Integrand size = 32, antiderivative size = 130 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \text {arctanh}(\cos (e+f x))}{16 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{16 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f} \]
1/16*a^2*c*arctanh(cos(f*x+e))/f-1/3*a^2*c*cot(f*x+e)^3/f-1/5*a^2*c*cot(f* x+e)^5/f+1/16*a^2*c*cot(f*x+e)*csc(f*x+e)/f+1/24*a^2*c*cot(f*x+e)*csc(f*x+ e)^3/f-1/6*a^2*c*cot(f*x+e)*csc(f*x+e)^5/f
Time = 0.12 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.57 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {2 a^2 c \cot (e+f x)}{15 f}+\frac {a^2 c \csc ^2\left (\frac {1}{2} (e+f x)\right )}{64 f}-\frac {a^2 c \csc ^6\left (\frac {1}{2} (e+f x)\right )}{384 f}+\frac {a^2 c \cot (e+f x) \csc ^2(e+f x)}{15 f}-\frac {a^2 c \cot (e+f x) \csc ^4(e+f x)}{5 f}+\frac {a^2 c \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{16 f}-\frac {a^2 c \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{16 f}-\frac {a^2 c \sec ^2\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {a^2 c \sec ^6\left (\frac {1}{2} (e+f x)\right )}{384 f} \]
(2*a^2*c*Cot[e + f*x])/(15*f) + (a^2*c*Csc[(e + f*x)/2]^2)/(64*f) - (a^2*c *Csc[(e + f*x)/2]^6)/(384*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x]^2)/(15*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^4)/(5*f) + (a^2*c*Log[Cos[(e + f*x)/2]] )/(16*f) - (a^2*c*Log[Sin[(e + f*x)/2]])/(16*f) - (a^2*c*Sec[(e + f*x)/2]^ 2)/(64*f) + (a^2*c*Sec[(e + f*x)/2]^6)/(384*f)
Time = 0.40 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3042, 3445, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \csc ^7(e+f x) (a \sin (e+f x)+a)^2 (c-c \sin (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))}{\sin (e+f x)^7}dx\) |
| \(\Big \downarrow \) 3445 |
| \(\displaystyle \int \left (a^2 c \csc ^7(e+f x)+a^2 c \csc ^6(e+f x)-a^2 c \csc ^5(e+f x)-a^2 c \csc ^4(e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 c \text {arctanh}(\cos (e+f x))}{16 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{16 f}\) |
(a^2*c*ArcTanh[Cos[e + f*x]])/(16*f) - (a^2*c*Cot[e + f*x]^3)/(3*f) - (a^2 *c*Cot[e + f*x]^5)/(5*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(16*f) + (a^2 *c*Cot[e + f*x]*Csc[e + f*x]^3)/(24*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^ 5)/(6*f)
3.1.11.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[si n[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; FreeQ[{ a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ [m] && IntegerQ[n]
Time = 1.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.96
| method | result | size |
| parallelrisch | \(-\frac {19 c \left (\frac {512 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{19}+\left (\sec \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\cos \left (f x +e \right )+\frac {17 \cos \left (3 f x +3 e \right )}{114}-\frac {\cos \left (5 f x +5 e \right )}{38}\right ) \csc \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {128 \cos \left (f x +e \right )}{57}+\frac {32 \cos \left (3 f x +3 e \right )}{57}-\frac {32 \cos \left (5 f x +5 e \right )}{285}\right ) \left (\sec ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\csc ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right ) a^{2}}{8192 f}\) | \(125\) |
| risch | \(-\frac {a^{2} c \left (15 \,{\mathrm e}^{11 i \left (f x +e \right )}-85 \,{\mathrm e}^{9 i \left (f x +e \right )}-570 \,{\mathrm e}^{7 i \left (f x +e \right )}+480 i {\mathrm e}^{8 i \left (f x +e \right )}-570 \,{\mathrm e}^{5 i \left (f x +e \right )}-320 i {\mathrm e}^{6 i \left (f x +e \right )}-85 \,{\mathrm e}^{3 i \left (f x +e \right )}+15 \,{\mathrm e}^{i \left (f x +e \right )}-192 i {\mathrm e}^{2 i \left (f x +e \right )}+32 i\right )}{120 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{6}}-\frac {a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{16 f}+\frac {a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{16 f}\) | \(171\) |
