Integrand size = 38, antiderivative size = 275 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {64 c^3 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{f (5+2 m) (7+2 m) \left (3+8 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}-\frac {16 c^2 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (7+2 m) \left (15+16 m+4 m^2\right )}-\frac {2 c (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)} \] Output:
-64*c^3*(B*(5-2*m)-A*(7+2*m))*cos(f*x+e)*(a+a*sin(f*x+e))^m/f/(5+2*m)/(7+2 *m)/(4*m^2+8*m+3)/(c-c*sin(f*x+e))^(1/2)-16*c^2*(B*(5-2*m)-A*(7+2*m))*cos( f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1/2)/f/(7+2*m)/(4*m^2+16*m+15) -2*c*(B*(5-2*m)-A*(7+2*m))*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^ (3/2)/f/(5+2*m)/(7+2*m)-2*B*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e)) ^(5/2)/f/(7+2*m)
Result contains complex when optimal does not.
Time = 14.94 (sec) , antiderivative size = 667, normalized size of antiderivative = 2.43 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {(a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^{5/2} \left (\frac {\left (2100 A-1575 B+1272 A m-110 B m+304 A m^2-68 B m^2+32 A m^3-8 B m^3\right ) \left (\left (\frac {1}{8}+\frac {i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )+\left (\frac {1}{8}-\frac {i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{(1+2 m) (3+2 m) (5+2 m) (7+2 m)}+\frac {\left (2100 A-1575 B+1272 A m-110 B m+304 A m^2-68 B m^2+32 A m^3-8 B m^3\right ) \left (\left (\frac {1}{8}-\frac {i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )+\left (\frac {1}{8}+\frac {i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{(1+2 m) (3+2 m) (5+2 m) (7+2 m)}+\frac {\left (350 A-385 B+184 A m-104 B m+24 A m^2-12 B m^2\right ) \left (\left (\frac {1}{8}-\frac {i}{8}\right ) \cos \left (\frac {3}{2} (e+f x)\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{(3+2 m) (5+2 m) (7+2 m)}+\frac {\left (350 A-385 B+184 A m-104 B m+24 A m^2-12 B m^2\right ) \left (\left (\frac {1}{8}+\frac {i}{8}\right ) \cos \left (\frac {3}{2} (e+f x)\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{(3+2 m) (5+2 m) (7+2 m)}+\frac {(14 A-35 B+4 A m-6 B m) \left (\left (-\frac {1}{8}+\frac {i}{8}\right ) \cos \left (\frac {5}{2} (e+f x)\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{(5+2 m) (7+2 m)}+\frac {(14 A-35 B+4 A m-6 B m) \left (\left (-\frac {1}{8}-\frac {i}{8}\right ) \cos \left (\frac {5}{2} (e+f x)\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{(5+2 m) (7+2 m)}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) B \cos \left (\frac {7}{2} (e+f x)\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) B \sin \left (\frac {7}{2} (e+f x)\right )}{7+2 m}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) B \cos \left (\frac {7}{2} (e+f x)\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) B \sin \left (\frac {7}{2} (e+f x)\right )}{7+2 m}\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \] Input:
Integrate[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]) ^(5/2),x]
Output:
((a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(5/2)*(((2100*A - 1575*B + 1272*A*m - 110*B*m + 304*A*m^2 - 68*B*m^2 + 32*A*m^3 - 8*B*m^3)*((1/8 + I/ 8)*Cos[(e + f*x)/2] + (1/8 - I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)* (5 + 2*m)*(7 + 2*m)) + ((2100*A - 1575*B + 1272*A*m - 110*B*m + 304*A*m^2 - 68*B*m^2 + 32*A*m^3 - 8*B*m^3)*((1/8 - I/8)*Cos[(e + f*x)/2] + (1/8 + I/ 8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((350*A - 385*B + 184*A*m - 104*B*m + 24*A*m^2 - 12*B*m^2)*((1/8 - I/8)*Cos[(3*(e + f*x))/2] - (1/8 + I/8)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((350*A - 385*B + 184*A*m - 104*B*m + 24*A*m^2 - 12*B*m^2)*((1/8 + I/8)*Cos[(3*(e + f*x))/2] - (1/8 - I/8)*Sin[(3*(e + f*x))/2]))/((3 + 2*m) *(5 + 2*m)*(7 + 2*m)) + ((14*A - 35*B + 4*A*m - 6*B*m)*((-1/8 + I/8)*Cos[( 5*(e + f*x))/2] - (1/8 + I/8)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)) + ((14*A - 35*B + 4*A*m - 6*B*m)*((-1/8 - I/8)*Cos[(5*(e + f*x))/2] - (1/ 8 - I/8)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)) + ((1/8 - I/8)*B*Cos [(7*(e + f*x))/2] - (1/8 + I/8)*B*Sin[(7*(e + f*x))/2])/(7 + 2*m) + ((1/8 + I/8)*B*Cos[(7*(e + f*x))/2] - (1/8 - I/8)*B*Sin[(7*(e + f*x))/2])/(7 + 2 *m)))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5)
Time = 1.02 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.84, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3452, 3042, 3219, 3042, 3219, 3042, 3217}
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.
