3.613 \(\int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=178 \[ -\frac {a^3 \cos ^3(c+d x)}{d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}+\frac {5 a^3 \cot (c+d x)}{d}+\frac {a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {45 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {45 a^3 x}{8} \]

[Out]

45/8*a^3*x+45/8*a^3*arctanh(cos(d*x+c))/d-5*a^3*cos(d*x+c)/d-a^3*cos(d*x+c)^3/d+5*a^3*cot(d*x+c)/d-a^3*cot(d*x
+c)^3/d-3/8*a^3*cot(d*x+c)*csc(d*x+c)/d-1/4*a^3*cot(d*x+c)*csc(d*x+c)^3/d+3/8*a^3*cos(d*x+c)*sin(d*x+c)/d+1/4*
a^3*cos(d*x+c)*sin(d*x+c)^3/d

________________________________________________________________________________________

Rubi [A]  time = 0.22, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2872, 3770, 3767, 8, 3768, 2638, 2633, 2635} \[ -\frac {a^3 \cos ^3(c+d x)}{d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}+\frac {5 a^3 \cot (c+d x)}{d}+\frac {a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {45 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {45 a^3 x}{8} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*Cot[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

(45*a^3*x)/8 + (45*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (5*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/d + (5*a^3
*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]*Csc[c
+ d*x]^3)/(4*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)

Rule 8

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

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

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 \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\int \left (6 a^9-6 a^9 \csc (c+d x)-8 a^9 \csc ^2(c+d x)+3 a^9 \csc ^4(c+d x)+a^9 \csc ^5(c+d x)+8 a^9 \sin (c+d x)-3 a^9 \sin ^3(c+d x)-a^9 \sin ^4(c+d x)\right ) \, dx}{a^6}\\ &=6 a^3 x+a^3 \int \csc ^5(c+d x) \, dx-a^3 \int \sin ^4(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^4(c+d x) \, dx-\left (3 a^3\right ) \int \sin ^3(c+d x) \, dx-\left (6 a^3\right ) \int \csc (c+d x) \, dx-\left (8 a^3\right ) \int \csc ^2(c+d x) \, dx+\left (8 a^3\right ) \int \sin (c+d x) \, dx\\ &=6 a^3 x+\frac {6 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {8 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{4} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{4} \left (3 a^3\right ) \int \sin ^2(c+d x) \, dx+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 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}\\ &=6 a^3 x+\frac {6 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{d}+\frac {5 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac {1}{8} \left (3 a^3\right ) \int 1 \, dx+\frac {1}{8} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac {45 a^3 x}{8}+\frac {45 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{d}+\frac {5 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.86, size = 235, normalized size = 1.32 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (360 (c+d x)+16 \sin (2 (c+d x))-2 \sin (4 (c+d x))-368 \cos (c+d x)-16 \cos (3 (c+d x))-192 \tan \left (\frac {1}{2} (c+d x)\right )+192 \cot \left (\frac {1}{2} (c+d x)\right )-\csc ^4\left (\frac {1}{2} (c+d x)\right )-6 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )+6 \sec ^2\left (\frac {1}{2} (c+d x)\right )-360 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+360 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+64 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)\right )}{64 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]*Cot[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(1 + Sin[c + d*x])^3*(360*(c + d*x) - 368*Cos[c + d*x] - 16*Cos[3*(c + d*x)] + 192*Cot[(c + d*x)/2] - 6*C
sc[(c + d*x)/2]^2 - Csc[(c + d*x)/2]^4 + 360*Log[Cos[(c + d*x)/2]] - 360*Log[Sin[(c + d*x)/2]] + 6*Sec[(c + d*
x)/2]^2 + Sec[(c + d*x)/2]^4 + 64*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 4*Csc[(c + d*x)/2]^4*Sin[c + d*x] + 16*S
in[2*(c + d*x)] - 2*Sin[4*(c + d*x)] - 192*Tan[(c + d*x)/2]))/(64*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 258, normalized size = 1.45 \[ -\frac {16 \, a^{3} \cos \left (d x + c\right )^{7} - 90 \, a^{3} d x \cos \left (d x + c\right )^{4} + 48 \, a^{3} \cos \left (d x + c\right )^{5} + 180 \, a^{3} d x \cos \left (d x + c\right )^{2} - 150 \, a^{3} \cos \left (d x + c\right )^{3} - 90 \, a^{3} d x + 90 \, a^{3} \cos \left (d x + c\right ) - 45 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 45 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{7} - 9 \, a^{3} \cos \left (d x + c\right )^{5} + 60 \, a^{3} \cos \left (d x + c\right )^{3} - 45 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/16*(16*a^3*cos(d*x + c)^7 - 90*a^3*d*x*cos(d*x + c)^4 + 48*a^3*cos(d*x + c)^5 + 180*a^3*d*x*cos(d*x + c)^2
- 150*a^3*cos(d*x + c)^3 - 90*a^3*d*x + 90*a^3*cos(d*x + c) - 45*(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) + 45*(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) + 2*(2*a^3*cos(d*x + c)^7 - 9*a^3*cos(d*x + c)^5 + 60*a^3*cos(d*x + c)^3 - 45*a^3*cos(d*x + c))*sin(d*x
+ c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)

