(2*a*b)/d, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e
+ f*x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 -
b^2, 0]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 4366

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dis
t[d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d]
, x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] &&
(EqQ[F, Sin] || EqQ[F, sin])

Rule 455

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\int \cot ^6(c+d x) \csc ^5(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^6 \left (a^2+b^2-b^2 x^2\right )}{\left (1-x^2\right )^6} \, dx,x,\cos (c+d x)\right )}{d}+\frac{(2 a b) \operatorname{Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac{\operatorname{Subst}\left (\int \frac{a^2+10 a^2 x^2+10 a^2 x^4-10 b^2 x^6}{\left (1-x^2\right )^5} \, dx,x,\cos (c+d x)\right )}{10 d}+\frac{(2 a b) \operatorname{Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{2 a b \cot ^7(c+d x)}{7 d}-\frac{2 a b \cot ^9(c+d x)}{9 d}+\frac{\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}-\frac{\operatorname{Subst}\left (\int \frac{13 a^2-10 b^2+80 \left (a^2-b^2\right ) x^2-80 b^2 x^4}{\left (1-x^2\right )^4} \, dx,x,\cos (c+d x)\right )}{80 d}\\ &=-\frac{2 a b \cot ^7(c+d x)}{7 d}-\frac{2 a b \cot ^9(c+d x)}{9 d}-\frac{\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac{\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac{\operatorname{Subst}\left (\int \frac{5 \left (3 a^2-22 b^2\right )-480 b^2 x^2}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{480 d}\\ &=-\frac{2 a b \cot ^7(c+d x)}{7 d}-\frac{2 a b \cot ^9(c+d x)}{9 d}+\frac{\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac{\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac{\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac{\left (3 a^2+10 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{128 d}\\ &=-\frac{2 a b \cot ^7(c+d x)}{7 d}-\frac{2 a b \cot ^9(c+d x)}{9 d}+\frac{\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}+\frac{\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac{\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac{\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac{\left (3 a^2+10 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{256 d}\\ &=\frac{\left (3 a^2+10 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{2 a b \cot ^7(c+d x)}{7 d}-\frac{2 a b \cot ^9(c+d x)}{9 d}+\frac{\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}+\frac{\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac{\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac{\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}\\ \end{align*}

Mathematica [A]  time = 1.43241, size = 244, normalized size = 1.16 \[ -\frac{80640 \left (3 a^2+10 b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-80640 \left (3 a^2+10 b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\csc ^{10}(c+d x) \left (630 \left (1879 a^2+290 b^2\right ) \cos (c+d x)+1260 \left (519 a^2-62 b^2\right ) \cos (3 (c+d x))+218484 a^2 \cos (5 (c+d x))+9135 a^2 \cos (7 (c+d x))-945 a^2 \cos (9 (c+d x))+537600 a b \sin (2 (c+d x))+522240 a b \sin (4 (c+d x))+207360 a b \sin (6 (c+d x))+25600 a b \sin (8 (c+d x))-2560 a b \sin (10 (c+d x))-24360 b^2 \cos (5 (c+d x))-77070 b^2 \cos (7 (c+d x))-3150 b^2 \cos (9 (c+d x))\right )}{20643840 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]

[Out]

-(-80640*(3*a^2 + 10*b^2)*Log[Cos[(c + d*x)/2]] + 80640*(3*a^2 + 10*b^2)*Log[Sin[(c + d*x)/2]] + Csc[c + d*x]^
10*(630*(1879*a^2 + 290*b^2)*Cos[c + d*x] + 1260*(519*a^2 - 62*b^2)*Cos[3*(c + d*x)] + 218484*a^2*Cos[5*(c + d
*x)] - 24360*b^2*Cos[5*(c + d*x)] + 9135*a^2*Cos[7*(c + d*x)] - 77070*b^2*Cos[7*(c + d*x)] - 945*a^2*Cos[9*(c
+ d*x)] - 3150*b^2*Cos[9*(c + d*x)] + 537600*a*b*Sin[2*(c + d*x)] + 522240*a*b*Sin[4*(c + d*x)] + 207360*a*b*S
in[6*(c + d*x)] + 25600*a*b*Sin[8*(c + d*x)] - 2560*a*b*Sin[10*(c + d*x)]))/(20643840*d)

________________________________________________________________________________________

Maple [B]  time = 0.101, size = 404, normalized size = 1.9 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{10\,d \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{160\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{640\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{1280\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{1280\,d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{256\,d}}-{\frac{3\,{a}^{2}\cos \left ( dx+c \right ) }{256\,d}}-{\frac{3\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{256\,d}}-{\frac{2\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{4\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{192\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d}}-{\frac{5\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{384\,d}}-{\frac{5\,{b}^{2}\cos \left ( dx+c \right ) }{128\,d}}-{\frac{5\,{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^11*(a+b*sin(d*x+c))^2,x)

[Out]

