Integrand size = 15, antiderivative size = 81 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=\frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}+\frac {(3 A-4 B) \cos (x)}{35 (1-\sin (x))^3}+\frac {2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))^2}+\frac {2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))} \]
1/7*(A+B)*cos(x)/(1-sin(x))^4+1/35*(3*A-4*B)*cos(x)/(1-sin(x))^3+2/105*(3* A-4*B)*cos(x)/(1-sin(x))^2+2/105*(3*A-4*B)*cos(x)/(1-sin(x))
Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.67 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=\frac {\cos (x) \left (36 A-13 B+(-39 A+52 B) \sin (x)+8 (3 A-4 B) \sin ^2(x)+(-6 A+8 B) \sin ^3(x)\right )}{105 (-1+\sin (x))^4} \]
(Cos[x]*(36*A - 13*B + (-39*A + 52*B)*Sin[x] + 8*(3*A - 4*B)*Sin[x]^2 + (- 6*A + 8*B)*Sin[x]^3))/(105*(-1 + Sin[x])^4)
Time = 0.40 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 3229, 3042, 3129, 3042, 3129, 3042, 3127}
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.
| \(\displaystyle \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin (x)}{(1-\sin (x))^4}dx\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle \frac {1}{7} (3 A-4 B) \int \frac {1}{(1-\sin (x))^3}dx+\frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} (3 A-4 B) \int \frac {1}{(1-\sin (x))^3}dx+\frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {1}{7} (3 A-4 B) \left (\frac {2}{5} \int \frac {1}{(1-\sin (x))^2}dx+\frac {\cos (x)}{5 (1-\sin (x))^3}\right )+\frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} (3 A-4 B) \left (\frac {2}{5} \int \frac {1}{(1-\sin (x))^2}dx+\frac {\cos (x)}{5 (1-\sin (x))^3}\right )+\frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {1}{7} (3 A-4 B) \left (\frac {2}{5} \left (\frac {1}{3} \int \frac {1}{1-\sin (x)}dx+\frac {\cos (x)}{3 (1-\sin (x))^2}\right )+\frac {\cos (x)}{5 (1-\sin (x))^3}\right )+\frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} (3 A-4 B) \left (\frac {2}{5} \left (\frac {1}{3} \int \frac {1}{1-\sin (x)}dx+\frac {\cos (x)}{3 (1-\sin (x))^2}\right )+\frac {\cos (x)}{5 (1-\sin (x))^3}\right )+\frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle \frac {(A+B) \cos (x)}{7 (1-\sin (x))^4}+\frac {1}{7} (3 A-4 B) \left (\frac {\cos (x)}{5 (1-\sin (x))^3}+\frac {2}{5} \left (\frac {\cos (x)}{3 (1-\sin (x))}+\frac {\cos (x)}{3 (1-\sin (x))^2}\right )\right )\) |
((3*A - 4*B)*((2*(Cos[x]/(3*(1 - Sin[x])^2) + Cos[x]/(3*(1 - Sin[x]))))/5 + Cos[x]/(5*(1 - Sin[x])^3)))/7 + ((A + B)*Cos[x])/(7*(1 - Sin[x])^4)
3.5.81.3.1 Defintions of rubi rules used
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]
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] + Simp[(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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99
| method | result | size |
| risch | \(\frac {\frac {8 B \,{\mathrm e}^{4 i x}}{3}-\frac {8 i B \,{\mathrm e}^{3 i x}}{3}+\frac {12 A \,{\mathrm e}^{2 i x}}{5}-\frac {4 i A \,{\mathrm e}^{i x}}{5}+4 i A \,{\mathrm e}^{3 i x}-\frac {16 B \,{\mathrm e}^{2 i x}}{5}+\frac {16 i B \,{\mathrm e}^{i x}}{15}-\frac {4 A}{35}+\frac {16 B}{105}}{\left ({\mathrm e}^{i x}-i\right )^{7}}\) | \(80\) |
| parallelrisch | \(\frac {-210 A \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\left (630 A -210 B \right ) \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+\left (-1260 A +350 B \right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (1260 A -560 B \right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (-882 A +336 B \right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\left (294 A -182 B \right ) \tan \left (\frac {x}{2}\right )-72 A +26 B}{105 \left (\tan \left (\frac {x}{2}\right )-1\right )^{7}}\) | \(95\) |
| default | \(-\frac {32 A +24 B}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {2 \left (36 A +32 B \right )}{5 \left (\tan \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {2 A}{\tan \left (\frac {x}{2}\right )-1}-\frac {2 \left (8 A +8 B \right )}{7 \left (\tan \left (\frac {x}{2}\right )-1\right )^{7}}-\frac {2 \left (18 A +10 B \right )}{3 \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {6 A +2 B}{\left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {24 A +24 B}{3 \left (\tan \left (\frac {x}{2}\right )-1\right )^{6}}\) | \(115\) |
| norman | \(\frac {-2 A \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+\left (-\frac {318 A}{35}+\frac {362 B}{105}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\left (-\frac {102 A}{5}+\frac {98 B}{15}\right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (\frac {14 A}{5}-\frac {26 B}{15}\right ) \tan \left (\frac {x}{2}\right )+\left (\frac {74 A}{5}-\frac {106 B}{15}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (-14 A +\frac {10 B}{3}\right ) \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\left (6 A -2 B \right ) \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+\left (18 A -\frac {22 B}{3}\right ) \left (\tan ^{5}\left (\frac {x}{2}\right )\right )-\frac {24 A}{35}+\frac {26 B}{105}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right )^{7}}\) | \(132\) |
4/105*(70*B*exp(4*I*x)-70*I*B*exp(3*I*x)+63*A*exp(2*I*x)-21*I*A*exp(I*x)+1 05*I*A*exp(3*I*x)-84*B*exp(2*I*x)+28*I*B*exp(I*x)-3*A+4*B)/(exp(I*x)-I)^7
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (65) = 130\).
Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.85 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=-\frac {2 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right )^{4} + 8 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right )^{3} - 9 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right )^{2} - 15 \, {\left (4 \, A - 3 \, B\right )} \cos \left (x\right ) - {\left (2 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right )^{3} - 6 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right )^{2} - 15 \, {\left (3 \, A - 4 \, B\right )} \cos \left (x\right ) + 15 \, A + 15 \, B\right )} \sin \left (x\right ) - 15 \, A - 15 \, B}{105 \, {\left (\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{3} + 4 \, \cos \left (x\right )^{2} - 4 \, \cos \left (x\right ) - 8\right )} \sin \left (x\right ) + 4 \, \cos \left (x\right ) + 8\right )}} \]
-1/105*(2*(3*A - 4*B)*cos(x)^4 + 8*(3*A - 4*B)*cos(x)^3 - 9*(3*A - 4*B)*co s(x)^2 - 15*(4*A - 3*B)*cos(x) - (2*(3*A - 4*B)*cos(x)^3 - 6*(3*A - 4*B)*c os(x)^2 - 15*(3*A - 4*B)*cos(x) + 15*A + 15*B)*sin(x) - 15*A - 15*B)/(cos( x)^4 - 3*cos(x)^3 - 8*cos(x)^2 + (cos(x)^3 + 4*cos(x)^2 - 4*cos(x) - 8)*si n(x) + 4*cos(x) + 8)
Leaf count of result is larger than twice the leaf count of optimal. 887 vs. \(2 (75) = 150\).
Time = 2.69 (sec) , antiderivative size = 887, normalized size of antiderivative = 10.95 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=\text {Too large to display} \]
-210*A*tan(x/2)**6/(105*tan(x/2)**7 - 735*tan(x/2)**6 + 2205*tan(x/2)**5 - 3675*tan(x/2)**4 + 3675*tan(x/2)**3 - 2205*tan(x/2)**2 + 735*tan(x/2) - 1 05) + 630*A*tan(x/2)**5/(105*tan(x/2)**7 - 735*tan(x/2)**6 + 2205*tan(x/2) **5 - 3675*tan(x/2)**4 + 3675*tan(x/2)**3 - 2205*tan(x/2)**2 + 735*tan(x/2 ) - 105) - 1260*A*tan(x/2)**4/(105*tan(x/2)**7 - 735*tan(x/2)**6 + 2205*ta n(x/2)**5 - 3675*tan(x/2)**4 + 3675*tan(x/2)**3 - 2205*tan(x/2)**2 + 735*t an(x/2) - 105) + 1260*A*tan(x/2)**3/(105*tan(x/2)**7 - 735*tan(x/2)**6 + 2 205*tan(x/2)**5 - 3675*tan(x/2)**4 + 3675*tan(x/2)**3 - 2205*tan(x/2)**2 + 735*tan(x/2) - 105) - 882*A*tan(x/2)**2/(105*tan(x/2)**7 - 735*tan(x/2)** 6 + 2205*tan(x/2)**5 - 3675*tan(x/2)**4 + 3675*tan(x/2)**3 - 2205*tan(x/2) **2 + 735*tan(x/2) - 105) + 294*A*tan(x/2)/(105*tan(x/2)**7 - 735*tan(x/2) **6 + 2205*tan(x/2)**5 - 3675*tan(x/2)**4 + 3675*tan(x/2)**3 - 2205*tan(x/ 2)**2 + 735*tan(x/2) - 105) - 72*A/(105*tan(x/2)**7 - 735*tan(x/2)**6 + 22 05*tan(x/2)**5 - 3675*tan(x/2)**4 + 3675*tan(x/2)**3 - 2205*tan(x/2)**2 + 735*tan(x/2) - 105) - 210*B*tan(x/2)**5/(105*tan(x/2)**7 - 735*tan(x/2)**6 + 2205*tan(x/2)**5 - 3675*tan(x/2)**4 + 3675*tan(x/2)**3 - 2205*tan(x/2)* *2 + 735*tan(x/2) - 105) + 350*B*tan(x/2)**4/(105*tan(x/2)**7 - 735*tan(x/ 2)**6 + 2205*tan(x/2)**5 - 3675*tan(x/2)**4 + 3675*tan(x/2)**3 - 2205*tan( x/2)**2 + 735*tan(x/2) - 105) - 560*B*tan(x/2)**3/(105*tan(x/2)**7 - 735*t an(x/2)**6 + 2205*tan(x/2)**5 - 3675*tan(x/2)**4 + 3675*tan(x/2)**3 - 2...
Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (65) = 130\).
Time = 0.20 (sec) , antiderivative size = 309, normalized size of antiderivative = 3.81 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=-\frac {2 \, B {\left (\frac {91 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {168 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {280 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {175 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - 13\right )}}{105 \, {\left (\frac {7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {21 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {35 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {21 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {7 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {\sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - 1\right )}} + \frac {2 \, A {\left (\frac {49 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {147 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {210 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {210 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {35 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - 12\right )}}{35 \, {\left (\frac {7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {21 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {35 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {21 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {7 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {\sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - 1\right )}} \]
-2/105*B*(91*sin(x)/(cos(x) + 1) - 168*sin(x)^2/(cos(x) + 1)^2 + 280*sin(x )^3/(cos(x) + 1)^3 - 175*sin(x)^4/(cos(x) + 1)^4 + 105*sin(x)^5/(cos(x) + 1)^5 - 13)/(7*sin(x)/(cos(x) + 1) - 21*sin(x)^2/(cos(x) + 1)^2 + 35*sin(x) ^3/(cos(x) + 1)^3 - 35*sin(x)^4/(cos(x) + 1)^4 + 21*sin(x)^5/(cos(x) + 1)^ 5 - 7*sin(x)^6/(cos(x) + 1)^6 + sin(x)^7/(cos(x) + 1)^7 - 1) + 2/35*A*(49* sin(x)/(cos(x) + 1) - 147*sin(x)^2/(cos(x) + 1)^2 + 210*sin(x)^3/(cos(x) + 1)^3 - 210*sin(x)^4/(cos(x) + 1)^4 + 105*sin(x)^5/(cos(x) + 1)^5 - 35*sin (x)^6/(cos(x) + 1)^6 - 12)/(7*sin(x)/(cos(x) + 1) - 21*sin(x)^2/(cos(x) + 1)^2 + 35*sin(x)^3/(cos(x) + 1)^3 - 35*sin(x)^4/(cos(x) + 1)^4 + 21*sin(x) ^5/(cos(x) + 1)^5 - 7*sin(x)^6/(cos(x) + 1)^6 + sin(x)^7/(cos(x) + 1)^7 - 1)
Time = 0.45 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.38 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=-\frac {2 \, {\left (105 \, A \tan \left (\frac {1}{2} \, x\right )^{6} - 315 \, A \tan \left (\frac {1}{2} \, x\right )^{5} + 105 \, B \tan \left (\frac {1}{2} \, x\right )^{5} + 630 \, A \tan \left (\frac {1}{2} \, x\right )^{4} - 175 \, B \tan \left (\frac {1}{2} \, x\right )^{4} - 630 \, A \tan \left (\frac {1}{2} \, x\right )^{3} + 280 \, B \tan \left (\frac {1}{2} \, x\right )^{3} + 441 \, A \tan \left (\frac {1}{2} \, x\right )^{2} - 168 \, B \tan \left (\frac {1}{2} \, x\right )^{2} - 147 \, A \tan \left (\frac {1}{2} \, x\right ) + 91 \, B \tan \left (\frac {1}{2} \, x\right ) + 36 \, A - 13 \, B\right )}}{105 \, {\left (\tan \left (\frac {1}{2} \, x\right ) - 1\right )}^{7}} \]
-2/105*(105*A*tan(1/2*x)^6 - 315*A*tan(1/2*x)^5 + 105*B*tan(1/2*x)^5 + 630 *A*tan(1/2*x)^4 - 175*B*tan(1/2*x)^4 - 630*A*tan(1/2*x)^3 + 280*B*tan(1/2* x)^3 + 441*A*tan(1/2*x)^2 - 168*B*tan(1/2*x)^2 - 147*A*tan(1/2*x) + 91*B*t an(1/2*x) + 36*A - 13*B)/(tan(1/2*x) - 1)^7
Time = 6.73 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20 \[ \int \frac {A+B \sin (x)}{(1-\sin (x))^4} \, dx=-\frac {2\,A\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+\left (2\,B-6\,A\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+\left (12\,A-\frac {10\,B}{3}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+\left (\frac {16\,B}{3}-12\,A\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\left (\frac {42\,A}{5}-\frac {16\,B}{5}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\left (\frac {26\,B}{15}-\frac {14\,A}{5}\right )\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {24\,A}{35}-\frac {26\,B}{105}}{{\left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )}^7} \]