\(\int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\) [612]

Optimal result

Integrand size = 29, antiderivative size = 176 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {25 a^3 x}{8}+\frac {13 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d} \]

[Out]

-25/8*a^3*x+13/2*a^3*arctanh(cos(d*x+c))/d-5*a^3*cos(d*x+c)/d-2/3*a^3*cos(d*x+c)^3/d+1/5*a^3*cos(d*x+c)^5/d-a^
3*cot(d*x+c)/d-1/3*a^3*cot(d*x+c)^3/d-3/2*a^3*cot(d*x+c)*csc(d*x+c)/d-23/8*a^3*cos(d*x+c)*sin(d*x+c)/d+3/4*a^3
*cos(d*x+c)*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2951, 3855, 3853, 3852, 2718, 2715, 8, 2713} \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {13 a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {2 a^3 \cos ^3(c+d x)}{3 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {23 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {25 a^3 x}{8} \]

[In]

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

[Out]

(-25*a^3*x)/8 + (13*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a^3*Cos[c + d*x])/d - (2*a^3*Cos[c + d*x]^3)/(3*d) +
(a^3*Cos[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c +
d*x])/(2*d) - (23*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (3*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 2713

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 2715

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] && Integ
erQ[2*n]

Rule 2718

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

Rule 2951

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 3852

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 3853

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 3855

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-6 a^9-8 a^9 \csc (c+d x)+3 a^9 \csc ^3(c+d x)+a^9 \csc ^4(c+d x)+6 a^9 \sin (c+d x)+8 a^9 \sin ^2(c+d x)-3 a^9 \sin ^4(c+d x)-a^9 \sin ^5(c+d x)\right ) \, dx}{a^6} \\ & = -6 a^3 x+a^3 \int \csc ^4(c+d x) \, dx-a^3 \int \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (3 a^3\right ) \int \sin ^4(c+d x) \, dx+\left (6 a^3\right ) \int \sin (c+d x) \, dx-\left (8 a^3\right ) \int \csc (c+d x) \, dx+\left (8 a^3\right ) \int \sin ^2(c+d x) \, dx \\ & = -6 a^3 x+\frac {8 a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {6 a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {4 a^3 \cos (c+d x) \sin (c+d x)}{d}+\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{2} \left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac {1}{4} \left (9 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (4 a^3\right ) \int 1 \, dx-\frac {a^3 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {a^3 \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -2 a^3 x+\frac {13 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac {1}{8} \left (9 a^3\right ) \int 1 \, dx \\ & = -\frac {25 a^3 x}{8}+\frac {13 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.48 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.24 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (-1500 (c+d x)-2580 \cos (c+d x)-50 \cos (3 (c+d x))+6 \cos (5 (c+d x))-160 \cot \left (\frac {1}{2} (c+d x)\right )-180 \csc ^2\left (\frac {1}{2} (c+d x)\right )+3120 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3120 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+180 \sec ^2\left (\frac {1}{2} (c+d x)\right )+160 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-10 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-600 \sin (2 (c+d x))-45 \sin (4 (c+d x))+160 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{480 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

[In]

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

[Out]

