\(\int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx\) [714]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.15 (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, 327, 212, 2671, 308, 209} \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {15 \text {arctanh}(\cos (c+d x))}{8 a d}+\frac {15 \cos (c+d x)}{8 a d}+\frac {5 \cot ^3(c+d x)}{6 a d}-\frac {5 \cot (c+d x)}{2 a d}-\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}+\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}-\frac {5 x}{2 a} \]

[In]

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

[Out]

(-5*x)/(2*a) - (15*ArcTanh[Cos[c + d*x]])/(8*a*d) + (15*Cos[c + d*x])/(8*a*d) - (5*Cot[c + d*x])/(2*a*d) + (5*
Cos[c + d*x]*Cot[c + d*x]^2)/(8*a*d) + (5*Cot[c + d*x]^3)/(6*a*d) - (Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*a*d) -
(Cos[c + d*x]*Cot[c + d*x]^4)/(4*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 ^2(c+d x) \cot ^4(c+d x) \, dx}{a}+\frac {\int \cos (c+d x) \cot ^5(c+d x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{a d} \\ & = -\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}+\frac {5 \text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{4 a d}+\frac {5 \text {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 a d} \\ & = \frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}-\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac {15 \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (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,\cot (c+d x)\right )}{2 a d} \\ & = \frac {15 \cos (c+d x)}{8 a d}-\frac {5 \cot (c+d x)}{2 a d}+\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac {5 \cot ^3(c+d x)}{6 a d}-\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac {15 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 a d}+\frac {5 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 a d} \\ & = -\frac {5 x}{2 a}-\frac {15 \text {arctanh}(\cos (c+d x))}{8 a d}+\frac {15 \cos (c+d x)}{8 a d}-\frac {5 \cot (c+d x)}{2 a d}+\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac {5 \cot ^3(c+d x)}{6 a d}-\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.68 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^4(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (180 c+180 d x-30 \cos (c+d x)+90 \cos (3 (c+d x))+60 c \cos (4 (c+d x))+60 d x \cos (4 (c+d x))-12 \cos (5 (c+d x))+135 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+45 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-60 \cos (2 (c+d x)) \left (4 c+4 d x+3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-135 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-45 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+95 \sin (2 (c+d x))-68 \sin (4 (c+d x))+3 \sin (6 (c+d x))\right )}{192 a d (1+\sin (c+d x))} \]

[In]

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

[Out]

-1/192*(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(180*c + 180*d*x - 30*Cos[c + d*x] + 90*Cos[3*(
c + d*x)] + 60*c*Cos[4*(c + d*x)] + 60*d*x*Cos[4*(c + d*x)] - 12*Cos[5*(c + d*x)] + 135*Log[Cos[(c + d*x)/2]]
+ 45*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 60*Cos[2*(c + d*x)]*(4*c + 4*d*x + 3*Log[Cos[(c + d*x)/2]] - 3*L
og[Sin[(c + d*x)/2]]) - 135*Log[Sin[(c + d*x)/2]] - 45*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 95*Sin[2*(c +
d*x)] - 68*Sin[4*(c + d*x)] + 3*Sin[6*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {18}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+30 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {32 \left (-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-1\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-80 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}\)	\(192\)
default	\(\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {18}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+30 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {32 \left (-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-1\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-80 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}\)	\(192\)
parallelrisch	\(\frac {2880 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-20 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+85\right ) \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 \cos \left (\frac {11 d x}{2}+\frac {11 c}{2}\right )-3 \cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )+65 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+65 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-30 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-30 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+720 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (-161+64 \cos \left (d x +c \right )\right ) \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3840 d x}{1536 d a}\)	\(202\)
risch	\(-\frac {5 x}{2 a}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d a}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 a d}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 a d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d a}-\frac {72 i {\mathrm e}^{6 i \left (d x +c \right )}+27 \,{\mathrm e}^{7 i \left (d x +c \right )}-168 i {\mathrm e}^{4 i \left (d x +c \right )}-3 \,{\mathrm e}^{5 i \left (d x +c \right )}+152 i {\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{3 i \left (d x +c \right )}-56 i+27 \,{\mathrm e}^{i \left (d x +c \right )}}{12 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}\)	\(222\)
norman	\(\frac {-\frac {1}{64 a d}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{192 d a}+\frac {47 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {51 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {51 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}-\frac {47 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {5 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {5 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {5 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {15 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {15 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {15 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {15 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {5 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {5 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {17 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {21 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {23 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {91 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}-\frac {189 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}-\frac {23 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}-\frac {89 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {15 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}\)	\(469\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^5/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/16/d/a*(1/4*tan(1/2*d*x+1/2*c)^4-2/3*tan(1/2*d*x+1/2*c)^3-4*tan(1/2*d*x+1/2*c)^2+18*tan(1/2*d*x+1/2*c)-1/4/t
an(1/2*d*x+1/2*c)^4+2/3/tan(1/2*d*x+1/2*c)^3+4/tan(1/2*d*x+1/2*c)^2-18/tan(1/2*d*x+1/2*c)+30*ln(tan(1/2*d*x+1/
2*c))-32*(-1/2*tan(1/2*d*x+1/2*c)^3-tan(1/2*d*x+1/2*c)^2+1/2*tan(1/2*d*x+1/2*c)-1)/(1+tan(1/2*d*x+1/2*c)^2)^2-
80*arctan(tan(1/2*d*x+1/2*c)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.27 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {120 \, d x \cos \left (d x + c\right )^{4} - 48 \, \cos \left (d x + c\right )^{5} - 240 \, d x \cos \left (d x + c\right )^{2} + 150 \, \cos \left (d x + c\right )^{3} + 120 \, d x + 45 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 45 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 8 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 90 \, \cos \left (d x + c\right )}{48 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/48*(120*d*x*cos(d*x + c)^4 - 48*cos(d*x + c)^5 - 240*d*x*cos(d*x + c)^2 + 150*cos(d*x + c)^3 + 120*d*x + 45
*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2) - 45*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 +
1)*log(-1/2*cos(d*x + c) + 1/2) + 8*(3*cos(d*x + c)^5 - 20*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c) - 9
0*cos(d*x + c))/(a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^2 + a*d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**5/(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. 340 vs. \(2 (134) = 268\).

