3.401 \(\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=132 \[ -\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{d}+\frac {3 a^3 \cot (c+d x)}{d}+\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+3 a^3 x \]

[Out]

3*a^3*x+3/8*a^3*arctanh(cos(d*x+c))/d-a^3*cos(d*x+c)/d+3*a^3*cot(d*x+c)/d-a^3*cot(d*x+c)^3/d-1/5*a^3*cot(d*x+c
)^5/d+11/8*a^3*cot(d*x+c)*csc(d*x+c)/d-3/4*a^3*cot(d*x+c)*csc(d*x+c)^3/d

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Rubi [A]  time = 0.22, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2872, 3770, 3767, 8, 3768, 2638} \[ -\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{d}+\frac {3 a^3 \cot (c+d x)}{d}+\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+3 a^3 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

3*a^3*x + (3*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (a^3*Cos[c + d*x])/d + (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d
*x]^3)/d - (a^3*Cot[c + d*x]^5)/(5*d) + (11*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c +
d*x]^3)/(4*d)

Rule 8

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

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2872

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 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 \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\int \left (3 a^7+a^7 \csc (c+d x)-5 a^7 \csc ^2(c+d x)-5 a^7 \csc ^3(c+d x)+a^7 \csc ^4(c+d x)+3 a^7 \csc ^5(c+d x)+a^7 \csc ^6(c+d x)+a^7 \sin (c+d x)\right ) \, dx}{a^4}\\ &=3 a^3 x+a^3 \int \csc (c+d x) \, dx+a^3 \int \csc ^4(c+d x) \, dx+a^3 \int \csc ^6(c+d x) \, dx+a^3 \int \sin (c+d x) \, dx+\left (3 a^3\right ) \int \csc ^5(c+d x) \, dx-\left (5 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (5 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=3 a^3 x-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a^3 \cos (c+d x)}{d}+\frac {5 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{2} \left (5 a^3\right ) \int \csc (c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {a^3 \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {\left (5 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=3 a^3 x+\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{8} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=3 a^3 x+\frac {3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]  time = 0.51, size = 216, normalized size = 1.64 \[ \frac {a^3 \left (-320 \cos (c+d x)-608 \tan \left (\frac {1}{2} (c+d x)\right )+608 \cot \left (\frac {1}{2} (c+d x)\right )-15 \csc ^4\left (\frac {1}{2} (c+d x)\right )+110 \csc ^2\left (\frac {1}{2} (c+d x)\right )+15 \sec ^4\left (\frac {1}{2} (c+d x)\right )-110 \sec ^2\left (\frac {1}{2} (c+d x)\right )-120 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+120 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )+64 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \csc ^5(c+d x)-13 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+208 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+960 c+960 d x\right )}{320 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(960*c + 960*d*x - 320*Cos[c + d*x] + 608*Cot[(c + d*x)/2] + 110*Csc[(c + d*x)/2]^2 - 15*Csc[(c + d*x)/2]
^4 + 120*Log[Cos[(c + d*x)/2]] - 120*Log[Sin[(c + d*x)/2]] - 110*Sec[(c + d*x)/2]^2 + 15*Sec[(c + d*x)/2]^4 +
208*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 64*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - 13*Csc[(c + d*x)/2]^4*Sin[c + d
*x] - Csc[(c + d*x)/2]^6*Sin[c + d*x] - 608*Tan[(c + d*x)/2]))/(320*d)

