\(\int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx\) [578]

Optimal result

Integrand size = 27, antiderivative size = 134 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {15 a x}{8}+\frac {5 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {5 a \cos (c+d x)}{2 d}-\frac {5 a \cos ^3(c+d x)}{6 d}-\frac {15 a \cot (c+d x)}{8 d}+\frac {5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}+\frac {a \cos ^4(c+d x) \cot (c+d x)}{4 d}-\frac {a \cos ^3(c+d x) \cot ^2(c+d x)}{2 d} \]

[Out]

-15/8*a*x+5/2*a*arctanh(cos(d*x+c))/d-5/2*a*cos(d*x+c)/d-5/6*a*cos(d*x+c)^3/d-15/8*a*cot(d*x+c)/d+5/8*a*cos(d*
x+c)^2*cot(d*x+c)/d+1/4*a*cos(d*x+c)^4*cot(d*x+c)/d-1/2*a*cos(d*x+c)^3*cot(d*x+c)^2/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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2917, 2672, 294, 308, 212, 2671, 327, 209} \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {5 a \cos ^3(c+d x)}{6 d}-\frac {5 a \cos (c+d x)}{2 d}-\frac {15 a \cot (c+d x)}{8 d}+\frac {a \cos ^4(c+d x) \cot (c+d x)}{4 d}-\frac {a \cos ^3(c+d x) \cot ^2(c+d x)}{2 d}+\frac {5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}-\frac {15 a x}{8} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 1.96 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.87 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \left (216 \cos (c+d x)+8 \cos (3 (c+d x))+3 \left (60 c+60 d x+32 \cot (c+d x)+4 \csc ^2\left (\frac {1}{2} (c+d x)\right )-80 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+80 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 \sec ^2\left (\frac {1}{2} (c+d x)\right )+16 \sin (2 (c+d x))+\sin (4 (c+d x))\right )\right )}{96 d} \]

[In]

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

[Out]

-1/96*(a*(216*Cos[c + d*x] + 8*Cos[3*(c + d*x)] + 3*(60*c + 60*d*x + 32*Cot[c + d*x] + 4*Csc[(c + d*x)/2]^2 -
80*Log[Cos[(c + d*x)/2]] + 80*Log[Sin[(c + d*x)/2]] - 4*Sec[(c + d*x)/2]^2 + 16*Sin[2*(c + d*x)] + Sin[4*(c +
d*x)])))/d

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\)	\(136\)
default	\(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\)	\(136\)
parallelrisch	\(\frac {15 \left (-\frac {128 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\left (\frac {32 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}-\frac {64}{5}\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+\frac {\cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )}{15}+\frac {\cos \left (\frac {11 d x}{2}+\frac {11 c}{2}\right )}{15}-\frac {16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{3}-\frac {16 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )}{3}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\frac {448 \cos \left (d x +c \right )}{9}-\frac {256 \cos \left (2 d x +2 c \right )}{45}+\frac {64 \cos \left (3 d x +3 c \right )}{45}-\frac {1664}{45}\right ) \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 d x \right ) a}{256 d}\)	\(179\)
risch	\(-\frac {15 a x}{8}-\frac {a \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 d}-\frac {9 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {9 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}-\frac {a \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {a \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}-2 i {\mathrm e}^{2 i \left (d x +c \right )}+2 i\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {a \sin \left (4 d x +4 c \right )}{32 d}\)	\(200\)
norman	\(\frac {-\frac {a}{8 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {15 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {5 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {5 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {15 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {15 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {15 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {45 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {15 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {15 a x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {367 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {59 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {17 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {17 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {5 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\)	\(312\)

[In]

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

[Out]

1/d*(a*(-1/sin(d*x+c)*cos(d*x+c)^7-(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)-15/8*d*x-15/8*c)
+a*(-1/2/sin(d*x+c)^2*cos(d*x+c)^7-1/2*cos(d*x+c)^5-5/6*cos(d*x+c)^3-5/2*cos(d*x+c)-5/2*ln(csc(d*x+c)-cot(d*x+
c))))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.21 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {8 \, a \cos \left (d x + c\right )^{5} + 45 \, a d x \cos \left (d x + c\right )^{2} + 40 \, a \cos \left (d x + c\right )^{3} - 45 \, a d x - 60 \, a \cos \left (d x + c\right ) - 30 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 30 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (2 \, a \cos \left (d x + c\right )^{5} + 5 \, a \cos \left (d x + c\right )^{3} - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {2 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a + 3 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a}{24 \, d} \]

[In]

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

[Out]

-1/24*(2*(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*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 25*tan(d*x + c)^2 + 8)/(tan(d*x + c)^5
+ 2*tan(d*x + c)^3 + tan(d*x + c)))*a)/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.60 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, {\left (d x + c\right )} a - 60 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3 \, {\left (30 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, {\left (27 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 152 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 56 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]

[In]

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

[Out]

1/24*(3*a*tan(1/2*d*x + 1/2*c)^2 - 45*(d*x + c)*a - 60*a*log(abs(tan(1/2*d*x + 1/2*c))) + 12*a*tan(1/2*d*x + 1
/2*c) + 3*(30*a*tan(1/2*d*x + 1/2*c)^2 - 4*a*tan(1/2*d*x + 1/2*c) - a)/tan(1/2*d*x + 1/2*c)^2 + 2*(27*a*tan(1/
2*d*x + 1/2*c)^7 - 72*a*tan(1/2*d*x + 1/2*c)^6 + 3*a*tan(1/2*d*x + 1/2*c)^5 - 168*a*tan(1/2*d*x + 1/2*c)^4 - 3
*a*tan(1/2*d*x + 1/2*c)^3 - 152*a*tan(1/2*d*x + 1/2*c)^2 - 27*a*tan(1/2*d*x + 1/2*c) - 56*a)/(tan(1/2*d*x + 1/
2*c)^2 + 1)^4)/d

Mupad [B] (verification not implemented)

Time = 9.85 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.40 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {5\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {49\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}+7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+58\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {161\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+17\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {62\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {15\,a\,\mathrm {atan}\left (\frac {225\,a^2}{16\,\left (\frac {75\,a^2}{4}-\frac {225\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {75\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {75\,a^2}{4}-\frac {225\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d} \]

[In]

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

[Out]

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