\(\int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx\) [611]

Optimal result

Integrand size = 29, antiderivative size = 181 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {85 a^3 x}{16}-\frac {a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d} \]

[Out]

-85/16*a^3*x-1/2*a^3*arctanh(cos(d*x+c))/d+a^3*cos(d*x+c)/d+2/3*a^3*cos(d*x+c)^3/d+3/5*a^3*cos(d*x+c)^5/d-3*a^
3*cot(d*x+c)/d-1/2*a^3*cot(d*x+c)*csc(d*x+c)/d-43/16*a^3*cos(d*x+c)*sin(d*x+c)/d+5/24*a^3*cos(d*x+c)*sin(d*x+c
)^3/d+1/6*a^3*cos(d*x+c)*sin(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2951, 3852, 8, 3853, 3855, 2718, 2715, 2713} \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}+\frac {a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac {5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {85 a^3 x}{16} \]

[In]

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

[Out]

(-85*a^3*x)/16 - (a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (a^3*Cos[c + d*x])/d + (2*a^3*Cos[c + d*x]^3)/(3*d) + (3*
a^3*Cos[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (43*a^3*Cos[c + d
*x]*Sin[c + d*x])/(16*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d
)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2951

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-8 a^9+3 a^9 \csc ^2(c+d x)+a^9 \csc ^3(c+d x)-6 a^9 \sin (c+d x)+6 a^9 \sin ^2(c+d x)+8 a^9 \sin ^3(c+d x)-3 a^9 \sin ^5(c+d x)-a^9 \sin ^6(c+d x)\right ) \, dx}{a^6} \\ & = -8 a^3 x+a^3 \int \csc ^3(c+d x) \, dx-a^3 \int \sin ^6(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (3 a^3\right ) \int \sin ^5(c+d x) \, dx-\left (6 a^3\right ) \int \sin (c+d x) \, dx+\left (6 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (8 a^3\right ) \int \sin ^3(c+d x) \, dx \\ & = -8 a^3 x+\frac {6 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{2} a^3 \int \csc (c+d x) \, dx-\frac {1}{6} \left (5 a^3\right ) \int \sin ^4(c+d x) \, dx+\left (3 a^3\right ) \int 1 \, dx-\frac {\left (3 a^3\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (8 a^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -5 a^3 x-\frac {a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{d}+\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {1}{8} \left (5 a^3\right ) \int \sin ^2(c+d x) \, dx \\ & = -5 a^3 x-\frac {a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {1}{16} \left (5 a^3\right ) \int 1 \, dx \\ & = -\frac {85 a^3 x}{16}-\frac {a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(664\) vs. \(2(181)=362\).

Time = 11.67 (sec) , antiderivative size = 664, normalized size of antiderivative = 3.67 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {85 (c+d x) (a+a \sin (c+d x))^3}{16 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {15 \cos (c+d x) (a+a \sin (c+d x))^3}{8 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {17 \cos (3 (c+d x)) (a+a \sin (c+d x))^3}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {3 \cos (5 (c+d x)) (a+a \sin (c+d x))^3}{80 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {3 \cot \left (\frac {1}{2} (c+d x)\right ) (a+a \sin (c+d x))^3}{2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sin (c+d x))^3}{8 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a+a \sin (c+d x))^3}{2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+a \sin (c+d x))^3}{2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sin (c+d x))^3}{8 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {81 (a+a \sin (c+d x))^3 \sin (2 (c+d x))}{64 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {3 (a+a \sin (c+d x))^3 \sin (4 (c+d x))}{64 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {(a+a \sin (c+d x))^3 \sin (6 (c+d x))}{192 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {3 (a+a \sin (c+d x))^3 \tan \left (\frac {1}{2} (c+d x)\right )}{2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

[In]

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

[Out]

