Integrand size = 13, antiderivative size = 75 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=-\frac {(A-B) \cos (x)}{7 (1+\sin (x))^4}-\frac {(3 A+4 B) \cos (x)}{35 (1+\sin (x))^3}-\frac {2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))^2}-\frac {2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))} \] Output:
-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*(3*A+4*B)*cos(x)/(105+105*sin(x))
Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=-\frac {\cos (x) \left (36 A+13 B+13 (3 A+4 B) \sin (x)+8 (3 A+4 B) \sin ^2(x)+(6 A+8 B) \sin ^3(x)\right )}{105 (1+\sin (x))^4} \] Input:
Integrate[(A + B*Sin[x])/(1 + Sin[x])^4,x]
Output:
-1/105*(Cos[x]*(36*A + 13*B + 13*(3*A + 4*B)*Sin[x] + 8*(3*A + 4*B)*Sin[x] ^2 + (6*A + 8*B)*Sin[x]^3))/(1 + Sin[x])^4
Time = 0.39 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, 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.
\(\displaystyle \int \frac {A+B \sin (x)}{(\sin (x)+1)^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \sin (x)}{(\sin (x)+1)^4}dx\) |
\(\Big \downarrow \) 3229 |
\(\displaystyle \frac {1}{7} (3 A+4 B) \int \frac {1}{(\sin (x)+1)^3}dx-\frac {(A-B) \cos (x)}{7 (\sin (x)+1)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} (3 A+4 B) \int \frac {1}{(\sin (x)+1)^3}dx-\frac {(A-B) \cos (x)}{7 (\sin (x)+1)^4}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {1}{7} (3 A+4 B) \left (\frac {2}{5} \int \frac {1}{(\sin (x)+1)^2}dx-\frac {\cos (x)}{5 (\sin (x)+1)^3}\right )-\frac {(A-B) \cos (x)}{7 (\sin (x)+1)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} (3 A+4 B) \left (\frac {2}{5} \int \frac {1}{(\sin (x)+1)^2}dx-\frac {\cos (x)}{5 (\sin (x)+1)^3}\right )-\frac {(A-B) \cos (x)}{7 (\sin (x)+1)^4}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {1}{7} (3 A+4 B) \left (\frac {2}{5} \left (\frac {1}{3} \int \frac {1}{\sin (x)+1}dx-\frac {\cos (x)}{3 (\sin (x)+1)^2}\right )-\frac {\cos (x)}{5 (\sin (x)+1)^3}\right )-\frac {(A-B) \cos (x)}{7 (\sin (x)+1)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} (3 A+4 B) \left (\frac {2}{5} \left (\frac {1}{3} \int \frac {1}{\sin (x)+1}dx-\frac {\cos (x)}{3 (\sin (x)+1)^2}\right )-\frac {\cos (x)}{5 (\sin (x)+1)^3}\right )-\frac {(A-B) \cos (x)}{7 (\sin (x)+1)^4}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle \frac {1}{7} (3 A+4 B) \left (\frac {2}{5} \left (-\frac {\cos (x)}{3 (\sin (x)+1)}-\frac {\cos (x)}{3 (\sin (x)+1)^2}\right )-\frac {\cos (x)}{5 (\sin (x)+1)^3}\right )-\frac {(A-B) \cos (x)}{7 (\sin (x)+1)^4}\) |
Input:
Int[(A + B*Sin[x])/(1 + Sin[x])^4,x]
Output:
-1/7*((A - B)*Cos[x])/(1 + Sin[x])^4 + ((3*A + 4*B)*(-1/5*Cos[x]/(1 + Sin[ x])^3 + (2*(-1/3*Cos[x]/(1 + Sin[x])^2 - Cos[x]/(3*(1 + Sin[x]))))/5))/7
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 = 0.47 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.07
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 | \(\frac {-210 A \tan \left (\frac {x}{2}\right )^{6}+\left (-630 A -210 B \right ) \tan \left (\frac {x}{2}\right )^{5}+\left (-1260 A -350 B \right ) \tan \left (\frac {x}{2}\right )^{4}+\left (-1260 A -560 B \right ) \tan \left (\frac {x}{2}\right )^{3}+\left (-882 A -336 B \right ) \tan \left (\frac {x}{2}\right )^{2}+\left (-294 A -182 B \right ) \tan \left (\frac {x}{2}\right )-72 A -26 B}{105 \left (\tan \left (\frac {x}{2}\right )+1\right )^{7}}\) | \(95\) |