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fricas [B]  time = 0.47, size = 252, normalized size = 1.91 \[ \frac {304 \, a^{3} \cos \left (d x + c\right )^{5} - 560 \, a^{3} \cos \left (d x + c\right )^{3} + 240 \, a^{3} \cos \left (d x + c\right ) + 15 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10 \, {\left (24 \, a^{3} d x \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{5} - 48 \, a^{3} d x \cos \left (d x + c\right )^{2} + 5 \, a^{3} \cos \left (d x + c\right )^{3} + 24 \, a^{3} d x - 3 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/80*(304*a^3*cos(d*x + c)^5 - 560*a^3*cos(d*x + c)^3 + 240*a^3*cos(d*x + c) + 15*(a^3*cos(d*x + c)^4 - 2*a^3*
cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 15*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2
+ a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 10*(24*a^3*d*x*cos(d*x + c)^4 - 8*a^3*cos(d*x + c)^5 - 48*
a^3*d*x*cos(d*x + c)^2 + 5*a^3*cos(d*x + c)^3 + 24*a^3*d*x - 3*a^3*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c
)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))

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giac [A]  time = 0.32, size = 226, normalized size = 1.71 \[ \frac {2 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 960 \, {\left (d x + c\right )} a^{3} - 120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 580 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {640 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {274 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 580 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 80 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/320*(2*a^3*tan(1/2*d*x + 1/2*c)^5 + 15*a^3*tan(1/2*d*x + 1/2*c)^4 + 30*a^3*tan(1/2*d*x + 1/2*c)^3 - 80*a^3*t
an(1/2*d*x + 1/2*c)^2 + 960*(d*x + c)*a^3 - 120*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 580*a^3*tan(1/2*d*x + 1/2
*c) - 640*a^3/(tan(1/2*d*x + 1/2*c)^2 + 1) + (274*a^3*tan(1/2*d*x + 1/2*c)^5 + 580*a^3*tan(1/2*d*x + 1/2*c)^4
+ 80*a^3*tan(1/2*d*x + 1/2*c)^3 - 30*a^3*tan(1/2*d*x + 1/2*c)^2 - 15*a^3*tan(1/2*d*x + 1/2*c) - 2*a^3)/tan(1/2
*d*x + 1/2*c)^5)/d

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maple [A]  time = 0.35, size = 173, normalized size = 1.31 \[ -\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{8 d}-\frac {3 a^{3} \cos \left (d x +c \right )}{8 d}-\frac {3 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {a^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{d}+3 a^{3} x +\frac {3 a^{3} \cot \left (d x +c \right )}{d}+\frac {3 a^{3} c}{d}-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x)

[Out]

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

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maxima [A]  time = 0.53, size = 180, normalized size = 1.36 \[ \frac {80 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{3} - 15 \, a^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 20 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {16 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{80 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/80*(80*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^3 - 15*a^3*(2*(5*cos(d*x + c)^3 - 3*cos(d*x +
c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) + 20*a^3*(2*
cos(d*x + c)/(cos(d*x + c)^2 - 1) - 4*cos(d*x + c) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) - 16*a
^3/tan(d*x + c)^5)/d

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mupad [B]  time = 10.11, size = 554, normalized size = 4.20 \[ -\frac {a^3\,\left (2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+65\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+550\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+80\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-550\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-65\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+120\,\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 )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+120\,\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 )}^5+1920\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+1920\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+8\,\sin \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 )}^5\right )}{320\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )} \]

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)^6,x)

[Out]

-(a^3*(2*cos(c/2 + (d*x)/2)^12 - 2*sin(c/2 + (d*x)/2)^12 - 15*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^11 + 15*co
s(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2) - 32*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 65*cos(c/2 + (d*x)/2)
^3*sin(c/2 + (d*x)/2)^9 + 550*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 + 80*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d
*x)/2)^7 + 560*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 - 550*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 - 65*
cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 + 32*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 120*log(sin(c/2 +
(d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 + 120*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)^5 + 1920*atan((8*cos(c/2 + (d*x)/2) - sin(c/2 + (d*x)/2))/(c
os(c/2 + (d*x)/2) + 8*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 + 1920*atan((8*cos(c/2 +
(d*x)/2) - sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2) + 8*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*
x)/2)^5))/(320*d*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^5*(cos(c/2 + (d*x)/2)^2 + sin(c/2 + (d*x)/2)^2))

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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)**6*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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