\(\int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [712]

Optimal result

Integrand size = 29, antiderivative size = 150 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {15 x}{8 a}+\frac {5 \text {arctanh}(\cos (c+d x))}{2 a d}-\frac {5 \cos (c+d x)}{2 a d}-\frac {5 \cos ^3(c+d x)}{6 a d}+\frac {15 \cot (c+d x)}{8 a d}-\frac {5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}-\frac {\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d} \]

[Out]

15/8*x/a+5/2*arctanh(cos(d*x+c))/a/d-5/2*cos(d*x+c)/a/d-5/6*cos(d*x+c)^3/a/d+15/8*cot(d*x+c)/a/d-5/8*cos(d*x+c
)^2*cot(d*x+c)/a/d-1/4*cos(d*x+c)^4*cot(d*x+c)/a/d-1/2*cos(d*x+c)^3*cot(d*x+c)^2/a/d

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2918, 2672, 294, 308, 212, 2671, 327, 209} \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 \text {arctanh}(\cos (c+d x))}{2 a d}-\frac {5 \cos ^3(c+d x)}{6 a d}-\frac {5 \cos (c+d x)}{2 a d}+\frac {15 \cot (c+d x)}{8 a d}-\frac {\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac {5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac {15 x}{8 a} \]

[In]

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

[Out]

(15*x)/(8*a) + (5*ArcTanh[Cos[c + d*x]])/(2*a*d) - (5*Cos[c + d*x])/(2*a*d) - (5*Cos[c + d*x]^3)/(6*a*d) + (15
*Cot[c + d*x])/(8*a*d) - (5*Cos[c + d*x]^2*Cot[c + d*x])/(8*a*d) - (Cos[c + d*x]^4*Cot[c + d*x])/(4*a*d) - (Co
s[c + d*x]^3*Cot[c + d*x]^2)/(2*a*d)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2671

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[b*(ff/f), Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff
)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]

Rule 2672

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 2918

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cos ^4(c+d x) \cot ^2(c+d x) \, dx}{a}+\frac {\int \cos ^3(c+d x) \cot ^3(c+d x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^3} \, dx,x,\cot (c+d x)\right )}{a d} \\ & = -\frac {\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac {5 \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{4 a d}+\frac {5 \text {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d} \\ & = -\frac {5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}-\frac {\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac {15 \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 a d}+\frac {5 \text {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{2 a d} \\ & = -\frac {5 \cos (c+d x)}{2 a d}-\frac {5 \cos ^3(c+d x)}{6 a d}+\frac {15 \cot (c+d x)}{8 a d}-\frac {5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}-\frac {\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac {15 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 a d}+\frac {5 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d} \\ & = \frac {15 x}{8 a}+\frac {5 \text {arctanh}(\cos (c+d x))}{2 a d}-\frac {5 \cos (c+d x)}{2 a d}-\frac {5 \cos ^3(c+d x)}{6 a d}+\frac {15 \cot (c+d x)}{8 a d}-\frac {5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}-\frac {\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.19 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-360 c-360 d x+400 \cos (c+d x)-200 \cos (3 (c+d x))-8 \cos (5 (c+d x))-480 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 \cos (2 (c+d x)) \left (3 c+3 d x+4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+480 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-285 \sin (2 (c+d x))+42 \sin (4 (c+d x))+3 \sin (6 (c+d x))\right )}{1536 a d (1+\sin (c+d x))} \]

[In]

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

[Out]

-1/1536*((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(-360*c - 360*d*x + 400*Cos[c + d*x] - 200*Cos[3*(c + d*x)] -
8*Cos[5*(c + d*x)] - 480*Log[Cos[(c + d*x)/2]] + 120*Cos[2*(c + d*x)]*(3*c + 3*d*x + 4*Log[Cos[(c + d*x)/2]]
- 4*Log[Sin[(c + d*x)/2]]) + 480*Log[Sin[(c + d*x)/2]] - 285*Sin[2*(c + d*x)] + 42*Sin[4*(c + d*x)] + 3*Sin[6*
(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.23

method	result	size
parallelrisch	\(\frac {-1920 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+96 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-6\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (-\cos \left (\frac {11 d x}{2}+\frac {11 c}{2}\right )-\cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )-15 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )-15 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+80 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+64 \left (-26+35 \cos \left (d x +c \right )-4 \cos \left (2 d x +2 c \right )+\cos \left (3 d x +3 c \right )\right ) \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1440 d x}{768 d a}\)	\(184\)
derivativedivides	\(\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {-9 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {152 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {56}{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+15 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}\)	\(192\)
default	\(\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {-9 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {152 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {56}{3}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+15 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}\)	\(192\)
risch	\(\frac {15 x}{8 a}-\frac {{\mathrm e}^{3 i \left (d x +c \right )}}{24 a d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{4 d a}-\frac {9 \,{\mathrm e}^{i \left (d x +c \right )}}{8 a d}-\frac {9 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d a}-\frac {{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a d}+\frac {{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}+2 i {\mathrm e}^{2 i \left (d x +c \right )}-2 i}{a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}+\frac {\sin \left (4 d x +4 c \right )}{32 d a}\)	\(222\)
norman	\(\frac {\frac {75 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {15 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {75 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {15 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {75 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {75 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {75 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {75 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {75 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {75 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {1}{8 a d}+\frac {15 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {15 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {39 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {59 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {685 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {205 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}-\frac {165 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {169 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {565 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {89 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}-\frac {3 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {21 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {15 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}}{\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 )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}\)	\(537\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/768*(-1920*ln(tan(1/2*d*x+1/2*c))+96*(sec(1/2*d*x+1/2*c)^2-6)*csc(1/2*d*x+1/2*c)^2+3*(-cos(11/2*d*x+11/2*c)-
cos(9/2*d*x+9/2*c)-15*cos(7/2*d*x+7/2*c)-15*cos(5/2*d*x+5/2*c)+80*cos(3/2*d*x+3/2*c)+80*cos(1/2*d*x+1/2*c))*se
c(1/2*d*x+1/2*c)^2*csc(1/2*d*x+1/2*c)+64*(-26+35*cos(d*x+c)-4*cos(2*d*x+2*c)+cos(3*d*x+3*c))*cot(1/2*d*x+1/2*c
)^2+1440*d*x)/d/a

