\(\int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx\) [234]

Optimal result

Integrand size = 32, antiderivative size = 130 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {3 a^3 A \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {2 a^3 A \cot ^5(c+d x)}{5 d}+\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3045, 3853, 3855, 3852} \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {3 a^3 A \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^3 A \cot ^5(c+d x)}{5 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d} \]

[In]

Int[Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]

[Out]

(3*a^3*A*ArcTanh[Cos[c + d*x]])/(16*d) - (2*a^3*A*Cot[c + d*x]^3)/(3*d) - (2*a^3*A*Cot[c + d*x]^5)/(5*d) + (3*
a^3*A*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5*a^3*A*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*A*Cot[c + d*x]*C
sc[c + d*x]^5)/(6*d)

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 (-a^3 A \csc ^3(c+d x)-2 a^3 A \csc ^4(c+d x)+2 a^3 A \csc ^6(c+d x)+a^3 A \csc ^7(c+d x)\right ) \, dx \\ & = -\left (\left (a^3 A\right ) \int \csc ^3(c+d x) \, dx\right )+\left (a^3 A\right ) \int \csc ^7(c+d x) \, dx-\left (2 a^3 A\right ) \int \csc ^4(c+d x) \, dx+\left (2 a^3 A\right ) \int \csc ^6(c+d x) \, dx \\ & = \frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {1}{2} \left (a^3 A\right ) \int \csc (c+d x) \, dx+\frac {1}{6} \left (5 a^3 A\right ) \int \csc ^5(c+d x) \, dx+\frac {\left (2 a^3 A\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (2 a^3 A\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {a^3 A \text {arctanh}(\cos (c+d x))}{2 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {2 a^3 A \cot ^5(c+d x)}{5 d}+\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{8} \left (5 a^3 A\right ) \int \csc ^3(c+d x) \, dx \\ & = \frac {a^3 A \text {arctanh}(\cos (c+d x))}{2 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {2 a^3 A \cot ^5(c+d x)}{5 d}+\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{16} \left (5 a^3 A\right ) \int \csc (c+d x) \, dx \\ & = \frac {3 a^3 A \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {2 a^3 A \cot ^5(c+d x)}{5 d}+\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(306\) vs. \(2(130)=260\).

Time = 0.29 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.35 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=a^3 A \left (\frac {2 \cot \left (\frac {1}{2} (c+d x)\right )}{15 d}+\frac {3 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{240 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{80 d}-\frac {\csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {3 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {2 \tan \left (\frac {1}{2} (c+d x)\right )}{15 d}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{240 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{80 d}\right ) \]

[In]

Integrate[Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]

[Out]

a^3*A*((2*Cot[(c + d*x)/2])/(15*d) + (3*Csc[(c + d*x)/2]^2)/(64*d) + (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24
0*d) - Csc[(c + d*x)/2]^4/(64*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(80*d) - Csc[(c + d*x)/2]^6/(384*d) +
(3*Log[Cos[(c + d*x)/2]])/(16*d) - (3*Log[Sin[(c + d*x)/2]])/(16*d) - (3*Sec[(c + d*x)/2]^2)/(64*d) + Sec[(c
+ d*x)/2]^4/(64*d) + Sec[(c + d*x)/2]^6/(384*d) - (2*Tan[(c + d*x)/2])/(15*d) - (Sec[(c + d*x)/2]^2*Tan[(c + d
*x)/2])/(240*d) + (Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(80*d))

Maple [A] (verified)

Time = 1.89 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(-\frac {25 a^{3} \left (\frac {1536 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{25}+\left (\sec ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )-\frac {13 \cos \left (3 d x +3 c \right )}{150}-\frac {3 \cos \left (5 d x +5 c \right )}{50}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {256 \cos \left (d x +c \right )}{75}+\frac {64 \cos \left (3 d x +3 c \right )}{75}-\frac {64 \cos \left (5 d x +5 c \right )}{375}\right ) \left (\csc ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) A}{8192 d}\)	\(125\)
derivativedivides	\(\frac {-A \,a^{3} \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-2 A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )+2 A \,a^{3} \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (d x +c \right )\right )}{15}\right ) \cot \left (d x +c \right )+A \,a^{3} \left (\left (-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(163\)
default	\(\frac {-A \,a^{3} \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-2 A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )+2 A \,a^{3} \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (d x +c \right )\right )}{15}\right ) \cot \left (d x +c \right )+A \,a^{3} \left (\left (-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(163\)
risch	\(-\frac {A \,a^{3} \left (45 \,{\mathrm e}^{11 i \left (d x +c \right )}+65 \,{\mathrm e}^{9 i \left (d x +c \right )}-750 \,{\mathrm e}^{7 i \left (d x +c \right )}+960 i {\mathrm e}^{8 i \left (d x +c \right )}-750 \,{\mathrm e}^{5 i \left (d x +c \right )}-640 i {\mathrm e}^{6 i \left (d x +c \right )}+65 \,{\mathrm e}^{3 i \left (d x +c \right )}+45 \,{\mathrm e}^{i \left (d x +c \right )}-384 i {\mathrm e}^{2 i \left (d x +c \right )}+64 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {3 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {3 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}\)	\(171\)