| \(\displaystyle \int (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m (A+B \sin (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3452 |
| \(\displaystyle -\frac {(B (5-2 m)-A (2 m+7)) \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2}dx}{2 m+7}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(B (5-2 m)-A (2 m+7)) \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2}dx}{2 m+7}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle -\frac {(B (5-2 m)-A (2 m+7)) \left (\frac {8 c \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2}dx}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(B (5-2 m)-A (2 m+7)) \left (\frac {8 c \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2}dx}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle -\frac {(B (5-2 m)-A (2 m+7)) \left (\frac {8 c \left (\frac {4 c \int (\sin (e+f x) a+a)^m \sqrt {c-c \sin (e+f x)}dx}{2 m+3}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(B (5-2 m)-A (2 m+7)) \left (\frac {8 c \left (\frac {4 c \int (\sin (e+f x) a+a)^m \sqrt {c-c \sin (e+f x)}dx}{2 m+3}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}\) |
| \(\Big \downarrow \) 3217 |
| \(\displaystyle -\frac {(B (5-2 m)-A (2 m+7)) \left (\frac {8 c \left (\frac {8 c^2 \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) (2 m+3) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}\) |
Input:
Int[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2) ,x]
Output:
(-2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*( 7 + 2*m)) - ((B*(5 - 2*m) - A*(7 + 2*m))*((2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)) + (8*c*((8*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x ]]) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/( f*(3 + 2*m))))/(5 + 2*m)))/(7 + 2*m)
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f _.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[a*((2*m - 1)/(m + n )) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I GtQ[m - 1/2, 0] && !LtQ[n, -1] && !(IGtQ[n - 1/2, 0] && LtQ[n, m]) && !( ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*(m + n + 1))), x] - Simp[(B*c*(m - n) - A*d*(m + n + 1))/(d*(m + n + 1)) Int[( a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 1, 0]
\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}d x\]
Input:
int((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x)
Output:
int((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (264) = 528\).
