\(\int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx\) [580]

Optimal result

Integrand size = 25, antiderivative size = 134 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a x}{2}-\frac {15 a \text {arctanh}(\cos (c+d x))}{8 d}+\frac {15 a \cos (c+d x)}{8 d}+\frac {5 a \cot (c+d x)}{2 d}+\frac {5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac {5 a \cot ^3(c+d x)}{6 d}+\frac {a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a \cos (c+d x) \cot ^4(c+d x)}{4 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2917, 2672, 294, 327, 212, 2671, 308, 209} \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {15 a \text {arctanh}(\cos (c+d x))}{8 d}+\frac {15 a \cos (c+d x)}{8 d}-\frac {5 a \cot ^3(c+d x)}{6 d}+\frac {5 a \cot (c+d x)}{2 d}+\frac {a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac {5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}+\frac {5 a x}{2} \]

[In]

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

[Out]

(5*a*x)/2 - (15*a*ArcTanh[Cos[c + d*x]])/(8*d) + (15*a*Cos[c + d*x])/(8*d) + (5*a*Cot[c + d*x])/(2*d) + (5*a*C
os[c + d*x]*Cot[c + d*x]^2)/(8*d) - (5*a*Cot[c + d*x]^3)/(6*d) + (a*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d) - (a*
Cos[c + d*x]*Cot[c + d*x]^4)/(4*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 2917

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

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

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.03 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (192 \cos (c+d x)-64 \cot (c+d x) \left (-7+\csc ^2(c+d x)\right )+3 \left (18 \csc ^2\left (\frac {1}{2} (c+d x)\right )-\csc ^4\left (\frac {1}{2} (c+d x)\right )+40 \left (4 c+4 d x-3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-18 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )+16 \sin (2 (c+d x))\right )\right )}{192 d} \]

[In]

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

[Out]

(a*(192*Cos[c + d*x] - 64*Cot[c + d*x]*(-7 + Csc[c + d*x]^2) + 3*(18*Csc[(c + d*x)/2]^2 - Csc[(c + d*x)/2]^4 +
40*(4*c + 4*d*x - 3*Log[Cos[(c + d*x)/2]] + 3*Log[Sin[(c + d*x)/2]]) - 18*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/
2]^4 + 16*Sin[2*(c + d*x)])))/(192*d)

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\)	\(172\)
default	\(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\)	\(172\)
parallelrisch	\(\frac {\left (960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-20 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+85\right ) \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )+\cos \left (\frac {11 d x}{2}+\frac {11 c}{2}\right )+10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+10 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-\frac {65 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )}{3}-\frac {65 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )}{3}\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (-161+64 \cos \left (d x +c \right )\right ) \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1280 d x \right ) a}{512 d}\)	\(196\)
risch	\(\frac {5 a x}{2}-\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {a \left (27 \,{\mathrm e}^{7 i \left (d x +c \right )}-3 \,{\mathrm e}^{5 i \left (d x +c \right )}-72 i {\mathrm e}^{6 i \left (d x +c \right )}-3 \,{\mathrm e}^{3 i \left (d x +c \right )}+168 i {\mathrm e}^{4 i \left (d x +c \right )}+27 \,{\mathrm e}^{i \left (d x +c \right )}-152 i {\mathrm e}^{2 i \left (d x +c \right )}+56 i\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\)	\(206\)
norman	\(\frac {-\frac {a}{64 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d}+\frac {7 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {25 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {25 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {25 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {25 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {7 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {5 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+5 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {95 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {95 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {15 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\)	\(282\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.51 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {120 \, a d x \cos \left (d x + c\right )^{4} + 48 \, a \cos \left (d x + c\right )^{5} - 240 \, a d x \cos \left (d x + c\right )^{2} - 150 \, a \cos \left (d x + c\right )^{3} + 120 \, a d x + 90 \, a \cos \left (d x + c\right ) - 45 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 45 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 8 \, {\left (3 \, a \cos \left (d x + c\right )^{5} - 20 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

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

[Out]

1/48*(120*a*d*x*cos(d*x + c)^4 + 48*a*cos(d*x + c)^5 - 240*a*d*x*cos(d*x + c)^2 - 150*a*cos(d*x + c)^3 + 120*a
*d*x + 90*a*cos(d*x + c) - 45*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*log(1/2*cos(d*x + c) + 1/2) + 45*(a*
cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*log(-1/2*cos(d*x + c) + 1/2) + 8*(3*a*cos(d*x + c)^5 - 20*a*cos(d*x +
c)^3 + 15*a*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**5*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.01 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {8 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a - 3 \, a {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]

[In]

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

[Out]

1/48*(8*(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*
a*(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(co
s(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.59 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 480 \, {\left (d x + c\right )} a + 360 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 216 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {192 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {750 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 216 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]

[In]

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

[Out]

1/192*(3*a*tan(1/2*d*x + 1/2*c)^4 + 8*a*tan(1/2*d*x + 1/2*c)^3 - 48*a*tan(1/2*d*x + 1/2*c)^2 + 480*(d*x + c)*a
+ 360*a*log(abs(tan(1/2*d*x + 1/2*c))) - 216*a*tan(1/2*d*x + 1/2*c) - 192*(a*tan(1/2*d*x + 1/2*c)^3 - 2*a*tan
(1/2*d*x + 1/2*c)^2 - a*tan(1/2*d*x + 1/2*c) - 2*a)/(tan(1/2*d*x + 1/2*c)^2 + 1)^2 - (750*a*tan(1/2*d*x + 1/2*
c)^4 - 216*a*tan(1/2*d*x + 1/2*c)^3 - 48*a*tan(1/2*d*x + 1/2*c)^2 + 8*a*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 = 10.08 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.24 \[ \int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}-\frac {9\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {15\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}+\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+36\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {154\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {159\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+\frac {50\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-\frac {a}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {5\,a\,\mathrm {atan}\left (\frac {25\,a^2}{\frac {75\,a^2}{4}-25\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {75\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {75\,a^2}{4}-25\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d} \]

[In]

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

[Out]

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