(-85*(c + d*x)*(a + a*Sin[c + d*x])^3)/(16*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (15*Cos[c + d*x]*(a +
a*Sin[c + d*x])^3)/(8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (17*Cos[3*(c + d*x)]*(a + a*Sin[c + d*x])^3
)/(48*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (3*Cos[5*(c + d*x)]*(a + a*Sin[c + d*x])^3)/(80*d*(Cos[(c +
d*x)/2] + Sin[(c + d*x)/2])^6) - (3*Cot[(c + d*x)/2]*(a + a*Sin[c + d*x])^3)/(2*d*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2])^6) - (Csc[(c + d*x)/2]^2*(a + a*Sin[c + d*x])^3)/(8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (L
og[Cos[(c + d*x)/2]]*(a + a*Sin[c + d*x])^3)/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (Log[Sin[(c + d*x
)/2]]*(a + a*Sin[c + d*x])^3)/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (Sec[(c + d*x)/2]^2*(a + a*Sin[c
+ d*x])^3)/(8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (81*(a + a*Sin[c + d*x])^3*Sin[2*(c + d*x)])/(64*d
*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (3*(a + a*Sin[c + d*x])^3*Sin[4*(c + d*x)])/(64*d*(Cos[(c + d*x)/2
] + Sin[(c + d*x)/2])^6) + ((a + a*Sin[c + d*x])^3*Sin[6*(c + d*x)])/(192*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/
2])^6) + (3*(a + a*Sin[c + d*x])^3*Tan[(c + d*x)/2])/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.99

method	result	size
parallelrisch	\(-\frac {a^{3} \left (-10200 d x \cos \left (2 d x +2 c \right )+960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (2 d x +2 c \right )+10200 d x -960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2912 \cos \left (2 d x +2 c \right )+460 \cos \left (d x +c \right )+8145 \sin \left (2 d x +2 c \right )+1156 \cos \left (3 d x +3 c \right )-1120 \sin \left (4 d x +4 c \right )-55 \sin \left (6 d x +6 c \right )+36 \cos \left (7 d x +7 c \right )+268 \cos \left (5 d x +5 c \right )+5 \sin \left (8 d x +8 c \right )-2912\right ) \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15360 d}\)	\(179\)
derivativedivides	\(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+3 a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\)	\(236\)
default	\(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+3 a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\)	\(236\)
risch	\(-\frac {85 a^{3} x}{16}+\frac {17 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{96 d}+\frac {81 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {15 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {15 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {81 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}+\frac {17 a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{96 d}+\frac {a^{3} \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}-6 i {\mathrm e}^{2 i \left (d x +c \right )}+6 i\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{80 d}-\frac {3 a^{3} \sin \left (4 d x +4 c \right )}{64 d}\)	\(256\)
norman	\(\frac {-\frac {a^{3}}{8 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {103 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {671 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {45 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {45 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {671 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {103 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {3 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a^{3} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {85 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {255 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {1275 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {425 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {1275 a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {255 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {85 a^{3} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {27 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {37 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {37 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {91 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}+\frac {91 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {179 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}}{\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 )^{2}}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\)	\(458\)

[In]

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

[Out]

-1/15360*a^3*(-10200*d*x*cos(2*d*x+2*c)+960*ln(tan(1/2*d*x+1/2*c))*cos(2*d*x+2*c)+10200*d*x-960*ln(tan(1/2*d*x
+1/2*c))+2912*cos(2*d*x+2*c)+460*cos(d*x+c)+8145*sin(2*d*x+2*c)+1156*cos(3*d*x+3*c)-1120*sin(4*d*x+4*c)-55*sin
(6*d*x+6*c)+36*cos(7*d*x+7*c)+268*cos(5*d*x+5*c)+5*sin(8*d*x+8*c)-2912)*sec(1/2*d*x+1/2*c)^2*csc(1/2*d*x+1/2*c
)^2/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.17 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {144 \, a^{3} \cos \left (d x + c\right )^{7} + 16 \, a^{3} \cos \left (d x + c\right )^{5} - 1275 \, a^{3} d x \cos \left (d x + c\right )^{2} + 80 \, a^{3} \cos \left (d x + c\right )^{3} + 1275 \, a^{3} d x - 120 \, a^{3} \cos \left (d x + c\right ) - 60 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 60 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{7} - 34 \, a^{3} \cos \left (d x + c\right )^{5} - 85 \, a^{3} \cos \left (d x + c\right )^{3} + 255 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

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

[Out]

1/240*(144*a^3*cos(d*x + c)^7 + 16*a^3*cos(d*x + c)^5 - 1275*a^3*d*x*cos(d*x + c)^2 + 80*a^3*cos(d*x + c)^3 +
1275*a^3*d*x - 120*a^3*cos(d*x + c) - 60*(a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2) + 60*(a^3*cos(
d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1/2) + 5*(8*a^3*cos(d*x + c)^7 - 34*a^3*cos(d*x + c)^5 - 85*a^3*cos(
d*x + c)^3 + 255*a^3*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.32 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {96 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 80 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 360 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3}}{960 \, d} \]

