3.1255 \(\int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=198 \[ -\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a b \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {3 a b \cot (c+d x) \csc (c+d x)}{128 d} \]

[Out]

3/128*a*b*arctanh(cos(d*x+c))/d-1/7*(a^2+b^2)*cot(d*x+c)^7/d-1/9*(2*a^2+b^2)*cot(d*x+c)^9/d-1/11*a^2*cot(d*x+c
)^11/d+3/128*a*b*cot(d*x+c)*csc(d*x+c)/d+1/64*a*b*cot(d*x+c)*csc(d*x+c)^3/d-1/16*a*b*cot(d*x+c)*csc(d*x+c)^5/d
+1/8*a*b*cot(d*x+c)^3*csc(d*x+c)^5/d-1/5*a*b*cot(d*x+c)^5*csc(d*x+c)^5/d

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Rubi [A]  time = 0.46, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2911, 2611, 3768, 3770, 448} \[ -\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a b \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {3 a b \cot (c+d x) \csc (c+d x)}{128 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*b*ArcTanh[Cos[c + d*x]])/(128*d) - ((a^2 + b^2)*Cot[c + d*x]^7)/(7*d) - ((2*a^2 + b^2)*Cot[c + d*x]^9)/(9
*d) - (a^2*Cot[c + d*x]^11)/(11*d) + (3*a*b*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a*b*Cot[c + d*x]*Csc[c + d*x
]^3)/(64*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) + (a*b*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) - (a*b*Cot[
c + d*x]^5*Csc[c + d*x]^5)/(5*d)

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

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 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]

Rubi steps

\begin {align*} \int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\int \cot ^6(c+d x) \csc ^6(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-(a b) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right ) \left (a^2+\left (a^2+b^2\right ) x^2\right )}{x^{12}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac {1}{8} (3 a b) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{x^{12}}+\frac {2 a^2+b^2}{x^{10}}+\frac {a^2+b^2}{x^8}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac {1}{16} (a b) \int \csc ^5(c+d x) \, dx\\ &=-\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {a b \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac {1}{64} (3 a b) \int \csc ^3(c+d x) \, dx\\ &=-\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a b \cot (c+d x) \csc (c+d x)}{128 d}+\frac {a b \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac {1}{128} (3 a b) \int \csc (c+d x) \, dx\\ &=\frac {3 a b \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a b \cot (c+d x) \csc (c+d x)}{128 d}+\frac {a b \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]  time = 1.67, size = 250, normalized size = 1.26 \[ -\frac {\csc ^{11}(c+d x) \left (1478400 \left (8 a^2+b^2\right ) \cos (c+d x)+42240 \left (160 a^2-b^2\right ) \cos (3 (c+d x))+1943040 a^2 \cos (5 (c+d x))+140800 a^2 \cos (7 (c+d x))-28160 a^2 \cos (9 (c+d x))+2560 a^2 \cos (11 (c+d x))+5828130 a b \sin (2 (c+d x))+4790016 a b \sin (4 (c+d x))+2302839 a b \sin (6 (c+d x))+110880 a b \sin (8 (c+d x))-10395 a b \sin (10 (c+d x))-865920 b^2 \cos (5 (c+d x))-499840 b^2 \cos (7 (c+d x))-77440 b^2 \cos (9 (c+d x))+7040 b^2 \cos (11 (c+d x))\right )+5322240 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-5322240 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{227082240 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/227082240*(-5322240*a*b*Log[Cos[(c + d*x)/2]] + 5322240*a*b*Log[Sin[(c + d*x)/2]] + Csc[c + d*x]^11*(147840
0*(8*a^2 + b^2)*Cos[c + d*x] + 42240*(160*a^2 - b^2)*Cos[3*(c + d*x)] + 1943040*a^2*Cos[5*(c + d*x)] - 865920*
b^2*Cos[5*(c + d*x)] + 140800*a^2*Cos[7*(c + d*x)] - 499840*b^2*Cos[7*(c + d*x)] - 28160*a^2*Cos[9*(c + d*x)]
- 77440*b^2*Cos[9*(c + d*x)] + 2560*a^2*Cos[11*(c + d*x)] + 7040*b^2*Cos[11*(c + d*x)] + 5828130*a*b*Sin[2*(c
+ d*x)] + 4790016*a*b*Sin[4*(c + d*x)] + 2302839*a*b*Sin[6*(c + d*x)] + 110880*a*b*Sin[8*(c + d*x)] - 10395*a*
b*Sin[10*(c + d*x)]))/d

