\(\int \frac {\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx\) [242]

Optimal result

Integrand size = 32, antiderivative size = 138 \[ \int \frac {\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=-\frac {19 A \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {4 A \cot (c+d x)}{a^3 d}-\frac {A \cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {29 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}+\frac {164 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))} \]

[Out]

-19/2*A*arctanh(cos(d*x+c))/a^3/d+4*A*cot(d*x+c)/a^3/d-1/2*A*cot(d*x+c)*csc(d*x+c)/a^3/d+2/5*A*cos(d*x+c)/a^3/
d/(1+sin(d*x+c))^3+29/15*A*cos(d*x+c)/a^3/d/(1+sin(d*x+c))^2+164/15*A*cos(d*x+c)/a^3/d/(1+sin(d*x+c))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {3045, 3855, 3852, 8, 3853, 2729, 2727} \[ \int \frac {\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=-\frac {19 A \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {4 A \cot (c+d x)}{a^3 d}+\frac {164 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}+\frac {29 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}+\frac {2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac {A \cot (c+d x) \csc (c+d x)}{2 a^3 d} \]

[In]

Int[(Csc[c + d*x]^3*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]

[Out]

(-19*A*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (4*A*Cot[c + d*x])/(a^3*d) - (A*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d)
+ (2*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^3) + (29*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x])^2) + (
164*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x]))

Rule 8

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

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2729

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d
*(2*n + 1))), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 3045

Int[sin[(e_.) + (f_.)*(x_)]^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; Fr
eeQ[{a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IntegerQ[n]

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}& = \int \left (\frac {9 A \csc (c+d x)}{a^3}-\frac {4 A \csc ^2(c+d x)}{a^3}+\frac {A \csc ^3(c+d x)}{a^3}-\frac {2 A}{a^3 (1+\sin (c+d x))^3}-\frac {5 A}{a^3 (1+\sin (c+d x))^2}-\frac {9 A}{a^3 (1+\sin (c+d x))}\right ) \, dx \\ & = \frac {A \int \csc ^3(c+d x) \, dx}{a^3}-\frac {(2 A) \int \frac {1}{(1+\sin (c+d x))^3} \, dx}{a^3}-\frac {(4 A) \int \csc ^2(c+d x) \, dx}{a^3}-\frac {(5 A) \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{a^3}+\frac {(9 A) \int \csc (c+d x) \, dx}{a^3}-\frac {(9 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3} \\ & = -\frac {9 A \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {A \cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {5 A \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}+\frac {9 A \cos (c+d x)}{a^3 d (1+\sin (c+d x))}+\frac {A \int \csc (c+d x) \, dx}{2 a^3}-\frac {(4 A) \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{5 a^3}-\frac {(5 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{3 a^3}+\frac {(4 A) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d} \\ & = -\frac {19 A \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {4 A \cot (c+d x)}{a^3 d}-\frac {A \cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {29 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}+\frac {32 A \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))}-\frac {(4 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{15 a^3} \\ & = -\frac {19 A \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {4 A \cot (c+d x)}{a^3 d}-\frac {A \cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac {29 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}+\frac {164 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.90 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.78 \[ \int \frac {\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=\frac {A \left (240 \cot \left (\frac {1}{2} (c+d x)\right )-15 \csc ^2\left (\frac {1}{2} (c+d x)\right )-1140 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1140 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+15 \sec ^2\left (\frac {1}{2} (c+d x)\right )-\frac {96 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {48}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}-\frac {464 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {232}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {2624 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}-240 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{120 a^3 d} \]

[In]

Integrate[(Csc[c + d*x]^3*(A - A*Sin[c + d*x]))/(a + a*Sin[c + d*x])^3,x]

[Out]

(A*(240*Cot[(c + d*x)/2] - 15*Csc[(c + d*x)/2]^2 - 1140*Log[Cos[(c + d*x)/2]] + 1140*Log[Sin[(c + d*x)/2]] + 1
5*Sec[(c + d*x)/2]^2 - (96*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 + 48/(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])^4 - (464*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 232/(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])^2 - (2624*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 240*Tan[(c + d*x)/2]))/
(120*a^3*d)

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {A \left (\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {64}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {32}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {208}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {72}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {128}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+38 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{4 d \,a^{3}}\)	\(148\)
default	\(\frac {A \left (\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {64}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {32}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {208}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {72}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {128}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+38 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{4 d \,a^{3}}\)	\(148\)
parallelrisch	\(-\frac {\left (-76 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-472 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1504 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {6056 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-11 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {3868 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3}-\frac {5284}{15}\right ) A}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\)	\(148\)
risch	\(\frac {A \left (1425 i {\mathrm e}^{7 i \left (d x +c \right )}+285 \,{\mathrm e}^{8 i \left (d x +c \right )}-5225 i {\mathrm e}^{5 i \left (d x +c \right )}-3325 \,{\mathrm e}^{6 i \left (d x +c \right )}+5635 i {\mathrm e}^{3 i \left (d x +c \right )}+6423 \,{\mathrm e}^{4 i \left (d x +c \right )}-1955 i {\mathrm e}^{i \left (d x +c \right )}-3951 \,{\mathrm e}^{2 i \left (d x +c \right )}+448\right )}{15 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} a^{3} d}-\frac {19 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}+\frac {19 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}\)	\(174\)
norman	\(\frac {\frac {57 A \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {A}{8 a d}+\frac {11 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {11 A \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}+\frac {A \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}+\frac {1943 A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}+\frac {2627 A \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}+\frac {8335 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d}+\frac {1493 A \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}+\frac {7427 A \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d}+\frac {35399 A \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 a d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {19 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3} d}\)	\(272\)