| derivativedivides | \(\frac {-a^{2} c \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )-a^{2} c \left (\left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+a^{2} c \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (f x +e \right )\right )}{15}\right ) \cot \left (f x +e \right )+a^{2} c \left (\left (-\frac {\left (\csc ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \csc \left (f x +e \right )}{16}\right ) \cot \left (f x +e \right )+\frac {5 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{16}\right )}{f}\) | \(174\) |
| default | \(\frac {-a^{2} c \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )-a^{2} c \left (\left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+a^{2} c \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (f x +e \right )\right )}{15}\right ) \cot \left (f x +e \right )+a^{2} c \left (\left (-\frac {\left (\csc ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \csc \left (f x +e \right )}{16}\right ) \cot \left (f x +e \right )+\frac {5 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{16}\right )}{f}\) | \(174\) |
-19/8192*c*(512/19*ln(tan(1/2*f*x+1/2*e))+(sec(1/2*f*x+1/2*e)*(cos(f*x+e)+ 17/114*cos(3*f*x+3*e)-1/38*cos(5*f*x+5*e))*csc(1/2*f*x+1/2*e)+128/57*cos(f *x+e)+32/57*cos(3*f*x+3*e)-32/285*cos(5*f*x+5*e))*sec(1/2*f*x+1/2*e)^5*csc (1/2*f*x+1/2*e)^5)*a^2/f
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (118) = 236\).
Time = 0.27 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.85 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=-\frac {30 \, a^{2} c \cos \left (f x + e\right )^{5} - 80 \, a^{2} c \cos \left (f x + e\right )^{3} - 30 \, a^{2} c \cos \left (f x + e\right ) - 15 \, {\left (a^{2} c \cos \left (f x + e\right )^{6} - 3 \, a^{2} c \cos \left (f x + e\right )^{4} + 3 \, a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 15 \, {\left (a^{2} c \cos \left (f x + e\right )^{6} - 3 \, a^{2} c \cos \left (f x + e\right )^{4} + 3 \, a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 32 \, {\left (2 \, a^{2} c \cos \left (f x + e\right )^{5} - 5 \, a^{2} c \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )}{480 \, {\left (f \cos \left (f x + e\right )^{6} - 3 \, f \cos \left (f x + e\right )^{4} + 3 \, f \cos \left (f x + e\right )^{2} - f\right )}} \]
-1/480*(30*a^2*c*cos(f*x + e)^5 - 80*a^2*c*cos(f*x + e)^3 - 30*a^2*c*cos(f *x + e) - 15*(a^2*c*cos(f*x + e)^6 - 3*a^2*c*cos(f*x + e)^4 + 3*a^2*c*cos( f*x + e)^2 - a^2*c)*log(1/2*cos(f*x + e) + 1/2) + 15*(a^2*c*cos(f*x + e)^6 - 3*a^2*c*cos(f*x + e)^4 + 3*a^2*c*cos(f*x + e)^2 - a^2*c)*log(-1/2*cos(f *x + e) + 1/2) + 32*(2*a^2*c*cos(f*x + e)^5 - 5*a^2*c*cos(f*x + e)^3)*sin( f*x + e))/(f*cos(f*x + e)^6 - 3*f*cos(f*x + e)^4 + 3*f*cos(f*x + e)^2 - f)
Timed out. \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.78 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {5 \, a^{2} c {\left (\frac {2 \, {\left (15 \, \cos \left (f x + e\right )^{5} - 40 \, \cos \left (f x + e\right )^{3} + 33 \, \cos \left (f x + e\right )\right )}}{\cos \left (f x + e\right )^{6} - 3 \, \cos \left (f x + e\right )^{4} + 3 \, \cos \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\cos \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - 30 \, a^{2} c {\left (\frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\cos \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + \frac {160 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{2} c}{\tan \left (f x + e\right )^{3}} - \frac {32 \, {\left (15 \, \tan \left (f x + e\right )^{4} + 10 \, \tan \left (f x + e\right )^{2} + 3\right )} a^{2} c}{\tan \left (f x + e\right )^{5}}}{480 \, f} \]
1/480*(5*a^2*c*(2*(15*cos(f*x + e)^5 - 40*cos(f*x + e)^3 + 33*cos(f*x + e) )/(cos(f*x + e)^6 - 3*cos(f*x + e)^4 + 3*cos(f*x + e)^2 - 1) - 15*log(cos( f*x + e) + 1) + 15*log(cos(f*x + e) - 1)) - 30*a^2*c*(2*(3*cos(f*x + e)^3 - 5*cos(f*x + e))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1) - 3*log(cos(f*x + e) + 1) + 3*log(cos(f*x + e) - 1)) + 160*(3*tan(f*x + e)^2 + 1)*a^2*c/ta n(f*x + e)^3 - 32*(15*tan(f*x + e)^4 + 10*tan(f*x + e)^2 + 3)*a^2*c/tan(f* x + e)^5)/f
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (118) = 236\).