[In]

int(csc(d*x+c)^7*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-25/8192*a^3*(1536/25*ln(tan(1/2*d*x+1/2*c))+sec(1/2*d*x+1/2*c)^5*(sec(1/2*d*x+1/2*c)*(cos(d*x+c)-13/150*cos(3
*d*x+3*c)-3/50*cos(5*d*x+5*c))*csc(1/2*d*x+1/2*c)+256/75*cos(d*x+c)+64/75*cos(3*d*x+3*c)-64/375*cos(5*d*x+5*c)
)*csc(1/2*d*x+1/2*c)^5)*A/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (118) = 236\).

Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.85 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {90 \, A a^{3} \cos \left (d x + c\right )^{5} - 80 \, A a^{3} \cos \left (d x + c\right )^{3} - 90 \, A a^{3} \cos \left (d x + c\right ) - 45 \, {\left (A a^{3} \cos \left (d x + c\right )^{6} - 3 \, A a^{3} \cos \left (d x + c\right )^{4} + 3 \, A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 45 \, {\left (A a^{3} \cos \left (d x + c\right )^{6} - 3 \, A a^{3} \cos \left (d x + c\right )^{4} + 3 \, A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 64 \, {\left (2 \, A a^{3} \cos \left (d x + c\right )^{5} - 5 \, A a^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(csc(d*x+c)^7*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/480*(90*A*a^3*cos(d*x + c)^5 - 80*A*a^3*cos(d*x + c)^3 - 90*A*a^3*cos(d*x + c) - 45*(A*a^3*cos(d*x + c)^6 -
3*A*a^3*cos(d*x + c)^4 + 3*A*a^3*cos(d*x + c)^2 - A*a^3)*log(1/2*cos(d*x + c) + 1/2) + 45*(A*a^3*cos(d*x + c)
^6 - 3*A*a^3*cos(d*x + c)^4 + 3*A*a^3*cos(d*x + c)^2 - A*a^3)*log(-1/2*cos(d*x + c) + 1/2) + 64*(2*A*a^3*cos(d
*x + c)^5 - 5*A*a^3*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2
- d)

Sympy [F(-1)]

Timed out. \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**7*(a+a*sin(d*x+c))**3*(A-A*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.59 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {5 \, A a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 120 \, A a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {320 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a^{3}}{\tan \left (d x + c\right )^{3}} - \frac {64 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} A a^{3}}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]

[In]

integrate(csc(d*x+c)^7*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/480*(5*A*a^3*(2*(15*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 33*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4
+ 3*cos(d*x + c)^2 - 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) - 120*A*a^3*(2*cos(d*x + c)/(c
os(d*x + c)^2 - 1) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) + 320*(3*tan(d*x + c)^2 + 1)*A*a^3/tan(d*x
+ c)^3 - 64*(15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 + 3)*A*a^3/tan(d*x + c)^5)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (118) = 236\).

Time = 0.32 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.86 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {5 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 360 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 240 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {882 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 240 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(csc(d*x+c)^7*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1920*(5*A*a^3*tan(1/2*d*x + 1/2*c)^6 + 24*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 45*A*a^3*tan(1/2*d*x + 1/2*c)^4 + 4
0*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 15*A*a^3*tan(1/2*d*x + 1/2*c)^2 - 360*A*a^3*log(abs(tan(1/2*d*x + 1/2*c))) -
240*A*a^3*tan(1/2*d*x + 1/2*c) + (882*A*a^3*tan(1/2*d*x + 1/2*c)^6 + 240*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 15*A*a
^3*tan(1/2*d*x + 1/2*c)^4 - 40*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 45*A*a^3*tan(1/2*d*x + 1/2*c)^2 - 24*A*a^3*tan(1
/2*d*x + 1/2*c) - 5*A*a^3)/tan(1/2*d*x + 1/2*c)^6)/d

Mupad [B] (verification not implemented)

Time = 12.87 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.62 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {A\,a^3\,\left (5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-24\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+360\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}{1920\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]

[In]

int(((A - A*sin(c + d*x))*(a + a*sin(c + d*x))^3)/sin(c + d*x)^7,x)

[Out]

-(A*a^3*(5*cos(c/2 + (d*x)/2)^12 - 5*sin(c/2 + (d*x)/2)^12 - 24*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^11 + 24*
cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2) - 45*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 - 40*cos(c/2 + (d*x)/
2)^3*sin(c/2 + (d*x)/2)^9 + 15*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 + 240*cos(c/2 + (d*x)/2)^5*sin(c/2 +
(d*x)/2)^7 - 240*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 - 15*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 + 40
*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 + 45*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 360*log(sin(c/2 +
(d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6))/(1920*d*cos(c/2 + (d*x)/2)^6*sin(c/2
+ (d*x)/2)^6)