________________________________________________________________________________________

giac [A]  time = 0.40, size = 313, normalized size = 1.76 \[ \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 360 \, {\left (d x + c\right )} a^{3} - 360 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 184 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {250 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 136 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 32 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 552 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 837 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1248 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1100 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 736 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 556 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 152 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{4}}}{64 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/64*(a^3*tan(1/2*d*x + 1/2*c)^4 + 8*a^3*tan(1/2*d*x + 1/2*c)^3 + 8*a^3*tan(1/2*d*x + 1/2*c)^2 + 360*(d*x + c)
*a^3 - 360*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 184*a^3*tan(1/2*d*x + 1/2*c) + (250*a^3*tan(1/2*d*x + 1/2*c)^1
2 + 136*a^3*tan(1/2*d*x + 1/2*c)^11 - 32*a^3*tan(1/2*d*x + 1/2*c)^10 + 552*a^3*tan(1/2*d*x + 1/2*c)^9 - 837*a^
3*tan(1/2*d*x + 1/2*c)^8 + 1248*a^3*tan(1/2*d*x + 1/2*c)^7 - 1100*a^3*tan(1/2*d*x + 1/2*c)^6 + 736*a^3*tan(1/2
*d*x + 1/2*c)^5 - 556*a^3*tan(1/2*d*x + 1/2*c)^4 + 152*a^3*tan(1/2*d*x + 1/2*c)^3 - 12*a^3*tan(1/2*d*x + 1/2*c
)^2 - 8*a^3*tan(1/2*d*x + 1/2*c) - a^3)/(tan(1/2*d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c))^4)/d

________________________________________________________________________________________

maple [A]  time = 0.43, size = 247, normalized size = 1.39 \[ \frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}+\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}+\frac {15 a^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4 d}+\frac {45 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {45 a^{3} x}{8}+\frac {45 a^{3} c}{8 d}-\frac {9 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}-\frac {9 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d}-\frac {15 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{8 d}-\frac {45 a^{3} \cos \left (d x +c \right )}{8 d}-\frac {45 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{3}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x)

[Out]

3/d*a^3/sin(d*x+c)*cos(d*x+c)^7+3*a^3*cos(d*x+c)^5*sin(d*x+c)/d+15/4*a^3*cos(d*x+c)^3*sin(d*x+c)/d+45/8*a^3*co
s(d*x+c)*sin(d*x+c)/d+45/8*a^3*x+45/8/d*a^3*c-9/8/d*a^3/sin(d*x+c)^2*cos(d*x+c)^7-9/8*a^3*cos(d*x+c)^5/d-15/8*
a^3*cos(d*x+c)^3/d-45/8*a^3*cos(d*x+c)/d-45/8/d*a^3*ln(csc(d*x+c)-cot(d*x+c))-1/d*a^3/sin(d*x+c)^3*cos(d*x+c)^
7-1/4/d*a^3/sin(d*x+c)^4*cos(d*x+c)^7

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 268, normalized size = 1.51 \[ -\frac {4 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} + 2 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} - 8 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3} + a^{3} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{16 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/16*(4*(4*cos(d*x + c)^3 - 6*cos(d*x + c)/(cos(d*x + c)^2 - 1) + 24*cos(d*x + c) - 15*log(cos(d*x + c) + 1)
+ 15*log(cos(d*x + c) - 1))*a^3 + 2*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 25*tan(d*x + c)^2 + 8)/(tan(d*x + c)
^5 + 2*tan(d*x + c)^3 + tan(d*x + c)))*a^3 - 8*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 - 2)/(t
an(d*x + c)^5 + tan(d*x + c)^3))*a^3 + 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)))/d

________________________________________________________________________________________

mupad [B]  time = 8.96, size = 419, normalized size = 2.35 \[ \frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {45\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}-\frac {-34\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+258\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-138\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {2337\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4}-312\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+525\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-184\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {403\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-38\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {45\,a^3\,\mathrm {atan}\left (\frac {2025\,a^6}{16\,\left (\frac {2025\,a^6}{16}+\frac {2025\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}-\frac {2025\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,\left (\frac {2025\,a^6}{16}+\frac {2025\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}-\frac {23\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]

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)^5,x)

[Out]

(a^3*tan(c/2 + (d*x)/2)^2)/(8*d) + (a^3*tan(c/2 + (d*x)/2)^3)/(8*d) + (a^3*tan(c/2 + (d*x)/2)^4)/(64*d) - (45*
a^3*log(tan(c/2 + (d*x)/2)))/(8*d) - (3*a^3*tan(c/2 + (d*x)/2)^2 - 38*a^3*tan(c/2 + (d*x)/2)^3 + (403*a^3*tan(
c/2 + (d*x)/2)^4)/2 - 184*a^3*tan(c/2 + (d*x)/2)^5 + 525*a^3*tan(c/2 + (d*x)/2)^6 - 312*a^3*tan(c/2 + (d*x)/2)
^7 + (2337*a^3*tan(c/2 + (d*x)/2)^8)/4 - 138*a^3*tan(c/2 + (d*x)/2)^9 + 258*a^3*tan(c/2 + (d*x)/2)^10 - 34*a^3
*tan(c/2 + (d*x)/2)^11 + a^3/4 + 2*a^3*tan(c/2 + (d*x)/2))/(d*(16*tan(c/2 + (d*x)/2)^4 + 64*tan(c/2 + (d*x)/2)
^6 + 96*tan(c/2 + (d*x)/2)^8 + 64*tan(c/2 + (d*x)/2)^10 + 16*tan(c/2 + (d*x)/2)^12)) - (45*a^3*atan((2025*a^6)
/(16*((2025*a^6)/16 + (2025*a^6*tan(c/2 + (d*x)/2))/16)) - (2025*a^6*tan(c/2 + (d*x)/2))/(16*((2025*a^6)/16 +
(2025*a^6*tan(c/2 + (d*x)/2))/16))))/(4*d) - (23*a^3*tan(c/2 + (d*x)/2))/(8*d)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________