Time = 0.13 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.04 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {2 \, {\left ({\left (8 \, B c^{2} m^{3} + 36 \, B c^{2} m^{2} + 46 \, B c^{2} m + 15 \, B c^{2}\right )} \cos \left (f x + e\right )^{4} + 64 \, {\left (A + B\right )} c^{2} m - {\left (8 \, {\left (A - 2 \, B\right )} c^{2} m^{3} + 4 \, {\left (11 \, A - 28 \, B\right )} c^{2} m^{2} + 2 \, {\left (31 \, A - 86 \, B\right )} c^{2} m + 3 \, {\left (7 \, A - 20 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{3} + 32 \, {\left (7 \, A - 5 \, B\right )} c^{2} + {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 4 \, {\left (19 \, A - 11 \, B\right )} c^{2} m^{2} + 190 \, {\left (A - B\right )} c^{2} m + {\left (77 \, A - 85 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 60 \, {\left (A - B\right )} c^{2} m^{2} + 2 \, {\left (79 \, A - 63 \, B\right )} c^{2} m + {\left (161 \, A - 145 \, B\right )} c^{2}\right )} \cos \left (f x + e\right ) + {\left (64 \, {\left (A + B\right )} c^{2} m - {\left (8 \, B c^{2} m^{3} + 36 \, B c^{2} m^{2} + 46 \, B c^{2} m + 15 \, B c^{2}\right )} \cos \left (f x + e\right )^{3} + 32 \, {\left (7 \, A - 5 \, B\right )} c^{2} - {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 4 \, {\left (11 \, A - 19 \, B\right )} c^{2} m^{2} + 2 \, {\left (31 \, A - 63 \, B\right )} c^{2} m + 3 \, {\left (7 \, A - 15 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 60 \, {\left (A - B\right )} c^{2} m^{2} + 2 \, {\left (63 \, A - 79 \, B\right )} c^{2} m + {\left (49 \, A - 65 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + {\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \cos \left (f x + e\right ) - {\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \sin \left (f x + e\right ) + 105 \, f} \] Input:
integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, al gorithm="fricas")
Output:
2*((8*B*c^2*m^3 + 36*B*c^2*m^2 + 46*B*c^2*m + 15*B*c^2)*cos(f*x + e)^4 + 6 4*(A + B)*c^2*m - (8*(A - 2*B)*c^2*m^3 + 4*(11*A - 28*B)*c^2*m^2 + 2*(31*A - 86*B)*c^2*m + 3*(7*A - 20*B)*c^2)*cos(f*x + e)^3 + 32*(7*A - 5*B)*c^2 + (8*(A - B)*c^2*m^3 + 4*(19*A - 11*B)*c^2*m^2 + 190*(A - B)*c^2*m + (77*A - 85*B)*c^2)*cos(f*x + e)^2 + 2*(8*(A - B)*c^2*m^3 + 60*(A - B)*c^2*m^2 + 2*(79*A - 63*B)*c^2*m + (161*A - 145*B)*c^2)*cos(f*x + e) + (64*(A + B)*c^ 2*m - (8*B*c^2*m^3 + 36*B*c^2*m^2 + 46*B*c^2*m + 15*B*c^2)*cos(f*x + e)^3 + 32*(7*A - 5*B)*c^2 - (8*(A - B)*c^2*m^3 + 4*(11*A - 19*B)*c^2*m^2 + 2*(3 1*A - 63*B)*c^2*m + 3*(7*A - 15*B)*c^2)*cos(f*x + e)^2 - 2*(8*(A - B)*c^2* m^3 + 60*(A - B)*c^2*m^2 + 2*(63*A - 79*B)*c^2*m + (49*A - 65*B)*c^2)*cos( f*x + e))*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/( 16*f*m^4 + 128*f*m^3 + 344*f*m^2 + 352*f*m + (16*f*m^4 + 128*f*m^3 + 344*f *m^2 + 352*f*m + 105*f)*cos(f*x + e) - (16*f*m^4 + 128*f*m^3 + 344*f*m^2 + 352*f*m + 105*f)*sin(f*x + e) + 105*f)
Timed out. \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(5/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 725 vs. \(2 (264) = 528\).
