\(\int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [711]

Optimal result

Integrand size = 29, antiderivative size = 137 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {15 x}{8 a}+\frac {\text {arctanh}(\cos (c+d x))}{a d}-\frac {\cos (c+d x)}{a d}-\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos ^5(c+d x)}{5 a d}-\frac {15 \cot (c+d x)}{8 a d}+\frac {5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac {\cos ^4(c+d x) \cot (c+d x)}{4 a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2918, 2671, 294, 327, 209, 2672, 308, 212} \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{a d}-\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos (c+d x)}{a d}-\frac {15 \cot (c+d x)}{8 a d}+\frac {\cos ^4(c+d x) \cot (c+d x)}{4 a d}+\frac {5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}-\frac {15 x}{8 a} \]

[In]

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

[Out]

(-15*x)/(8*a) + ArcTanh[Cos[c + d*x]]/(a*d) - Cos[c + d*x]/(a*d) - Cos[c + d*x]^3/(3*a*d) - Cos[c + d*x]^5/(5*
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)

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

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (1200 \cos (c+d x)-225 \cos (3 (c+d x))-15 \cos (5 (c+d x))+1800 c \sin (c+d x)+1800 d x \sin (c+d x)-960 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+960 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+590 \sin (2 (c+d x))+64 \sin (4 (c+d x))+6 \sin (6 (c+d x))\right )}{1920 a d} \]

[In]

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

[Out]

-1/1920*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(1200*Cos[c + d*x] - 225*Cos[3*(c + d*x)] - 15*Cos[5*(c + d*x)] + 1
800*c*Sin[c + d*x] + 1800*d*x*Sin[c + d*x] - 960*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 960*Log[Sin[(c + d*x)/2]
]*Sin[c + d*x] + 590*Sin[2*(c + d*x)] + 64*Sin[4*(c + d*x)] + 6*Sin[6*(c + d*x)]))/(a*d)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.94

method	result	size
parallelrisch	\(\frac {-736-480 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (-15-34 \cos \left (d x +c \right )+18 \cos \left (2 d x +2 c \right )-2 \cos \left (3 d x +3 c \right )+\cos \left (4 d x +4 c \right )\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+240 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-900 d x -660 \cos \left (d x +c \right )-70 \cos \left (3 d x +3 c \right )-6 \cos \left (5 d x +5 c \right )}{480 d a}\)	\(129\)
derivativedivides	\(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 \left (-\frac {9 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+6 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {28 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {14 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\frac {23}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {15 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{2 d a}\)	\(177\)
default	\(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 \left (-\frac {9 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+6 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {28 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {14 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\frac {23}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {15 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{2 d a}\)	\(177\)
risch	\(-\frac {15 x}{8 a}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{4 d a}-\frac {11 \,{\mathrm e}^{i \left (d x +c \right )}}{16 a d}-\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 a d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d a}-\frac {2 i}{a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d a}-\frac {\cos \left (5 d x +5 c \right )}{80 a d}-\frac {\sin \left (4 d x +4 c \right )}{32 d a}-\frac {7 \cos \left (3 d x +3 c \right )}{48 a d}\)	\(190\)
norman	\(\frac {-\frac {225 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {45 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {75 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {15 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {225 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {45 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {75 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {225 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {225 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {45 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {1}{2 a d}-\frac {45 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {15 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}-\frac {499 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d a}-\frac {15 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {17 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {61 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {137 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {41 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {403 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d a}-\frac {107 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {179 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {81 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {493 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d a}-\frac {65 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {17 \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 )^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\)	\(569\)

[In]

int(cos(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/480*(-736-480*ln(tan(1/2*d*x+1/2*c))+15*(-15-34*cos(d*x+c)+18*cos(2*d*x+2*c)-2*cos(3*d*x+3*c)+cos(4*d*x+4*c)
)*cot(1/2*d*x+1/2*c)+240*sec(1/2*d*x+1/2*c)*csc(1/2*d*x+1/2*c)-900*d*x-660*cos(d*x+c)-70*cos(3*d*x+3*c)-6*cos(
5*d*x+5*c))/d/a

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {30 \, \cos \left (d x + c\right )^{5} + 75 \, \cos \left (d x + c\right )^{3} - {\left (24 \, \cos \left (d x + c\right )^{5} + 40 \, \cos \left (d x + c\right )^{3} + 225 \, d x + 120 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 60 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 60 \, \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 225 \, \cos \left (d x + c\right )}{120 \, a d \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/120*(30*cos(d*x + c)^5 + 75*cos(d*x + c)^3 - (24*cos(d*x + c)^5 + 40*cos(d*x + c)^3 + 225*d*x + 120*cos(d*x
+ c))*sin(d*x + c) + 60*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 60*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c
) - 225*cos(d*x + c))/(a*d*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*csc(d*x+c)**2/(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. 379 vs. \(2 (125) = 250\).

