3.749 \(\int \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=140 \[ -\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]

[Out]

-5/16*arctanh(cos(d*x+c))/a^3/d-4/3*cot(d*x+c)^3/a^3/d-cot(d*x+c)^5/a^3/d-1/7*cot(d*x+c)^7/a^3/d-5/16*cot(d*x+
c)*csc(d*x+c)/a^3/d+1/8*cot(d*x+c)*csc(d*x+c)^3/a^3/d+1/2*cot(d*x+c)*csc(d*x+c)^5/a^3/d

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Rubi [A]  time = 0.23, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2709, 3768, 3770, 3767} \[ -\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]

[Out]

(-5*ArcTanh[Cos[c + d*x]])/(16*a^3*d) - (4*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^5/(a^3*d) - Cot[c + d*x]^7
/(7*a^3*d) - (5*Cot[c + d*x]*Csc[c + d*x])/(16*a^3*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^3*d) + (Cot[c + d*x
]*Csc[c + d*x]^5)/(2*a^3*d)

Rule 2709

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Dist[a^p, Int[Expan
dIntegrand[(Sin[e + f*x]^p*(a + b*Sin[e + f*x])^(m - p/2))/(a - b*Sin[e + f*x])^(p/2), x], x], x] /; FreeQ[{a,
b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])

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 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 \frac {\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \left (a^5 \csc ^3(c+d x)-3 a^5 \csc ^4(c+d x)+2 a^5 \csc ^5(c+d x)+2 a^5 \csc ^6(c+d x)-3 a^5 \csc ^7(c+d x)+a^5 \csc ^8(c+d x)\right ) \, dx}{a^8}\\ &=\frac {\int \csc ^3(c+d x) \, dx}{a^3}+\frac {\int \csc ^8(c+d x) \, dx}{a^3}+\frac {2 \int \csc ^5(c+d x) \, dx}{a^3}+\frac {2 \int \csc ^6(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^4(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^7(c+d x) \, dx}{a^3}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac {\int \csc (c+d x) \, dx}{2 a^3}+\frac {3 \int \csc ^3(c+d x) \, dx}{2 a^3}-\frac {5 \int \csc ^5(c+d x) \, dx}{2 a^3}-\frac {\operatorname {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}-\frac {2 \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{4 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac {3 \int \csc (c+d x) \, dx}{4 a^3}-\frac {15 \int \csc ^3(c+d x) \, dx}{8 a^3}\\ &=-\frac {5 \tanh ^{-1}(\cos (c+d x))}{4 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}-\frac {15 \int \csc (c+d x) \, dx}{16 a^3}\\ &=-\frac {5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{a^3 d}-\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}\\ \end {align*}

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Mathematica [A]  time = 0.99, size = 251, normalized size = 1.79 \[ \frac {\csc ^7(c+d x) \left (4998 \sin (2 (c+d x))+504 \sin (4 (c+d x))-210 \sin (6 (c+d x))-4704 \cos (c+d x)+672 \cos (3 (c+d x))+1120 \cos (5 (c+d x))-160 \cos (7 (c+d x))+3675 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2205 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+735 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 \sin (7 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-3675 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2205 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-735 \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+105 \sin (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{21504 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]

[Out]

(Csc[c + d*x]^7*(-4704*Cos[c + d*x] + 672*Cos[3*(c + d*x)] + 1120*Cos[5*(c + d*x)] - 160*Cos[7*(c + d*x)] - 36
75*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 3675*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 4998*Sin[2*(c + d*x)] + 2205
*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 2205*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 504*Sin[4*(c + d*x)] -
735*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 735*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 210*Sin[6*(c + d*x)
] + 105*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] - 105*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(21504*a^3*d)

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fricas [A]  time = 0.46, size = 226, normalized size = 1.61 \[ \frac {320 \, \cos \left (d x + c\right )^{7} - 1120 \, \cos \left (d x + c\right )^{5} + 896 \, \cos \left (d x + c\right )^{3} - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 42 \, {\left (5 \, \cos \left (d x + c\right )^{5} - 8 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \, {\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/672*(320*cos(d*x + c)^7 - 1120*cos(d*x + c)^5 + 896*cos(d*x + c)^3 - 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4
+ 3*cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*
cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 42*(5*cos(d*x + c)^5 - 8*cos(d*x + c)^3 - 5*co
s(d*x + c))*sin(d*x + c))/((a^3*d*cos(d*x + c)^6 - 3*a^3*d*cos(d*x + c)^4 + 3*a^3*d*cos(d*x + c)^2 - a^3*d)*si
n(d*x + c))

