3.405 \(\int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=194 \[ -\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {5 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {17 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]

[Out]

-17/128*a^3*arctanh(cos(d*x+c))/d-4/5*a^3*cot(d*x+c)^5/d-5/7*a^3*cot(d*x+c)^7/d-1/9*a^3*cot(d*x+c)^9/d-17/128*
a^3*cot(d*x+c)*csc(d*x+c)/d+5/64*a^3*cot(d*x+c)*csc(d*x+c)^3/d-1/6*a^3*cot(d*x+c)^3*csc(d*x+c)^3/d+3/16*a^3*co
t(d*x+c)*csc(d*x+c)^5/d-3/8*a^3*cot(d*x+c)^3*csc(d*x+c)^5/d

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Rubi [A]  time = 0.35, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2611, 3768, 3770, 2607, 14, 270} \[ -\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {5 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {17 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]

[Out]

(-17*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (5*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*C
ot[c + d*x]^9)/(9*d) - (17*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d)
- (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (3*a^3*Cot[c + d*x
]^3*Csc[c + d*x]^5)/(8*d)

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 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 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 2873

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 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 ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^4(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^5(c+d x)+a^3 \cot ^4(c+d x) \csc ^6(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {1}{2} a^3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx-\frac {1}{8} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{8} a^3 \int \csc ^3(c+d x) \, dx+\frac {1}{16} \left (3 a^3\right ) \int \csc ^5(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {5 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{16} a^3 \int \csc (c+d x) \, dx+\frac {1}{64} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {5 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{128} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac {17 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {5 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\\ \end {align*}

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Mathematica [A]  time = 1.33, size = 313, normalized size = 1.61 \[ -\frac {a^3 \csc ^9(c+d x) \left (669060 \sin (2 (c+d x))+676620 \sin (4 (c+d x))-14700 \sin (6 (c+d x))-10710 \sin (8 (c+d x))+1161216 \cos (c+d x)+247296 \cos (3 (c+d x))-198144 \cos (5 (c+d x))-71424 \cos (7 (c+d x))+7936 \cos (9 (c+d x))-674730 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+449820 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-192780 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+48195 \sin (7 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-5355 \sin (9 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+674730 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-449820 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+192780 \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-48195 \sin (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+5355 \sin (9 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{10321920 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]

[Out]

-1/10321920*(a^3*Csc[c + d*x]^9*(1161216*Cos[c + d*x] + 247296*Cos[3*(c + d*x)] - 198144*Cos[5*(c + d*x)] - 71
424*Cos[7*(c + d*x)] + 7936*Cos[9*(c + d*x)] + 674730*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 674730*Log[Sin[(c +
d*x)/2]]*Sin[c + d*x] + 669060*Sin[2*(c + d*x)] - 449820*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 449820*Log[
Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 676620*Sin[4*(c + d*x)] + 192780*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] -
192780*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 14700*Sin[6*(c + d*x)] - 48195*Log[Cos[(c + d*x)/2]]*Sin[7*(c
+ d*x)] + 48195*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 10710*Sin[8*(c + d*x)] + 5355*Log[Cos[(c + d*x)/2]]*
Sin[9*(c + d*x)] - 5355*Log[Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/d

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fricas [A]  time = 0.48, size = 304, normalized size = 1.57 \[ -\frac {15872 \, a^{3} \cos \left (d x + c\right )^{9} - 71424 \, a^{3} \cos \left (d x + c\right )^{7} + 64512 \, a^{3} \cos \left (d x + c\right )^{5} + 5355 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 5355 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 210 \, {\left (51 \, a^{3} \cos \left (d x + c\right )^{7} - 59 \, a^{3} \cos \left (d x + c\right )^{5} - 187 \, a^{3} \cos \left (d x + c\right )^{3} + 51 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^10*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/80640*(15872*a^3*cos(d*x + c)^9 - 71424*a^3*cos(d*x + c)^7 + 64512*a^3*cos(d*x + c)^5 + 5355*(a^3*cos(d*x +
c)^8 - 4*a^3*cos(d*x + c)^6 + 6*a^3*cos(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2)*
sin(d*x + c) - 5355*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6 + 6*a^3*cos(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 +
a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 210*(51*a^3*cos(d*x + c)^7 - 59*a^3*cos(d*x + c)^5 - 187*a^3
*cos(d*x + c)^3 + 51*a^3*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c
)^4 - 4*d*cos(d*x + c)^2 + d)*sin(d*x + c))