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fricas [B]  time = 0.82, size = 363, normalized size = 1.83 \[ \frac {2560 \, {\left (4 \, a^{2} + 11 \, b^{2}\right )} \cos \left (d x + c\right )^{11} - 14080 \, {\left (4 \, a^{2} + 11 \, b^{2}\right )} \cos \left (d x + c\right )^{9} + 126720 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} + 10395 \, {\left (a b \cos \left (d x + c\right )^{10} - 5 \, a b \cos \left (d x + c\right )^{8} + 10 \, a b \cos \left (d x + c\right )^{6} - 10 \, a b \cos \left (d x + c\right )^{4} + 5 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 10395 \, {\left (a b \cos \left (d x + c\right )^{10} - 5 \, a b \cos \left (d x + c\right )^{8} + 10 \, a b \cos \left (d x + c\right )^{6} - 10 \, a b \cos \left (d x + c\right )^{4} + 5 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 1386 \, {\left (15 \, a b \cos \left (d x + c\right )^{9} - 70 \, a b \cos \left (d x + c\right )^{7} - 128 \, a b \cos \left (d x + c\right )^{5} + 70 \, a b \cos \left (d x + c\right )^{3} - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/887040*(2560*(4*a^2 + 11*b^2)*cos(d*x + c)^11 - 14080*(4*a^2 + 11*b^2)*cos(d*x + c)^9 + 126720*(a^2 + b^2)*c
os(d*x + c)^7 + 10395*(a*b*cos(d*x + c)^10 - 5*a*b*cos(d*x + c)^8 + 10*a*b*cos(d*x + c)^6 - 10*a*b*cos(d*x + c
)^4 + 5*a*b*cos(d*x + c)^2 - a*b)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 10395*(a*b*cos(d*x + c)^10 - 5*a*
b*cos(d*x + c)^8 + 10*a*b*cos(d*x + c)^6 - 10*a*b*cos(d*x + c)^4 + 5*a*b*cos(d*x + c)^2 - a*b)*log(-1/2*cos(d*
x + c) + 1/2)*sin(d*x + c) - 1386*(15*a*b*cos(d*x + c)^9 - 70*a*b*cos(d*x + c)^7 - 128*a*b*cos(d*x + c)^5 + 70
*a*b*cos(d*x + c)^3 - 15*a*b*cos(d*x + c))*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)*sin(d*x + c))

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giac [B]  time = 0.36, size = 502, normalized size = 2.54 \[ \frac {315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1386 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 385 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1540 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3465 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 2475 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5940 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 6930 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3465 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 27720 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 11550 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36960 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13860 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 166320 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 34650 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 83160 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {502266 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 34650 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 83160 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 13860 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 11550 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 36960 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 27720 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3465 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2475 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 5940 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 385 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1540 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1386 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 315 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{7096320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/7096320*(315*a^2*tan(1/2*d*x + 1/2*c)^11 + 1386*a*b*tan(1/2*d*x + 1/2*c)^10 - 385*a^2*tan(1/2*d*x + 1/2*c)^9
+ 1540*b^2*tan(1/2*d*x + 1/2*c)^9 - 3465*a*b*tan(1/2*d*x + 1/2*c)^8 - 2475*a^2*tan(1/2*d*x + 1/2*c)^7 - 5940*
b^2*tan(1/2*d*x + 1/2*c)^7 - 6930*a*b*tan(1/2*d*x + 1/2*c)^6 + 3465*a^2*tan(1/2*d*x + 1/2*c)^5 + 27720*a*b*tan
(1/2*d*x + 1/2*c)^4 + 11550*a^2*tan(1/2*d*x + 1/2*c)^3 + 36960*b^2*tan(1/2*d*x + 1/2*c)^3 + 13860*a*b*tan(1/2*
d*x + 1/2*c)^2 - 166320*a*b*log(abs(tan(1/2*d*x + 1/2*c))) - 34650*a^2*tan(1/2*d*x + 1/2*c) - 83160*b^2*tan(1/
2*d*x + 1/2*c) + (502266*a*b*tan(1/2*d*x + 1/2*c)^11 + 34650*a^2*tan(1/2*d*x + 1/2*c)^10 + 83160*b^2*tan(1/2*d
*x + 1/2*c)^10 - 13860*a*b*tan(1/2*d*x + 1/2*c)^9 - 11550*a^2*tan(1/2*d*x + 1/2*c)^8 - 36960*b^2*tan(1/2*d*x +
1/2*c)^8 - 27720*a*b*tan(1/2*d*x + 1/2*c)^7 - 3465*a^2*tan(1/2*d*x + 1/2*c)^6 + 6930*a*b*tan(1/2*d*x + 1/2*c)
^5 + 2475*a^2*tan(1/2*d*x + 1/2*c)^4 + 5940*b^2*tan(1/2*d*x + 1/2*c)^4 + 3465*a*b*tan(1/2*d*x + 1/2*c)^3 + 385
*a^2*tan(1/2*d*x + 1/2*c)^2 - 1540*b^2*tan(1/2*d*x + 1/2*c)^2 - 1386*a*b*tan(1/2*d*x + 1/2*c) - 315*a^2)/tan(1
/2*d*x + 1/2*c)^11)/d