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giac [A]  time = 0.35, size = 244, normalized size = 1.74 \[ \frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {2178 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 609 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 91 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} + \frac {3 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 21 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 63 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 91 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 63 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 609 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{21}}}{2688 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/2688*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (2178*tan(1/2*d*x + 1/2*c)^7 - 609*tan(1/2*d*x + 1/2*c)^6 + 6
3*tan(1/2*d*x + 1/2*c)^5 + 91*tan(1/2*d*x + 1/2*c)^4 - 105*tan(1/2*d*x + 1/2*c)^3 + 63*tan(1/2*d*x + 1/2*c)^2
- 21*tan(1/2*d*x + 1/2*c) + 3)/(a^3*tan(1/2*d*x + 1/2*c)^7) + (3*a^18*tan(1/2*d*x + 1/2*c)^7 - 21*a^18*tan(1/2
*d*x + 1/2*c)^6 + 63*a^18*tan(1/2*d*x + 1/2*c)^5 - 105*a^18*tan(1/2*d*x + 1/2*c)^4 + 91*a^18*tan(1/2*d*x + 1/2
*c)^3 + 63*a^18*tan(1/2*d*x + 1/2*c)^2 - 609*a^18*tan(1/2*d*x + 1/2*c))/a^21)/d

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maple [B]  time = 0.73, size = 284, normalized size = 2.03 \[ \frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{896 d \,a^{3}}-\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 d \,a^{3}}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{3}}-\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{3}}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d \,a^{3}}+\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{3} d}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 d \,a^{3}}+\frac {1}{128 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {29}{128 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{3}}-\frac {3}{128 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {1}{896 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {3}{128 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5}{128 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {13}{384 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x)

[Out]

1/896/d/a^3*tan(1/2*d*x+1/2*c)^7-1/128/d/a^3*tan(1/2*d*x+1/2*c)^6+3/128/d/a^3*tan(1/2*d*x+1/2*c)^5-5/128/d/a^3
*tan(1/2*d*x+1/2*c)^4+13/384/d/a^3*tan(1/2*d*x+1/2*c)^3+3/128/d/a^3*tan(1/2*d*x+1/2*c)^2-29/128/d/a^3*tan(1/2*
d*x+1/2*c)+1/128/d/a^3/tan(1/2*d*x+1/2*c)^6+29/128/d/a^3/tan(1/2*d*x+1/2*c)+5/16/d/a^3*ln(tan(1/2*d*x+1/2*c))-
3/128/d/a^3/tan(1/2*d*x+1/2*c)^5-1/896/d/a^3/tan(1/2*d*x+1/2*c)^7-3/128/d/a^3/tan(1/2*d*x+1/2*c)^2+5/128/d/a^3
/tan(1/2*d*x+1/2*c)^4-13/384/d/a^3/tan(1/2*d*x+1/2*c)^3

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maxima [B]  time = 0.33, size = 315, normalized size = 2.25 \[ -\frac {\frac {\frac {609 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {91 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {21 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3}} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {63 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {105 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {91 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {609 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{3} \sin \left (d x + c\right )^{7}}}{2688 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^8/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2688*((609*sin(d*x + c)/(cos(d*x + c) + 1) - 63*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 91*sin(d*x + c)^3/(co
s(d*x + c) + 1)^3 + 105*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 21*sin(
d*x + c)^6/(cos(d*x + c) + 1)^6 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^3 - 840*log(sin(d*x + c)/(cos(d*x +
c) + 1))/a^3 - (21*sin(d*x + c)/(cos(d*x + c) + 1) - 63*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 105*sin(d*x + c
)^3/(cos(d*x + c) + 1)^3 - 91*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 6
09*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 3)*(cos(d*x + c) + 1)^7/(a^3*sin(d*x + c)^7))/d

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mupad [B]  time = 10.86, size = 387, normalized size = 2.76 \[ \frac {3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-21\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+91\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-609\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+609\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-91\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2688\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^8*(a + a*sin(c + d*x))^3),x)

[Out]

(3*sin(c/2 + (d*x)/2)^14 - 3*cos(c/2 + (d*x)/2)^14 - 21*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^13 + 21*cos(c/2
+ (d*x)/2)^13*sin(c/2 + (d*x)/2) + 63*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 - 105*cos(c/2 + (d*x)/2)^3*si
n(c/2 + (d*x)/2)^11 + 91*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 + 63*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/
2)^9 - 609*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 609*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 - 63*cos(
c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5 - 91*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4 + 105*cos(c/2 + (d*x)/2)
^11*sin(c/2 + (d*x)/2)^3 - 63*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^2 + 840*log(sin(c/2 + (d*x)/2)/cos(c/2
+ (d*x)/2))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7)/(2688*a^3*d*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7)

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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)**8*csc(d*x+c)**8/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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