default | \(-\frac {2 \left (36 A -32 B \right )}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {-24 A +24 B}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{6}}-\frac {-6 A +2 B}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {2 A}{\tan \left (\frac {x}{2}\right )+1}-\frac {2 \left (18 A -10 B \right )}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2 \left (8 A -8 B \right )}{7 \left (\tan \left (\frac {x}{2}\right )+1\right )^{7}}-\frac {-32 A +24 B}{2 \left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}\) | \(115\) |
norman | \(\frac {-2 A \tan \left (\frac {x}{2}\right )^{8}+\left (-\frac {318 A}{35}-\frac {362 B}{105}\right ) \tan \left (\frac {x}{2}\right )^{2}+\left (-\frac {102 A}{5}-\frac {98 B}{15}\right ) \tan \left (\frac {x}{2}\right )^{4}+\left (-\frac {74 A}{5}-\frac {106 B}{15}\right ) \tan \left (\frac {x}{2}\right )^{3}+\left (-\frac {14 A}{5}-\frac {26 B}{15}\right ) \tan \left (\frac {x}{2}\right )+\left (-6 A -2 B \right ) \tan \left (\frac {x}{2}\right )^{7}+\left (-14 A -\frac {10 B}{3}\right ) \tan \left (\frac {x}{2}\right )^{6}+\left (-18 A -\frac {22 B}{3}\right ) \tan \left (\frac {x}{2}\right )^{5}-\frac {24 A}{35}-\frac {26 B}{105}}{\left (\tan \left (\frac {x}{2}\right )^{2}+1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )^{7}}\) | \(132\) |
Input:
int((A+B*sin(x))/(1+sin(x))^4,x,method=_RETURNVERBOSE)
Output:
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. \(2 (67) = 134\).
Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.00 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=\frac {2 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{4} + 8 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{3} - 9 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{2} - 15 \, {\left (4 \, A + 3 \, B\right )} \cos \left (x\right ) + {\left (2 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{3} - 6 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{2} - 15 \, {\left (3 \, A + 4 \, B\right )} \cos \left (x\right ) + 15 \, A - 15 \, B\right )} \sin \left (x\right ) - 15 \, A + 15 \, B}{105 \, {\left (\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} + 4 \, \cos \left (x\right )^{2} - 4 \, \cos \left (x\right ) - 8\right )} \sin \left (x\right ) + 4 \, \cos \left (x\right ) + 8\right )}} \] Input:
integrate((A+B*sin(x))/(1+sin(x))^4,x, algorithm="fricas")
Output:
1/105*(2*(3*A + 4*B)*cos(x)^4 + 8*(3*A + 4*B)*cos(x)^3 - 9*(3*A + 4*B)*cos (x)^2 - 15*(4*A + 3*B)*cos(x) + (2*(3*A + 4*B)*cos(x)^3 - 6*(3*A + 4*B)*co s(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)*sin (x) + 4*cos(x) + 8)
Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (73) = 146\).
Time = 3.52 (sec) , antiderivative size = 889, normalized size of antiderivative = 11.85 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(x))/(1+sin(x))**4,x)
Output:
-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. \(2 (67) = 134\).