Time = 0.31 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.27 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {216 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a} + \frac {\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {42 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {477 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {616 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {432 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {24 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 3}{\frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {960 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{192 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/192*((216*sin(d*x + c)/(cos(d*x + c) + 1) - 48*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 8*sin(d*x + c)^3/(cos(d
*x + c) + 1)^3 + 3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)/a + (8*sin(d*x + c)/(cos(d*x + c) + 1) + 42*sin(d*x +
c)^2/(cos(d*x + c) + 1)^2 - 200*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 477*sin(d*x + c)^4/(cos(d*x + c) + 1)^4
- 616*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 432*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 24*sin(d*x + c)^7/(cos(d
*x + c) + 1)^7 - 3)/(a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 2*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a*sin(d
*x + c)^8/(cos(d*x + c) + 1)^8) - 960*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 360*log(sin(d*x + c)/(cos(d*
x + c) + 1))/a)/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.49 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {480 \, {\left (d x + c\right )}}{a} - \frac {360 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {192 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a} - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 216 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} + \frac {750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 216 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/192*(480*(d*x + c)/a - 360*log(abs(tan(1/2*d*x + 1/2*c)))/a - 192*(tan(1/2*d*x + 1/2*c)^3 + 2*tan(1/2*d*x +
1/2*c)^2 - tan(1/2*d*x + 1/2*c) + 2)/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a) - (3*a^3*tan(1/2*d*x + 1/2*c)^4 - 8*a
^3*tan(1/2*d*x + 1/2*c)^3 - 48*a^3*tan(1/2*d*x + 1/2*c)^2 + 216*a^3*tan(1/2*d*x + 1/2*c))/a^4 + (750*tan(1/2*d
*x + 1/2*c)^4 + 216*tan(1/2*d*x + 1/2*c)^3 - 48*tan(1/2*d*x + 1/2*c)^2 - 8*tan(1/2*d*x + 1/2*c) + 3)/(a*tan(1/
2*d*x + 1/2*c)^4))/d

Mupad [B] (verification not implemented)

Time = 10.26 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.91 \[ \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a\,d}+\frac {5\,\mathrm {atan}\left (\frac {25}{25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {75}{4}}-\frac {75\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {75}{4}\right )}\right )}{a\,d}+\frac {15\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a\,d}+\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {154\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {159\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}-\frac {50\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-\frac {1}{4}}{d\,\left (16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+32\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^5*(a + a*sin(c + d*x))),x)

[Out]

tan(c/2 + (d*x)/2)^4/(64*a*d) - tan(c/2 + (d*x)/2)^3/(24*a*d) - tan(c/2 + (d*x)/2)^2/(4*a*d) + (5*atan(25/(25*
tan(c/2 + (d*x)/2) + 75/4) - (75*tan(c/2 + (d*x)/2))/(4*(25*tan(c/2 + (d*x)/2) + 75/4))))/(a*d) + (15*log(tan(
c/2 + (d*x)/2)))/(8*a*d) + ((2*tan(c/2 + (d*x)/2))/3 + (7*tan(c/2 + (d*x)/2)^2)/2 - (50*tan(c/2 + (d*x)/2)^3)/
3 + (159*tan(c/2 + (d*x)/2)^4)/4 - (154*tan(c/2 + (d*x)/2)^5)/3 + 36*tan(c/2 + (d*x)/2)^6 - 2*tan(c/2 + (d*x)/
2)^7 - 1/4)/(d*(16*a*tan(c/2 + (d*x)/2)^4 + 32*a*tan(c/2 + (d*x)/2)^6 + 16*a*tan(c/2 + (d*x)/2)^8)) + (9*tan(c
/2 + (d*x)/2))/(8*a*d)