Time = 0.29 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.77 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {184 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {285 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {560 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {450 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1120 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {300 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {360 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {105 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 30}{\frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {10 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {10 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {5 \, a \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {a \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}} + \frac {225 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {30 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{60 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*((184*sin(d*x + c)/(cos(d*x + c) + 1) + 285*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 560*sin(d*x + c)^3/(co
s(d*x + c) + 1)^3 + 450*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1120*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 300*s
in(d*x + c)^6/(cos(d*x + c) + 1)^6 + 720*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 360*sin(d*x + c)^9/(cos(d*x + c
) + 1)^9 - 105*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 30)/(a*sin(d*x + c)/(cos(d*x + c) + 1) + 5*a*sin(d*x +
c)^3/(cos(d*x + c) + 1)^3 + 10*a*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 10*a*sin(d*x + c)^7/(cos(d*x + c) + 1)^
7 + 5*a*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + a*sin(d*x + c)^11/(cos(d*x + c) + 1)^11) + 225*arctan(sin(d*x +
c)/(cos(d*x + c) + 1))/a + 60*log(sin(d*x + c)/(cos(d*x + c) + 1))/a - 30*sin(d*x + c)/(a*(cos(d*x + c) + 1)))
/d

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {225 \, {\left (d x + c\right )}}{a} + \frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {60 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {60 \, {\left (2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {2 \, {\left (135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 150 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 150 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 184\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a}}{120 \, d} \]

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/120*(225*(d*x + c)/a + 120*log(abs(tan(1/2*d*x + 1/2*c)))/a - 60*tan(1/2*d*x + 1/2*c)/a - 60*(2*tan(1/2*d*x
+ 1/2*c) - 1)/(a*tan(1/2*d*x + 1/2*c)) - 2*(135*tan(1/2*d*x + 1/2*c)^9 - 360*tan(1/2*d*x + 1/2*c)^8 + 150*tan
(1/2*d*x + 1/2*c)^7 - 720*tan(1/2*d*x + 1/2*c)^6 - 1120*tan(1/2*d*x + 1/2*c)^4 - 150*tan(1/2*d*x + 1/2*c)^3 -
560*tan(1/2*d*x + 1/2*c)^2 - 135*tan(1/2*d*x + 1/2*c) - 184)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*a))/d

Mupad [B] (verification not implemented)

Time = 9.95 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.16 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {15\,\mathrm {atan}\left (\frac {15\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {225\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}-\frac {15}{2}\right )}+\frac {225}{16\,\left (\frac {225\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}-\frac {15}{2}\right )}\right )}{4\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{2}+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {112\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {19\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {92\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+1}{d\,\left (2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+10\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+10\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d} \]

[In]

int(cos(c + d*x)^8/(sin(c + d*x)^2*(a + a*sin(c + d*x))),x)

[Out]

(15*atan((15*tan(c/2 + (d*x)/2))/(2*((225*tan(c/2 + (d*x)/2))/16 - 15/2)) + 225/(16*((225*tan(c/2 + (d*x)/2))/
16 - 15/2))))/(4*a*d) - log(tan(c/2 + (d*x)/2))/(a*d) - ((92*tan(c/2 + (d*x)/2))/15 + (19*tan(c/2 + (d*x)/2)^2
)/2 + (56*tan(c/2 + (d*x)/2)^3)/3 + 15*tan(c/2 + (d*x)/2)^4 + (112*tan(c/2 + (d*x)/2)^5)/3 + 10*tan(c/2 + (d*x
)/2)^6 + 24*tan(c/2 + (d*x)/2)^7 + 12*tan(c/2 + (d*x)/2)^9 - (7*tan(c/2 + (d*x)/2)^10)/2 + 1)/(d*(2*a*tan(c/2
+ (d*x)/2) + 10*a*tan(c/2 + (d*x)/2)^3 + 20*a*tan(c/2 + (d*x)/2)^5 + 20*a*tan(c/2 + (d*x)/2)^7 + 10*a*tan(c/2
+ (d*x)/2)^9 + 2*a*tan(c/2 + (d*x)/2)^11)) + tan(c/2 + (d*x)/2)/(2*a*d)