-1/10/d*a^2/sin(d*x+c)^10*cos(d*x+c)^7-3/80/d*a^2/sin(d*x+c)^8*cos(d*x+c)^7-1/160/d*a^2/sin(d*x+c)^6*cos(d*x+c
)^7+1/640/d*a^2/sin(d*x+c)^4*cos(d*x+c)^7-3/1280/d*a^2/sin(d*x+c)^2*cos(d*x+c)^7-3/1280*a^2*cos(d*x+c)^5/d-1/2
56*a^2*cos(d*x+c)^3/d-3/256*a^2*cos(d*x+c)/d-3/256/d*a^2*ln(csc(d*x+c)-cot(d*x+c))-2/9/d*a*b/sin(d*x+c)^9*cos(
d*x+c)^7-4/63/d*a*b/sin(d*x+c)^7*cos(d*x+c)^7-1/8/d*b^2/sin(d*x+c)^8*cos(d*x+c)^7-1/48/d*b^2/sin(d*x+c)^6*cos(
d*x+c)^7+1/192/d*b^2/sin(d*x+c)^4*cos(d*x+c)^7-1/128/d*b^2/sin(d*x+c)^2*cos(d*x+c)^7-1/128*b^2*cos(d*x+c)^5/d-
5/384*b^2*cos(d*x+c)^3/d-5/128*b^2*cos(d*x+c)/d-5/128/d*b^2*ln(csc(d*x+c)-cot(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 0.992257, size = 367, normalized size = 1.75 \begin{align*} -\frac{63 \, a^{2}{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 210 \, b^{2}{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{5120 \,{\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a b}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/161280*(63*a^2*(2*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 - 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos(
d*x + c))/(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1)
- 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 210*b^2*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 5
5*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 +
1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 5120*(9*tan(d*x + c)^2 + 7)*a*b/tan(d*x + c)^9)/d

________________________________________________________________________________________

Fricas [B]  time = 1.96849, size = 1138, normalized size = 5.42 \begin{align*} -\frac{630 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{9} - 420 \,{\left (21 \, a^{2} - 58 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 5376 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{5} + 2940 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 630 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right ) - 315 \,{\left ({\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{10} - 5 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{8} + 10 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 10 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 5 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - 10 \, b^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 315 \,{\left ({\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{10} - 5 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{8} + 10 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 10 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 5 \,{\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - 10 \, b^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 5120 \,{\left (2 \, a b \cos \left (d x + c\right )^{9} - 9 \, a b \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \,{\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, 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)^6*csc(d*x+c)^11*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/161280*(630*(3*a^2 + 10*b^2)*cos(d*x + c)^9 - 420*(21*a^2 - 58*b^2)*cos(d*x + c)^7 - 5376*(3*a^2 + 10*b^2)*
cos(d*x + c)^5 + 2940*(3*a^2 + 10*b^2)*cos(d*x + c)^3 - 630*(3*a^2 + 10*b^2)*cos(d*x + c) - 315*((3*a^2 + 10*b
^2)*cos(d*x + c)^10 - 5*(3*a^2 + 10*b^2)*cos(d*x + c)^8 + 10*(3*a^2 + 10*b^2)*cos(d*x + c)^6 - 10*(3*a^2 + 10*
b^2)*cos(d*x + c)^4 + 5*(3*a^2 + 10*b^2)*cos(d*x + c)^2 - 3*a^2 - 10*b^2)*log(1/2*cos(d*x + c) + 1/2) + 315*((
3*a^2 + 10*b^2)*cos(d*x + c)^10 - 5*(3*a^2 + 10*b^2)*cos(d*x + c)^8 + 10*(3*a^2 + 10*b^2)*cos(d*x + c)^6 - 10*
(3*a^2 + 10*b^2)*cos(d*x + c)^4 + 5*(3*a^2 + 10*b^2)*cos(d*x + c)^2 - 3*a^2 - 10*b^2)*log(-1/2*cos(d*x + c) +
1/2) + 5120*(2*a*b*cos(d*x + c)^9 - 9*a*b*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^
8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)

________________________________________________________________________________________

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)**6*csc(d*x+c)**11*(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.30524, size = 632, normalized size = 3.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1290240*(126*a^2*tan(1/2*d*x + 1/2*c)^10 + 560*a*b*tan(1/2*d*x + 1/2*c)^9 - 315*a^2*tan(1/2*d*x + 1/2*c)^8 +
630*b^2*tan(1/2*d*x + 1/2*c)^8 - 2160*a*b*tan(1/2*d*x + 1/2*c)^7 - 630*a^2*tan(1/2*d*x + 1/2*c)^6 - 3360*b^2*
tan(1/2*d*x + 1/2*c)^6 + 2520*a^2*tan(1/2*d*x + 1/2*c)^4 + 5040*b^2*tan(1/2*d*x + 1/2*c)^4 + 13440*a*b*tan(1/2
*d*x + 1/2*c)^3 + 1260*a^2*tan(1/2*d*x + 1/2*c)^2 + 10080*b^2*tan(1/2*d*x + 1/2*c)^2 - 30240*a*b*tan(1/2*d*x +
1/2*c) - 5040*(3*a^2 + 10*b^2)*log(abs(tan(1/2*d*x + 1/2*c))) + (44286*a^2*tan(1/2*d*x + 1/2*c)^10 + 147620*b
^2*tan(1/2*d*x + 1/2*c)^10 + 30240*a*b*tan(1/2*d*x + 1/2*c)^9 - 1260*a^2*tan(1/2*d*x + 1/2*c)^8 - 10080*b^2*ta
n(1/2*d*x + 1/2*c)^8 - 13440*a*b*tan(1/2*d*x + 1/2*c)^7 - 2520*a^2*tan(1/2*d*x + 1/2*c)^6 - 5040*b^2*tan(1/2*d
*x + 1/2*c)^6 + 630*a^2*tan(1/2*d*x + 1/2*c)^4 + 3360*b^2*tan(1/2*d*x + 1/2*c)^4 + 2160*a*b*tan(1/2*d*x + 1/2*
c)^3 + 315*a^2*tan(1/2*d*x + 1/2*c)^2 - 630*b^2*tan(1/2*d*x + 1/2*c)^2 - 560*a*b*tan(1/2*d*x + 1/2*c) - 126*a^
2)/tan(1/2*d*x + 1/2*c)^10)/d