(a^3*(1 + Sin[c + d*x])^3*(-1500*(c + d*x) - 2580*Cos[c + d*x] - 50*Cos[3*(c + d*x)] + 6*Cos[5*(c + d*x)] - 16
0*Cot[(c + d*x)/2] - 180*Csc[(c + d*x)/2]^2 + 3120*Log[Cos[(c + d*x)/2]] - 3120*Log[Sin[(c + d*x)/2]] + 180*Se
c[(c + d*x)/2]^2 + 160*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 10*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 600*Sin[2*(c +
d*x)] - 45*Sin[4*(c + d*x)] + 160*Tan[(c + d*x)/2]))/(480*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\(\frac {a^{3} \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (6240 \left (-3 \sin \left (d x +c \right )+\sin \left (3 d x +3 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3000 d x \sin \left (3 d x +3 c \right )-9000 d x \sin \left (d x +c \right )-7884 \sin \left (2 d x +2 c \right )+7048 \sin \left (3 d x +3 c \right )+2412 \sin \left (4 d x +4 c \right )+68 \sin \left (6 d x +6 c \right )-6 \sin \left (8 d x +8 c \right )-3075 \cos \left (d x +c \right )+2305 \cos \left (3 d x +3 c \right )-465 \cos \left (5 d x +5 c \right )-45 \cos \left (7 d x +7 c \right )-21144 \sin \left (d x +c \right )\right )}{30720 d}\)	\(189\)
risch	\(-\frac {25 a^{3} x}{8}+\frac {a^{3} {\mathrm e}^{5 i \left (d x +c \right )}}{160 d}+\frac {5 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {43 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {43 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {5 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a^{3} {\mathrm e}^{-5 i \left (d x +c \right )}}{160 d}+\frac {a^{3} \left (9 \,{\mathrm e}^{5 i \left (d x +c \right )}+12 i {\mathrm e}^{2 i \left (d x +c \right )}-4 i-9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {3 a^{3} \sin \left (4 d x +4 c \right )}{32 d}-\frac {5 a^{3} \cos \left (3 d x +3 c \right )}{48 d}\)	\(244\)
derivativedivides	\(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\)	\(272\)
default	\(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\)	\(272\)
norman	\(\frac {-\frac {a^{3}}{24 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {7 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {23 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {31 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {31 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {23 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {7 a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {3 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{3} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {25 a^{3} x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {125 a^{3} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {125 a^{3} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {125 a^{3} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {125 a^{3} x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {25 a^{3} x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {18 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {665 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {519 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {881 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d}-\frac {1967 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {13 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\)	\(422\)

[In]

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

[Out]

1/30720*a^3*csc(1/2*d*x+1/2*c)^3*sec(1/2*d*x+1/2*c)^3*(6240*(-3*sin(d*x+c)+sin(3*d*x+3*c))*ln(tan(1/2*d*x+1/2*
c))+3000*d*x*sin(3*d*x+3*c)-9000*d*x*sin(d*x+c)-7884*sin(2*d*x+2*c)+7048*sin(3*d*x+3*c)+2412*sin(4*d*x+4*c)+68
*sin(6*d*x+6*c)-6*sin(8*d*x+8*c)-3075*cos(d*x+c)+2305*cos(3*d*x+3*c)-465*cos(5*d*x+5*c)-45*cos(7*d*x+7*c)-2114
4*sin(d*x+c))/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.31 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {90 \, a^{3} \cos \left (d x + c\right )^{7} + 75 \, a^{3} \cos \left (d x + c\right )^{5} - 500 \, a^{3} \cos \left (d x + c\right )^{3} + 375 \, a^{3} \cos \left (d x + c\right ) + 390 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 390 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + {\left (24 \, a^{3} \cos \left (d x + c\right )^{7} - 104 \, a^{3} \cos \left (d x + c\right )^{5} - 375 \, a^{3} d x \cos \left (d x + c\right )^{2} - 520 \, a^{3} \cos \left (d x + c\right )^{3} + 375 \, a^{3} d x + 780 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

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

[Out]

1/120*(90*a^3*cos(d*x + c)^7 + 75*a^3*cos(d*x + c)^5 - 500*a^3*cos(d*x + c)^3 + 375*a^3*cos(d*x + c) + 390*(a^
3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 390*(a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos
(d*x + c) + 1/2)*sin(d*x + c) + (24*a^3*cos(d*x + c)^7 - 104*a^3*cos(d*x + c)^5 - 375*a^3*d*x*cos(d*x + c)^2 -
520*a^3*cos(d*x + c)^3 + 375*a^3*d*x + 780*a^3*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x +
c))

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.40 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {4 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 30 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 45 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} + 20 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3}}{120 \, d} \]

[In]

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

[Out]