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maple [A]  time = 0.46, size = 303, normalized size = 1.53 \[ -\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{11 d \sin \left (d x +c \right )^{11}}-\frac {4 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{99 d \sin \left (d x +c \right )^{9}}-\frac {8 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{693 d \sin \left (d x +c \right )^{7}}-\frac {a b \left (\cos ^{7}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{10}}-\frac {3 a b \left (\cos ^{7}\left (d x +c \right )\right )}{40 d \sin \left (d x +c \right )^{8}}-\frac {a b \left (\cos ^{7}\left (d x +c \right )\right )}{80 d \sin \left (d x +c \right )^{6}}+\frac {a b \left (\cos ^{7}\left (d x +c \right )\right )}{320 d \sin \left (d x +c \right )^{4}}-\frac {3 a b \left (\cos ^{7}\left (d x +c \right )\right )}{640 d \sin \left (d x +c \right )^{2}}-\frac {3 a b \left (\cos ^{5}\left (d x +c \right )\right )}{640 d}-\frac {a b \left (\cos ^{3}\left (d x +c \right )\right )}{128 d}-\frac {3 a b \cos \left (d x +c \right )}{128 d}-\frac {3 a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d}-\frac {b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{9 d \sin \left (d x +c \right )^{9}}-\frac {2 b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{63 d \sin \left (d x +c \right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x)

[Out]

-1/11/d*a^2/sin(d*x+c)^11*cos(d*x+c)^7-4/99/d*a^2/sin(d*x+c)^9*cos(d*x+c)^7-8/693/d*a^2/sin(d*x+c)^7*cos(d*x+c
)^7-1/5/d*a*b/sin(d*x+c)^10*cos(d*x+c)^7-3/40/d*a*b/sin(d*x+c)^8*cos(d*x+c)^7-1/80/d*a*b/sin(d*x+c)^6*cos(d*x+
c)^7+1/320/d*a*b/sin(d*x+c)^4*cos(d*x+c)^7-3/640/d*a*b/sin(d*x+c)^2*cos(d*x+c)^7-3/640*a*b*cos(d*x+c)^5/d-1/12
8*a*b*cos(d*x+c)^3/d-3/128*a*b*cos(d*x+c)/d-3/128/d*a*b*ln(csc(d*x+c)-cot(d*x+c))-1/9/d*b^2/sin(d*x+c)^9*cos(d
*x+c)^7-2/63/d*b^2/sin(d*x+c)^7*cos(d*x+c)^7

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maxima [A]  time = 0.34, size = 196, normalized size = 0.99 \[ -\frac {693 \, a b {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {14080 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} b^{2}}{\tan \left (d x + c\right )^{9}} + \frac {1280 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{2}}{\tan \left (d x + c\right )^{11}}}{887040 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/887040*(693*a*b*(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)) + 14080*(9*tan(d*x + c)^2 + 7)*b^2/tan(d*x + c)^9 + 12
80*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^2/tan(d*x + c)^11)/d

