3.616 \(\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=172 \[ \frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-3 a^3 x \]

[Out]

-3*a^3*x-15/16*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-3/5*a^3*cot(d*
x+c)^5/d-1/7*a^3*cot(d*x+c)^7/d-15/16*a^3*cot(d*x+c)*csc(d*x+c)/d+11/8*a^3*cot(d*x+c)*csc(d*x+c)^3/d-1/2*a^3*c
ot(d*x+c)*csc(d*x+c)^5/d

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Rubi [A]  time = 0.29, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2872, 3767, 8, 3768, 3770, 2638} \[ \frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-3 a^3 x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

-3*a^3*x - (15*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cos[c + d*x])/d - (3*a^3*Cot[c + d*x])/d + (a^3*Cot[c
+ d*x]^3)/d - (3*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) - (15*a^3*Cot[c + d*x]*Csc[c + d*x])/(
16*d) + (11*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(2*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 ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\int \left (-3 a^9+8 a^9 \csc ^2(c+d x)+6 a^9 \csc ^3(c+d x)-6 a^9 \csc ^4(c+d x)-8 a^9 \csc ^5(c+d x)+3 a^9 \csc ^7(c+d x)+a^9 \csc ^8(c+d x)-a^9 \sin (c+d x)\right ) \, dx}{a^6}\\ &=-3 a^3 x+a^3 \int \csc ^8(c+d x) \, dx-a^3 \int \sin (c+d x) \, dx+\left (3 a^3\right ) \int \csc ^7(c+d x) \, dx+\left (6 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (6 a^3\right ) \int \csc ^4(c+d x) \, dx+\left (8 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (8 a^3\right ) \int \csc ^5(c+d x) \, dx\\ &=-3 a^3 x+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{d}+\frac {2 a^3 \cot (c+d x) \csc ^3(c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {1}{2} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \csc (c+d x) \, dx-\left (6 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {\left (6 a^3\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (8 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))}{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 {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {1}{8} \left (15 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=-3 a^3 x+\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 {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac {1}{16} \left (15 a^3\right ) \int \csc (c+d x) \, dx\\ &=-3 a^3 x-\frac {15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 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 {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}\\ \end {align*}

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Mathematica [A]  time = 1.40, size = 292, normalized size = 1.70 \[ \frac {a^3 \left (4480 \cos (c+d x)+9984 \tan \left (\frac {1}{2} (c+d x)\right )-9984 \cot \left (\frac {1}{2} (c+d x)\right )-35 \csc ^6\left (\frac {1}{2} (c+d x)\right )+350 \csc ^4\left (\frac {1}{2} (c+d x)\right )-1050 \csc ^2\left (\frac {1}{2} (c+d x)\right )+35 \sec ^6\left (\frac {1}{2} (c+d x)\right )-350 \sec ^4\left (\frac {1}{2} (c+d x)\right )+1050 \sec ^2\left (\frac {1}{2} (c+d x)\right )+4200 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4200 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\frac {5}{2} \sin (c+d x) \csc ^8\left (\frac {1}{2} (c+d x)\right )-17 \sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )+479 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )-7664 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+5 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right )+34 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )-13440 c-13440 d x\right )}{4480 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(-13440*c - 13440*d*x + 4480*Cos[c + d*x] - 9984*Cot[(c + d*x)/2] - 1050*Csc[(c + d*x)/2]^2 + 350*Csc[(c
+ d*x)/2]^4 - 35*Csc[(c + d*x)/2]^6 - 4200*Log[Cos[(c + d*x)/2]] + 4200*Log[Sin[(c + d*x)/2]] + 1050*Sec[(c +
d*x)/2]^2 - 350*Sec[(c + d*x)/2]^4 + 35*Sec[(c + d*x)/2]^6 - 7664*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 479*Csc[
(c + d*x)/2]^4*Sin[c + d*x] - 17*Csc[(c + d*x)/2]^6*Sin[c + d*x] - (5*Csc[(c + d*x)/2]^8*Sin[c + d*x])/2 + 998
4*Tan[(c + d*x)/2] + 34*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] + 5*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2]))/(4480*d)

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fricas [B]  time = 0.76, size = 336, normalized size = 1.95 \[ -\frac {4992 \, a^{3} \cos \left (d x + c\right )^{7} - 12992 \, a^{3} \cos \left (d x + c\right )^{5} + 11200 \, a^{3} \cos \left (d x + c\right )^{3} - 3360 \, a^{3} \cos \left (d x + c\right ) + 525 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 525 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (48 \, a^{3} d x \cos \left (d x + c\right )^{6} - 16 \, a^{3} \cos \left (d x + c\right )^{7} - 144 \, a^{3} d x \cos \left (d x + c\right )^{4} + 33 \, a^{3} \cos \left (d x + c\right )^{5} + 144 \, a^{3} d x \cos \left (d x + c\right )^{2} - 40 \, a^{3} \cos \left (d x + c\right )^{3} - 48 \, a^{3} d x + 15 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1120 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/1120*(4992*a^3*cos(d*x + c)^7 - 12992*a^3*cos(d*x + c)^5 + 11200*a^3*cos(d*x + c)^3 - 3360*a^3*cos(d*x + c)
+ 525*(a^3*cos(d*x + c)^6 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2)*si
n(d*x + c) - 525*(a^3*cos(d*x + c)^6 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c
) + 1/2)*sin(d*x + c) + 70*(48*a^3*d*x*cos(d*x + c)^6 - 16*a^3*cos(d*x + c)^7 - 144*a^3*d*x*cos(d*x + c)^4 + 3
3*a^3*cos(d*x + c)^5 + 144*a^3*d*x*cos(d*x + c)^2 - 40*a^3*cos(d*x + c)^3 - 48*a^3*d*x + 15*a^3*cos(d*x + c))*
sin(d*x + c))/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))