Time = 0.38 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.86 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {5 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 12 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 20 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 120 \, a^{2} c \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - 120 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {294 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 120 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 20 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 5 \, a^{2} c}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}}{1920 \, f} \]
1/1920*(5*a^2*c*tan(1/2*f*x + 1/2*e)^6 + 12*a^2*c*tan(1/2*f*x + 1/2*e)^5 + 15*a^2*c*tan(1/2*f*x + 1/2*e)^4 + 20*a^2*c*tan(1/2*f*x + 1/2*e)^3 - 15*a^ 2*c*tan(1/2*f*x + 1/2*e)^2 - 120*a^2*c*log(abs(tan(1/2*f*x + 1/2*e))) - 12 0*a^2*c*tan(1/2*f*x + 1/2*e) + (294*a^2*c*tan(1/2*f*x + 1/2*e)^6 + 120*a^2 *c*tan(1/2*f*x + 1/2*e)^5 + 15*a^2*c*tan(1/2*f*x + 1/2*e)^4 - 20*a^2*c*tan (1/2*f*x + 1/2*e)^3 - 15*a^2*c*tan(1/2*f*x + 1/2*e)^2 - 12*a^2*c*tan(1/2*f *x + 1/2*e) - 5*a^2*c)/tan(1/2*f*x + 1/2*e)^6)/f
Time = 12.23 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.62 \[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=-\frac {a^2\,c\,\left (5\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-12\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+12\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-20\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+120\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7-120\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+20\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+120\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{1920\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6} \]
-(a^2*c*(5*cos(e/2 + (f*x)/2)^12 - 5*sin(e/2 + (f*x)/2)^12 - 12*cos(e/2 + (f*x)/2)*sin(e/2 + (f*x)/2)^11 + 12*cos(e/2 + (f*x)/2)^11*sin(e/2 + (f*x)/ 2) - 15*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^10 - 20*cos(e/2 + (f*x)/2) ^3*sin(e/2 + (f*x)/2)^9 + 15*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2)^8 + 1 20*cos(e/2 + (f*x)/2)^5*sin(e/2 + (f*x)/2)^7 - 120*cos(e/2 + (f*x)/2)^7*si n(e/2 + (f*x)/2)^5 - 15*cos(e/2 + (f*x)/2)^8*sin(e/2 + (f*x)/2)^4 + 20*cos (e/2 + (f*x)/2)^9*sin(e/2 + (f*x)/2)^3 + 15*cos(e/2 + (f*x)/2)^10*sin(e/2 + (f*x)/2)^2 + 120*cos(e/2 + (f*x)/2)^6*log(sin(e/2 + (f*x)/2)/cos(e/2 + ( f*x)/2))*sin(e/2 + (f*x)/2)^6))/(1920*f*cos(e/2 + (f*x)/2)^6*sin(e/2 + (f* x)/2)^6)