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mupad [B]  time = 16.33, size = 448, normalized size = 2.26 \[ -\frac {\frac {5\,a^2\,\cos \left (c+d\,x\right )}{96}+\frac {5\,b^2\,\cos \left (c+d\,x\right )}{768}+\frac {5\,a^2\,\cos \left (3\,c+3\,d\,x\right )}{168}+\frac {23\,a^2\,\cos \left (5\,c+5\,d\,x\right )}{2688}+\frac {5\,a^2\,\cos \left (7\,c+7\,d\,x\right )}{8064}-\frac {a^2\,\cos \left (9\,c+9\,d\,x\right )}{8064}+\frac {a^2\,\cos \left (11\,c+11\,d\,x\right )}{88704}-\frac {b^2\,\cos \left (3\,c+3\,d\,x\right )}{5376}-\frac {41\,b^2\,\cos \left (5\,c+5\,d\,x\right )}{10752}-\frac {71\,b^2\,\cos \left (7\,c+7\,d\,x\right )}{32256}-\frac {11\,b^2\,\cos \left (9\,c+9\,d\,x\right )}{32256}+\frac {b^2\,\cos \left (11\,c+11\,d\,x\right )}{32256}+\frac {841\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{32768}+\frac {27\,a\,b\,\sin \left (4\,c+4\,d\,x\right )}{1280}+\frac {3323\,a\,b\,\sin \left (6\,c+6\,d\,x\right )}{327680}+\frac {a\,b\,\sin \left (8\,c+8\,d\,x\right )}{2048}-\frac {3\,a\,b\,\sin \left (10\,c+10\,d\,x\right )}{65536}+\frac {693\,a\,b\,\sin \left (c+d\,x\right )\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{65536}-\frac {495\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (3\,c+3\,d\,x\right )}{65536}+\frac {495\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (5\,c+5\,d\,x\right )}{131072}-\frac {165\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (7\,c+7\,d\,x\right )}{131072}+\frac {33\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (9\,c+9\,d\,x\right )}{131072}-\frac {3\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (11\,c+11\,d\,x\right )}{131072}}{d\,{\sin \left (c+d\,x\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^6*(a + b*sin(c + d*x))^2)/sin(c + d*x)^12,x)

[Out]

-((5*a^2*cos(c + d*x))/96 + (5*b^2*cos(c + d*x))/768 + (5*a^2*cos(3*c + 3*d*x))/168 + (23*a^2*cos(5*c + 5*d*x)
)/2688 + (5*a^2*cos(7*c + 7*d*x))/8064 - (a^2*cos(9*c + 9*d*x))/8064 + (a^2*cos(11*c + 11*d*x))/88704 - (b^2*c
os(3*c + 3*d*x))/5376 - (41*b^2*cos(5*c + 5*d*x))/10752 - (71*b^2*cos(7*c + 7*d*x))/32256 - (11*b^2*cos(9*c +
9*d*x))/32256 + (b^2*cos(11*c + 11*d*x))/32256 + (841*a*b*sin(2*c + 2*d*x))/32768 + (27*a*b*sin(4*c + 4*d*x))/
1280 + (3323*a*b*sin(6*c + 6*d*x))/327680 + (a*b*sin(8*c + 8*d*x))/2048 - (3*a*b*sin(10*c + 10*d*x))/65536 + (
693*a*b*sin(c + d*x)*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/65536 - (495*a*b*log(sin(c/2 + (d*x)/2)/cos(c
/2 + (d*x)/2))*sin(3*c + 3*d*x))/65536 + (495*a*b*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*sin(5*c + 5*d*x))
/131072 - (165*a*b*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*sin(7*c + 7*d*x))/131072 + (33*a*b*log(sin(c/2 +
(d*x)/2)/cos(c/2 + (d*x)/2))*sin(9*c + 9*d*x))/131072 - (3*a*b*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*sin
(11*c + 11*d*x))/131072)/(d*sin(c + d*x)^11)

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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)**6*csc(d*x+c)**12*(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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