Time = 0.04 (sec) , antiderivative size = 309, normalized size of antiderivative = 4.12 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx =\text {Too large to display} \] Input:
integrate((A+B*sin(x))/(1+sin(x))^4,x, algorithm="maxima")
Output:
-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 = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.49 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=-\frac {2 \, {\left (105 \, A \tan \left (\frac {1}{2} \, x\right )^{6} + 315 \, A \tan \left (\frac {1}{2} \, x\right )^{5} + 105 \, B \tan \left (\frac {1}{2} \, x\right )^{5} + 630 \, A \tan \left (\frac {1}{2} \, x\right )^{4} + 175 \, B \tan \left (\frac {1}{2} \, x\right )^{4} + 630 \, A \tan \left (\frac {1}{2} \, x\right )^{3} + 280 \, B \tan \left (\frac {1}{2} \, x\right )^{3} + 441 \, A \tan \left (\frac {1}{2} \, x\right )^{2} + 168 \, B \tan \left (\frac {1}{2} \, x\right )^{2} + 147 \, A \tan \left (\frac {1}{2} \, x\right ) + 91 \, B \tan \left (\frac {1}{2} \, x\right ) + 36 \, A + 13 \, B\right )}}{105 \, {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{7}} \] Input:
integrate((A+B*sin(x))/(1+sin(x))^4,x, algorithm="giac")
Output:
-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 = 16.64 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.25 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=-\frac {2\,A\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+\left (6\,A+2\,B\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+\left (12\,A+\frac {10\,B}{3}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+\left (12\,A+\frac {16\,B}{3}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\left (\frac {42\,A}{5}+\frac {16\,B}{5}\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\left (\frac {14\,A}{5}+\frac {26\,B}{15}\right )\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {24\,A}{35}+\frac {26\,B}{105}}{{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^7} \] Input:
int((A + B*sin(x))/(sin(x) + 1)^4,x)
Output:
-((24*A)/35 + (26*B)/105 + 2*A*tan(x/2)^6 + tan(x/2)*((14*A)/5 + (26*B)/15 ) + tan(x/2)^5*(6*A + 2*B) + tan(x/2)^4*(12*A + (10*B)/3) + tan(x/2)^3*(12 *A + (16*B)/3) + tan(x/2)^2*((42*A)/5 + (16*B)/5))/(tan(x/2) + 1)^7
Time = 0.20 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.07 \[ \int \frac {A+B \sin (x)}{(1+\sin (x))^4} \, dx=\frac {-5 \cos \left (x \right ) \sin \left (x \right )^{3} b +6 \cos \left (x \right ) \sin \left (x \right )^{2} a -7 \cos \left (x \right ) \sin \left (x \right )^{2} b +21 \cos \left (x \right ) \sin \left (x \right ) a +13 \cos \left (x \right ) \sin \left (x \right ) b +30 \cos \left (x \right ) a +12 \sin \left (x \right )^{4} a +21 \sin \left (x \right )^{4} b +42 \sin \left (x \right )^{3} a +76 \sin \left (x \right )^{3} b +51 \sin \left (x \right )^{2} a +98 \sin \left (x \right )^{2} b +21 \sin \left (x \right ) a +13 \sin \left (x \right ) b -30 a}{105 \cos \left (x \right ) \sin \left (x \right )^{3}+315 \cos \left (x \right ) \sin \left (x \right )^{2}+315 \cos \left (x \right ) \sin \left (x \right )+105 \cos \left (x \right )-105 \sin \left (x \right )^{4}-420 \sin \left (x \right )^{3}-630 \sin \left (x \right )^{2}-420 \sin \left (x \right )-105} \] Input:
int((A+B*sin(x))/(1+sin(x))^4,x)
Output:
( - 5*cos(x)*sin(x)**3*b + 6*cos(x)*sin(x)**2*a - 7*cos(x)*sin(x)**2*b + 2 1*cos(x)*sin(x)*a + 13*cos(x)*sin(x)*b + 30*cos(x)*a + 12*sin(x)**4*a + 21 *sin(x)**4*b + 42*sin(x)**3*a + 76*sin(x)**3*b + 51*sin(x)**2*a + 98*sin(x )**2*b + 21*sin(x)*a + 13*sin(x)*b - 30*a)/(105*(cos(x)*sin(x)**3 + 3*cos( x)*sin(x)**2 + 3*cos(x)*sin(x) + cos(x) - sin(x)**4 - 4*sin(x)**3 - 6*sin( x)**2 - 4*sin(x) - 1))