1/120*(4*(6*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 30*cos(d*x + c) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x +
c) - 1))*a^3 - 30*(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 - 45*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 25*tan(d*x + c)^2 + 8)/(t
an(d*x + c)^5 + 2*tan(d*x + c)^3 + tan(d*x + c)))*a^3 + 20*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x +
c)^2 - 2)/(tan(d*x + c)^5 + tan(d*x + c)^3))*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.66 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 375 \, {\left (d x + c\right )} a^{3} - 780 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {5 \, {\left (286 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {2 \, {\left (345 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 330 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 330 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 345 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 656 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]

[In]

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

[Out]

1/120*(5*a^3*tan(1/2*d*x + 1/2*c)^3 + 45*a^3*tan(1/2*d*x + 1/2*c)^2 - 375*(d*x + c)*a^3 - 780*a^3*log(abs(tan(
1/2*d*x + 1/2*c))) + 45*a^3*tan(1/2*d*x + 1/2*c) + 5*(286*a^3*tan(1/2*d*x + 1/2*c)^3 - 9*a^3*tan(1/2*d*x + 1/2
*c)^2 - 9*a^3*tan(1/2*d*x + 1/2*c) - a^3)/tan(1/2*d*x + 1/2*c)^3 + 2*(345*a^3*tan(1/2*d*x + 1/2*c)^9 - 720*a^3
*tan(1/2*d*x + 1/2*c)^8 + 330*a^3*tan(1/2*d*x + 1/2*c)^7 - 2880*a^3*tan(1/2*d*x + 1/2*c)^6 - 3680*a^3*tan(1/2*
d*x + 1/2*c)^4 - 330*a^3*tan(1/2*d*x + 1/2*c)^3 - 2560*a^3*tan(1/2*d*x + 1/2*c)^2 - 345*a^3*tan(1/2*d*x + 1/2*
c) - 656*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d

Mupad [B] (verification not implemented)

Time = 11.16 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.44 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,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}{24\,d}-\frac {13\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {-43\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+99\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-\frac {86\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}+399\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {95\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {1562\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {232\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {1114\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {193\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {1537\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}+\frac {14\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {25\,a^3\,\mathrm {atan}\left (\frac {625\,a^6}{16\,\left (\frac {325\,a^6}{4}-\frac {625\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {325\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {325\,a^6}{4}-\frac {625\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}+\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]

[In]

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

[Out]

(3*a^3*tan(c/2 + (d*x)/2)^2)/(8*d) + (a^3*tan(c/2 + (d*x)/2)^3)/(24*d) - (13*a^3*log(tan(c/2 + (d*x)/2)))/(2*d
) - ((14*a^3*tan(c/2 + (d*x)/2)^2)/3 + (1537*a^3*tan(c/2 + (d*x)/2)^3)/15 + (193*a^3*tan(c/2 + (d*x)/2)^4)/3 +
(1114*a^3*tan(c/2 + (d*x)/2)^5)/3 + (232*a^3*tan(c/2 + (d*x)/2)^6)/3 + (1562*a^3*tan(c/2 + (d*x)/2)^7)/3 + (9
5*a^3*tan(c/2 + (d*x)/2)^8)/3 + 399*a^3*tan(c/2 + (d*x)/2)^9 - (86*a^3*tan(c/2 + (d*x)/2)^10)/3 + 99*a^3*tan(c
/2 + (d*x)/2)^11 - 43*a^3*tan(c/2 + (d*x)/2)^12 + a^3/3 + 3*a^3*tan(c/2 + (d*x)/2))/(d*(8*tan(c/2 + (d*x)/2)^3
+ 40*tan(c/2 + (d*x)/2)^5 + 80*tan(c/2 + (d*x)/2)^7 + 80*tan(c/2 + (d*x)/2)^9 + 40*tan(c/2 + (d*x)/2)^11 + 8*
tan(c/2 + (d*x)/2)^13)) - (25*a^3*atan((625*a^6)/(16*((325*a^6)/4 - (625*a^6*tan(c/2 + (d*x)/2))/16)) + (325*a
^6*tan(c/2 + (d*x)/2))/(4*((325*a^6)/4 - (625*a^6*tan(c/2 + (d*x)/2))/16))))/(4*d) + (3*a^3*tan(c/2 + (d*x)/2)
)/(8*d)