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giac [A]  time = 0.37, size = 325, normalized size = 1.68 \[ \frac {140 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 2340 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4032 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5040 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 85680 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 52920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {242386 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 52920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 5040 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 16800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4032 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2340 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 140 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{645120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^10*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/645120*(140*a^3*tan(1/2*d*x + 1/2*c)^9 + 945*a^3*tan(1/2*d*x + 1/2*c)^8 + 2340*a^3*tan(1/2*d*x + 1/2*c)^7 +
1680*a^3*tan(1/2*d*x + 1/2*c)^6 - 4032*a^3*tan(1/2*d*x + 1/2*c)^5 - 12600*a^3*tan(1/2*d*x + 1/2*c)^4 - 16800*a
^3*tan(1/2*d*x + 1/2*c)^3 - 5040*a^3*tan(1/2*d*x + 1/2*c)^2 + 85680*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 52920
*a^3*tan(1/2*d*x + 1/2*c) - (242386*a^3*tan(1/2*d*x + 1/2*c)^9 + 52920*a^3*tan(1/2*d*x + 1/2*c)^8 - 5040*a^3*t
an(1/2*d*x + 1/2*c)^7 - 16800*a^3*tan(1/2*d*x + 1/2*c)^6 - 12600*a^3*tan(1/2*d*x + 1/2*c)^5 - 4032*a^3*tan(1/2
*d*x + 1/2*c)^4 + 1680*a^3*tan(1/2*d*x + 1/2*c)^3 + 2340*a^3*tan(1/2*d*x + 1/2*c)^2 + 945*a^3*tan(1/2*d*x + 1/
2*c) + 140*a^3)/tan(1/2*d*x + 1/2*c)^9)/d

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maple [A]  time = 0.41, size = 224, normalized size = 1.15 \[ -\frac {17 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{6}}-\frac {17 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{192 d \sin \left (d x +c \right )^{4}}+\frac {17 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{384 d \sin \left (d x +c \right )^{2}}+\frac {17 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{384 d}+\frac {17 a^{3} \cos \left (d x +c \right )}{128 d}+\frac {17 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d}-\frac {31 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{63 d \sin \left (d x +c \right )^{7}}-\frac {62 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{315 d \sin \left (d x +c \right )^{5}}-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{9 d \sin \left (d x +c \right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^10*(a+a*sin(d*x+c))^3,x)

[Out]

-17/48/d*a^3/sin(d*x+c)^6*cos(d*x+c)^5-17/192/d*a^3/sin(d*x+c)^4*cos(d*x+c)^5+17/384/d*a^3/sin(d*x+c)^2*cos(d*
x+c)^5+17/384*a^3*cos(d*x+c)^3/d+17/128*a^3*cos(d*x+c)/d+17/128/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-31/63/d*a^3/si
n(d*x+c)^7*cos(d*x+c)^5-62/315/d*a^3/sin(d*x+c)^5*cos(d*x+c)^5-3/8/d*a^3/sin(d*x+c)^8*cos(d*x+c)^5-1/9/d*a^3/s
in(d*x+c)^9*cos(d*x+c)^5

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maxima [A]  time = 0.35, size = 268, normalized size = 1.38 \[ \frac {945 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 840 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {6912 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}} - \frac {256 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{80640 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^10*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/80640*(945*a^3*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(d*x + c)^3 + 3*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) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c)
- 1)) + 840*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)) - 6912*(7*tan(d*x + c)^2 + 5)*a^3
/tan(d*x + c)^7 - 256*(63*tan(d*x + c)^4 + 90*tan(d*x + c)^2 + 35)*a^3/tan(d*x + c)^9)/d

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mupad [B]  time = 9.32, size = 357, normalized size = 1.84 \[ \frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}+\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {13\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {17\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d}-\frac {21\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}+\frac {21\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^3)/sin(c + d*x)^10,x)

[Out]

(a^3*cot(c/2 + (d*x)/2)^2)/(128*d) + (5*a^3*cot(c/2 + (d*x)/2)^3)/(192*d) + (5*a^3*cot(c/2 + (d*x)/2)^4)/(256*
d) + (a^3*cot(c/2 + (d*x)/2)^5)/(160*d) - (a^3*cot(c/2 + (d*x)/2)^6)/(384*d) - (13*a^3*cot(c/2 + (d*x)/2)^7)/(
3584*d) - (3*a^3*cot(c/2 + (d*x)/2)^8)/(2048*d) - (a^3*cot(c/2 + (d*x)/2)^9)/(4608*d) - (a^3*tan(c/2 + (d*x)/2
)^2)/(128*d) - (5*a^3*tan(c/2 + (d*x)/2)^3)/(192*d) - (5*a^3*tan(c/2 + (d*x)/2)^4)/(256*d) - (a^3*tan(c/2 + (d
*x)/2)^5)/(160*d) + (a^3*tan(c/2 + (d*x)/2)^6)/(384*d) + (13*a^3*tan(c/2 + (d*x)/2)^7)/(3584*d) + (3*a^3*tan(c
/2 + (d*x)/2)^8)/(2048*d) + (a^3*tan(c/2 + (d*x)/2)^9)/(4608*d) + (17*a^3*log(tan(c/2 + (d*x)/2)))/(128*d) - (
21*a^3*cot(c/2 + (d*x)/2))/(256*d) + (21*a^3*tan(c/2 + (d*x)/2))/(256*d)

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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)**4*csc(d*x+c)**10*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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