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giac [A]  time = 0.42, size = 291, normalized size = 1.69 \[ \frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 49 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 245 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 875 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 455 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 13440 \, {\left (d x + c\right )} a^{3} + 4200 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 9065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {8960 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {10890 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 455 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 875 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 245 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 49 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{4480 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/4480*(5*a^3*tan(1/2*d*x + 1/2*c)^7 + 35*a^3*tan(1/2*d*x + 1/2*c)^6 + 49*a^3*tan(1/2*d*x + 1/2*c)^5 - 245*a^3
*tan(1/2*d*x + 1/2*c)^4 - 875*a^3*tan(1/2*d*x + 1/2*c)^3 + 455*a^3*tan(1/2*d*x + 1/2*c)^2 - 13440*(d*x + c)*a^
3 + 4200*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 9065*a^3*tan(1/2*d*x + 1/2*c) + 8960*a^3/(tan(1/2*d*x + 1/2*c)^2
+ 1) - (10890*a^3*tan(1/2*d*x + 1/2*c)^7 + 9065*a^3*tan(1/2*d*x + 1/2*c)^6 + 455*a^3*tan(1/2*d*x + 1/2*c)^5 -
875*a^3*tan(1/2*d*x + 1/2*c)^4 - 245*a^3*tan(1/2*d*x + 1/2*c)^3 + 49*a^3*tan(1/2*d*x + 1/2*c)^2 + 35*a^3*tan(
1/2*d*x + 1/2*c) + 5*a^3)/tan(1/2*d*x + 1/2*c)^7)/d

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maple [A]  time = 0.38, size = 228, normalized size = 1.33 \[ -\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{4}}+\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{2}}+\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{16 d}+\frac {5 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{16 d}+\frac {15 a^{3} \cos \left (d x +c \right )}{16 d}+\frac {15 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {3 a^{3} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{d}-\frac {3 a^{3} \cot \left (d x +c \right )}{d}-3 a^{3} x -\frac {3 a^{3} c}{d}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{6}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x)

[Out]

-1/8/d*a^3/sin(d*x+c)^4*cos(d*x+c)^7+3/16/d*a^3/sin(d*x+c)^2*cos(d*x+c)^7+3/16*a^3*cos(d*x+c)^5/d+5/16*a^3*cos
(d*x+c)^3/d+15/16*a^3*cos(d*x+c)/d+15/16/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-3/5*a^3*cot(d*x+c)^5/d+a^3*cot(d*x+c)
^3/d-3*a^3*cot(d*x+c)/d-3*a^3*x-3/d*a^3*c-1/2/d*a^3/sin(d*x+c)^6*cos(d*x+c)^7-1/7/d*a^3/sin(d*x+c)^7*cos(d*x+c
)^7

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maxima [A]  time = 0.42, size = 233, normalized size = 1.35 \[ -\frac {224 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} - 35 \, a^{3} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 70 \, a^{3} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {160 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{1120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/1120*(224*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a^3 - 35*a^3*(2*(33*c
os(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1
) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 70*a^3*(2*(9*cos(d*x + c)^3 - 7*cos(d*x + c))/(cos(
d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - 16*cos(d*x + c) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) +
160*a^3/tan(d*x + c)^7)/d

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mupad [B]  time = 9.04, size = 388, normalized size = 2.26 \[ \frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {25\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {15\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16\,d}+\frac {6\,a^3\,\mathrm {atan}\left (\frac {36\,a^6}{\frac {45\,a^6}{4}+36\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {45\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {45\,a^6}{4}+36\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}-\frac {259\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-243\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+234\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {118\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {54\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}+a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{7}}{d\,\left (128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}+\frac {259\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^6*(a + a*sin(c + d*x))^3)/sin(c + d*x)^8,x)

[Out]

(13*a^3*tan(c/2 + (d*x)/2)^2)/(128*d) - (25*a^3*tan(c/2 + (d*x)/2)^3)/(128*d) - (7*a^3*tan(c/2 + (d*x)/2)^4)/(
128*d) + (7*a^3*tan(c/2 + (d*x)/2)^5)/(640*d) + (a^3*tan(c/2 + (d*x)/2)^6)/(128*d) + (a^3*tan(c/2 + (d*x)/2)^7
)/(896*d) + (15*a^3*log(tan(c/2 + (d*x)/2)))/(16*d) + (6*a^3*atan((36*a^6)/((45*a^6)/4 + 36*a^6*tan(c/2 + (d*x
)/2)) - (45*a^6*tan(c/2 + (d*x)/2))/(4*((45*a^6)/4 + 36*a^6*tan(c/2 + (d*x)/2)))))/d - ((54*a^3*tan(c/2 + (d*x
)/2)^2)/35 - 6*a^3*tan(c/2 + (d*x)/2)^3 - (118*a^3*tan(c/2 + (d*x)/2)^4)/5 + 6*a^3*tan(c/2 + (d*x)/2)^5 + 234*
a^3*tan(c/2 + (d*x)/2)^6 - 243*a^3*tan(c/2 + (d*x)/2)^7 + 259*a^3*tan(c/2 + (d*x)/2)^8 + a^3/7 + a^3*tan(c/2 +
(d*x)/2))/(d*(128*tan(c/2 + (d*x)/2)^7 + 128*tan(c/2 + (d*x)/2)^9)) + (259*a^3*tan(c/2 + (d*x)/2))/(128*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)**6*csc(d*x+c)**8*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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