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.99 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {8 \, \cos \left (d x + c\right )^{5} - 45 \, d x \cos \left (d x + c\right )^{2} + 40 \, \cos \left (d x + c\right )^{3} + 45 \, d x - 30 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 30 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (2 \, \cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 60 \, \cos \left (d x + c\right )}{24 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \]

[In]

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

[Out]

-1/24*(8*cos(d*x + c)^5 - 45*d*x*cos(d*x + c)^2 + 40*cos(d*x + c)^3 + 45*d*x - 30*(cos(d*x + c)^2 - 1)*log(1/2
*cos(d*x + c) + 1/2) + 30*(cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2) - 3*(2*cos(d*x + c)^5 + 5*cos(d*x
+ c)^3 - 15*cos(d*x + c))*sin(d*x + c) - 60*cos(d*x + c))/(a*d*cos(d*x + c)^2 - a*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (134) = 268\).

Time = 0.30 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.55 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {124 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {102 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {322 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {78 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {348 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {42 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {147 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {42 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 3}{\frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {4 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {3 \, {\left (\frac {4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{a} + \frac {90 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{24 \, d} \]

[In]

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

[Out]

1/24*((12*sin(d*x + c)/(cos(d*x + c) + 1) - 124*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 102*sin(d*x + c)^3/(cos(
d*x + c) + 1)^3 - 322*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 78*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 348*sin(d
*x + c)^6/(cos(d*x + c) + 1)^6 + 42*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 147*sin(d*x + c)^8/(cos(d*x + c) + 1
)^8 - 42*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 3)/(a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 4*a*sin(d*x + c)^4/
(cos(d*x + c) + 1)^4 + 6*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 4*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a*s
in(d*x + c)^10/(cos(d*x + c) + 1)^10) - 3*(4*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^2/(cos(d*x + c) +
1)^2)/a + 90*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a - 60*log(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {45 \, {\left (d x + c\right )}}{a} - \frac {60 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{2}} + \frac {3 \, {\left (30 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {2 \, {\left (27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 168 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 152 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 56\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]

[In]

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

[Out]

1/24*(45*(d*x + c)/a - 60*log(abs(tan(1/2*d*x + 1/2*c)))/a + 3*(a*tan(1/2*d*x + 1/2*c)^2 - 4*a*tan(1/2*d*x + 1
/2*c))/a^2 + 3*(30*tan(1/2*d*x + 1/2*c)^2 + 4*tan(1/2*d*x + 1/2*c) - 1)/(a*tan(1/2*d*x + 1/2*c)^2) - 2*(27*tan
(1/2*d*x + 1/2*c)^7 + 72*tan(1/2*d*x + 1/2*c)^6 + 3*tan(1/2*d*x + 1/2*c)^5 + 168*tan(1/2*d*x + 1/2*c)^4 - 3*ta
n(1/2*d*x + 1/2*c)^3 + 152*tan(1/2*d*x + 1/2*c)^2 - 27*tan(1/2*d*x + 1/2*c) + 56)/((tan(1/2*d*x + 1/2*c)^2 + 1
)^4*a))/d

Mupad [B] (verification not implemented)

Time = 10.73 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.02 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {15\,\mathrm {atan}\left (\frac {225}{16\,\left (\frac {225\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {75}{4}\right )}-\frac {75\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {225\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {75}{4}\right )}\right )}{4\,a\,d}-\frac {5\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a\,d}-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {49\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+58\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {161\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {62\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1}{2}}{d\,\left (4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+24\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d} \]

[In]

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

[Out]

tan(c/2 + (d*x)/2)^2/(8*a*d) - (15*atan(225/(16*((225*tan(c/2 + (d*x)/2))/16 + 75/4)) - (75*tan(c/2 + (d*x)/2)
)/(4*((225*tan(c/2 + (d*x)/2))/16 + 75/4))))/(4*a*d) - (5*log(tan(c/2 + (d*x)/2)))/(2*a*d) - ((62*tan(c/2 + (d
*x)/2)^2)/3 - 2*tan(c/2 + (d*x)/2) - 17*tan(c/2 + (d*x)/2)^3 + (161*tan(c/2 + (d*x)/2)^4)/3 - 13*tan(c/2 + (d*
x)/2)^5 + 58*tan(c/2 + (d*x)/2)^6 - 7*tan(c/2 + (d*x)/2)^7 + (49*tan(c/2 + (d*x)/2)^8)/2 + 7*tan(c/2 + (d*x)/2
)^9 + 1/2)/(d*(4*a*tan(c/2 + (d*x)/2)^2 + 16*a*tan(c/2 + (d*x)/2)^4 + 24*a*tan(c/2 + (d*x)/2)^6 + 16*a*tan(c/2
+ (d*x)/2)^8 + 4*a*tan(c/2 + (d*x)/2)^10)) - tan(c/2 + (d*x)/2)/(2*a*d)