[In]

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

[Out]

1/960*(96*(6*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 30*cos(d*x + c) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x
+ c) - 1))*a^3 - 80*(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 - 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48
*sin(2*d*x + 2*c))*a^3 - 360*(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^3)/d

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.69 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1275 \, {\left (d x + c\right )} a^{3} + 120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 360 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {30 \, {\left (6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, {\left (645 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1440 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1735 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3360 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5440 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1735 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1824 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 645 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 544 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]

[In]

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

[Out]

1/240*(30*a^3*tan(1/2*d*x + 1/2*c)^2 - 1275*(d*x + c)*a^3 + 120*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 360*a^3*t
an(1/2*d*x + 1/2*c) - 30*(6*a^3*tan(1/2*d*x + 1/2*c)^2 + 12*a^3*tan(1/2*d*x + 1/2*c) + a^3)/tan(1/2*d*x + 1/2*
c)^2 + 2*(645*a^3*tan(1/2*d*x + 1/2*c)^11 + 1440*a^3*tan(1/2*d*x + 1/2*c)^10 + 1735*a^3*tan(1/2*d*x + 1/2*c)^9
+ 3360*a^3*tan(1/2*d*x + 1/2*c)^8 + 450*a^3*tan(1/2*d*x + 1/2*c)^7 + 5440*a^3*tan(1/2*d*x + 1/2*c)^6 - 450*a^
3*tan(1/2*d*x + 1/2*c)^5 + 4800*a^3*tan(1/2*d*x + 1/2*c)^4 - 1735*a^3*tan(1/2*d*x + 1/2*c)^3 + 1824*a^3*tan(1/
2*d*x + 1/2*c)^2 - 645*a^3*tan(1/2*d*x + 1/2*c) + 544*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d

Mupad [B] (verification not implemented)

Time = 11.16 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.42 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {31\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{2}+\frac {95\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{2}+\frac {131\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{6}+109\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-75\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {1043\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{6}-135\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+150\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {887\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{6}+\frac {533\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{10}-\frac {115\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {227\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-6\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {a^3}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+60\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+60\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+24\,{\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 {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}+\frac {85\,a^3\,\mathrm {atan}\left (\frac {7225\,a^6}{64\,\left (\frac {85\,a^6}{8}+\frac {7225\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}-\frac {85\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,\left (\frac {85\,a^6}{8}+\frac {7225\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}\right )}{8\,d}+\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d} \]

[In]

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

[Out]

((227*a^3*tan(c/2 + (d*x)/2)^2)/15 - (115*a^3*tan(c/2 + (d*x)/2)^3)/2 + (533*a^3*tan(c/2 + (d*x)/2)^4)/10 - (8
87*a^3*tan(c/2 + (d*x)/2)^5)/6 + 150*a^3*tan(c/2 + (d*x)/2)^6 - 135*a^3*tan(c/2 + (d*x)/2)^7 + (1043*a^3*tan(c
/2 + (d*x)/2)^8)/6 - 75*a^3*tan(c/2 + (d*x)/2)^9 + 109*a^3*tan(c/2 + (d*x)/2)^10 + (131*a^3*tan(c/2 + (d*x)/2)
^11)/6 + (95*a^3*tan(c/2 + (d*x)/2)^12)/2 + (31*a^3*tan(c/2 + (d*x)/2)^13)/2 - a^3/2 - 6*a^3*tan(c/2 + (d*x)/2
))/(d*(4*tan(c/2 + (d*x)/2)^2 + 24*tan(c/2 + (d*x)/2)^4 + 60*tan(c/2 + (d*x)/2)^6 + 80*tan(c/2 + (d*x)/2)^8 +
60*tan(c/2 + (d*x)/2)^10 + 24*tan(c/2 + (d*x)/2)^12 + 4*tan(c/2 + (d*x)/2)^14)) + (a^3*tan(c/2 + (d*x)/2)^2)/(
8*d) + (a^3*log(tan(c/2 + (d*x)/2)))/(2*d) + (85*a^3*atan((7225*a^6)/(64*((85*a^6)/8 + (7225*a^6*tan(c/2 + (d*
x)/2))/64)) - (85*a^6*tan(c/2 + (d*x)/2))/(8*((85*a^6)/8 + (7225*a^6*tan(c/2 + (d*x)/2))/64))))/(8*d) + (3*a^3
*tan(c/2 + (d*x)/2))/(2*d)