[In]

int(csc(d*x+c)^3*(A-A*sin(d*x+c))/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/4/d*A/a^3*(1/2*tan(1/2*d*x+1/2*c)^2-8*tan(1/2*d*x+1/2*c)+64/5/(tan(1/2*d*x+1/2*c)+1)^5-32/(tan(1/2*d*x+1/2*c
)+1)^4+208/3/(tan(1/2*d*x+1/2*c)+1)^3-72/(tan(1/2*d*x+1/2*c)+1)^2+128/(tan(1/2*d*x+1/2*c)+1)-1/2/tan(1/2*d*x+1
/2*c)^2+8/tan(1/2*d*x+1/2*c)+38*ln(tan(1/2*d*x+1/2*c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (128) = 256\).

Time = 0.28 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.61 \[ \int \frac {\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=\frac {896 \, A \cos \left (d x + c\right )^{5} - 1222 \, A \cos \left (d x + c\right )^{4} - 3218 \, A \cos \left (d x + c\right )^{3} + 1168 \, A \cos \left (d x + c\right )^{2} + 2292 \, A \cos \left (d x + c\right ) - 285 \, {\left (A \cos \left (d x + c\right )^{5} + 3 \, A \cos \left (d x + c\right )^{4} - 3 \, A \cos \left (d x + c\right )^{3} - 7 \, A \cos \left (d x + c\right )^{2} + 2 \, A \cos \left (d x + c\right ) + {\left (A \cos \left (d x + c\right )^{4} - 2 \, A \cos \left (d x + c\right )^{3} - 5 \, A \cos \left (d x + c\right )^{2} + 2 \, A \cos \left (d x + c\right ) + 4 \, A\right )} \sin \left (d x + c\right ) + 4 \, A\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 285 \, {\left (A \cos \left (d x + c\right )^{5} + 3 \, A \cos \left (d x + c\right )^{4} - 3 \, A \cos \left (d x + c\right )^{3} - 7 \, A \cos \left (d x + c\right )^{2} + 2 \, A \cos \left (d x + c\right ) + {\left (A \cos \left (d x + c\right )^{4} - 2 \, A \cos \left (d x + c\right )^{3} - 5 \, A \cos \left (d x + c\right )^{2} + 2 \, A \cos \left (d x + c\right ) + 4 \, A\right )} \sin \left (d x + c\right ) + 4 \, A\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (448 \, A \cos \left (d x + c\right )^{4} + 1059 \, A \cos \left (d x + c\right )^{3} - 550 \, A \cos \left (d x + c\right )^{2} - 1134 \, A \cos \left (d x + c\right ) + 12 \, A\right )} \sin \left (d x + c\right ) + 24 \, A}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} - 3 \, a^{3} d \cos \left (d x + c\right )^{3} - 7 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 5 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(csc(d*x+c)^3*(A-A*sin(d*x+c))/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/60*(896*A*cos(d*x + c)^5 - 1222*A*cos(d*x + c)^4 - 3218*A*cos(d*x + c)^3 + 1168*A*cos(d*x + c)^2 + 2292*A*co
s(d*x + c) - 285*(A*cos(d*x + c)^5 + 3*A*cos(d*x + c)^4 - 3*A*cos(d*x + c)^3 - 7*A*cos(d*x + c)^2 + 2*A*cos(d*
x + c) + (A*cos(d*x + c)^4 - 2*A*cos(d*x + c)^3 - 5*A*cos(d*x + c)^2 + 2*A*cos(d*x + c) + 4*A)*sin(d*x + c) +
4*A)*log(1/2*cos(d*x + c) + 1/2) + 285*(A*cos(d*x + c)^5 + 3*A*cos(d*x + c)^4 - 3*A*cos(d*x + c)^3 - 7*A*cos(d
*x + c)^2 + 2*A*cos(d*x + c) + (A*cos(d*x + c)^4 - 2*A*cos(d*x + c)^3 - 5*A*cos(d*x + c)^2 + 2*A*cos(d*x + c)
+ 4*A)*sin(d*x + c) + 4*A)*log(-1/2*cos(d*x + c) + 1/2) - 2*(448*A*cos(d*x + c)^4 + 1059*A*cos(d*x + c)^3 - 55
0*A*cos(d*x + c)^2 - 1134*A*cos(d*x + c) + 12*A)*sin(d*x + c) + 24*A)/(a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x
+ c)^4 - 3*a^3*d*cos(d*x + c)^3 - 7*a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + 4*a^3*d + (a^3*d*cos(d*x + c
)^4 - 2*a^3*d*cos(d*x + c)^3 - 5*a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + 4*a^3*d)*sin(d*x + c))

Sympy [F]

\[ \int \frac {\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=- \frac {A \left (\int \left (- \frac {\csc ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\right )\, dx + \int \frac {\sin {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx\right )}{a^{3}} \]

[In]

integrate(csc(d*x+c)**3*(A-A*sin(d*x+c))/(a+a*sin(d*x+c))**3,x)

[Out]

-A*(Integral(-csc(c + d*x)**3/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x) + Integral(sin(c
+ d*x)*csc(c + d*x)**3/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x))/a**3

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (128) = 256\).

Time = 0.27 (sec) , antiderivative size = 622, normalized size of antiderivative = 4.51 \[ \int \frac {\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=\frac {12 \, A {\left (\frac {\frac {121 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {410 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {610 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {425 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {125 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 5}{\frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {30 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {5 \, \sin \left (d x + c\right )}{a^{3} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + A {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2782 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {9410 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {13645 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {9285 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2580 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 15}{\frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} - \frac {15 \, {\left (\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{a^{3}} + \frac {780 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{120 \, d} \]

[In]

integrate(csc(d*x+c)^3*(A-A*sin(d*x+c))/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/120*(12*A*((121*sin(d*x + c)/(cos(d*x + c) + 1) + 410*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 610*sin(d*x + c)
^3/(cos(d*x + c) + 1)^3 + 425*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 125*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 +
5)/(a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 5*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*a^3*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 + 10*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 5*a^3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 +
a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + 30*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - 5*sin(d*x + c)/(a^3*(
cos(d*x + c) + 1))) + A*((105*sin(d*x + c)/(cos(d*x + c) + 1) + 2782*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 941
0*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 13645*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 9285*sin(d*x + c)^5/(cos(d
*x + c) + 1)^5 + 2580*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 15)/(a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 5*a
^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 10*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*a^3*sin(d*x + c)^5/(c
os(d*x + c) + 1)^5 + 5*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a^3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7) - 15
*(12*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/a^3 + 780*log(sin(d*x + c)/(cos(d*
x + c) + 1))/a^3))/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.30 \[ \int \frac {\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {1140 \, A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {15 \, {\left (114 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {15 \, {\left (A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{6}} + \frac {16 \, {\left (240 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 825 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1165 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 755 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 199 \, A\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \]

[In]

integrate(csc(d*x+c)^3*(A-A*sin(d*x+c))/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/120*(1140*A*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - 15*(114*A*tan(1/2*d*x + 1/2*c)^2 - 16*A*tan(1/2*d*x + 1/2*c
) + A)/(a^3*tan(1/2*d*x + 1/2*c)^2) + 15*(A*a^3*tan(1/2*d*x + 1/2*c)^2 - 16*A*a^3*tan(1/2*d*x + 1/2*c))/a^6 +
16*(240*A*tan(1/2*d*x + 1/2*c)^4 + 825*A*tan(1/2*d*x + 1/2*c)^3 + 1165*A*tan(1/2*d*x + 1/2*c)^2 + 755*A*tan(1/
2*d*x + 1/2*c) + 199*A)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^5))/d

Mupad [B] (verification not implemented)

Time = 16.64 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.09 \[ \int \frac {\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx=\frac {A\,\left (165\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4234\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+14090\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+19780\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+12060\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+1830\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-1050\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-165\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+1140\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+5700\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+11400\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+11400\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+5700\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1140\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-15\right )}{120\,a^3\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \]

[In]

int((A - A*sin(c + d*x))/(sin(c + d*x)^3*(a + a*sin(c + d*x))^3),x)

[Out]

(A*(165*tan(c/2 + (d*x)/2) + 4234*tan(c/2 + (d*x)/2)^2 + 14090*tan(c/2 + (d*x)/2)^3 + 19780*tan(c/2 + (d*x)/2)
^4 + 12060*tan(c/2 + (d*x)/2)^5 + 1830*tan(c/2 + (d*x)/2)^6 - 1050*tan(c/2 + (d*x)/2)^7 - 165*tan(c/2 + (d*x)/
2)^8 + 15*tan(c/2 + (d*x)/2)^9 + 1140*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2)^2 + 5700*log(tan(c/2 + (d*x)/
2))*tan(c/2 + (d*x)/2)^3 + 11400*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2)^4 + 11400*log(tan(c/2 + (d*x)/2))*
tan(c/2 + (d*x)/2)^5 + 5700*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2)^6 + 1140*log(tan(c/2 + (d*x)/2))*tan(c/
2 + (d*x)/2)^7 - 15))/(120*a^3*d*tan(c/2 + (d*x)/2)^2*(tan(c/2 + (d*x)/2) + 1)^5)