Time = 0.17 (sec) , antiderivative size = 725, normalized size of antiderivative = 2.64 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, al gorithm="maxima")
Output:
-2*(((4*m^2 + 24*m + 43)*a^m*c^(5/2) - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*si n(f*x + e)/(cos(f*x + e) + 1) + 2*(4*m^2 + 8*m + 35)*a^m*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2*(4*m^2 + 8*m + 35)*a^m*c^(5/2)*sin(f*x + e) ^3/(cos(f*x + e) + 1)^3 - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*sin(f*x + e)^4/ (cos(f*x + e) + 1)^4 + (4*m^2 + 24*m + 43)*a^m*c^(5/2)*sin(f*x + e)^5/(cos (f*x + e) + 1)^5)*A*e^(2*m*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1) - m*lo g(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/((8*m^3 + 36*m^2 + 46*m + 15)* (sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(5/2)) - 2*((4*m^2 + 40*m + 115) *a^m*c^(5/2) - 2*(4*m^3 + 40*m^2 + 115*m)*a^m*c^(5/2)*sin(f*x + e)/(cos(f* x + e) + 1) + 2*(12*m^3 + 76*m^2 + 97*m + 175)*a^m*c^(5/2)*sin(f*x + e)^2/ (cos(f*x + e) + 1)^2 - (16*m^3 + 76*m^2 + 260*m - 175)*a^m*c^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - (16*m^3 + 76*m^2 + 260*m - 175)*a^m*c^(5/2) *sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 2*(12*m^3 + 76*m^2 + 97*m + 175)*a^ m*c^(5/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 2*(4*m^3 + 40*m^2 + 115*m) *a^m*c^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + (4*m^2 + 40*m + 115)*a^ m*c^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)*B*e^(2*m*log(sin(f*x + e)/( cos(f*x + e) + 1) + 1) - m*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/( (16*m^4 + 128*m^3 + 344*m^2 + 352*m + (16*m^4 + 128*m^3 + 344*m^2 + 352*m + 105)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 105)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(5/2)))/f
\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, al gorithm="giac")
Output:
integrate((B*sin(f*x + e) + A)*(-c*sin(f*x + e) + c)^(5/2)*(a*sin(f*x + e) + a)^m, x)
Time = 48.17 (sec) , antiderivative size = 749, normalized size of antiderivative = 2.72 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(5/2) ,x)
Output:
-((c - c*sin(e + f*x))^(1/2)*((B*c^2*(a + a*sin(e + f*x))^m*(m*46i + m^2*3 6i + m^3*8i + 15i))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) - (c^ 2*exp(e*3i + f*x*3i)*(a + a*sin(e + f*x))^m*(2100*A - 1575*B + 1272*A*m - 110*B*m + 304*A*m^2 + 32*A*m^3 - 68*B*m^2 - 8*B*m^3))/(4*f*(352*m + 344*m^ 2 + 128*m^3 + 16*m^4 + 105)) + (c^2*exp(e*4i + f*x*4i)*(a + a*sin(e + f*x) )^m*(A*2100i - B*1575i + A*m*1272i - B*m*110i + A*m^2*304i + A*m^3*32i - B *m^2*68i - B*m^3*8i))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) - ( c^2*exp(e*5i + f*x*5i)*(2*m + 1)*(a + a*sin(e + f*x))^m*(350*A - 385*B + 1 84*A*m - 104*B*m + 24*A*m^2 - 12*B*m^2))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) + (c^2*exp(e*2i + f*x*2i)*(2*m + 1)*(a + a*sin(e + f*x))^m *(A*350i - B*385i + A*m*184i - B*m*104i + A*m^2*24i - B*m^2*12i))/(4*f*(35 2*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) - (B*c^2*exp(e*7i + f*x*7i)*(a + a*sin(e + f*x))^m*(46*m + 36*m^2 + 8*m^3 + 15))/(4*f*(352*m + 344*m^2 + 12 8*m^3 + 16*m^4 + 105)) + (c^2*exp(e*1i + f*x*1i)*(a + a*sin(e + f*x))^m*(8 *m + 4*m^2 + 3)*(14*A - 35*B + 4*A*m - 6*B*m))/(4*f*(352*m + 344*m^2 + 128 *m^3 + 16*m^4 + 105)) - (c^2*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^m*(8* m + 4*m^2 + 3)*(A*14i - B*35i + A*m*4i - B*m*6i))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105))))/(exp(e*4i + f*x*4i) - (exp(e*3i + f*x*3i)*(m*35 2i + m^2*344i + m^3*128i + m^4*16i + 105i))/(352*m + 344*m^2 + 128*m^3 + 1 6*m^4 + 105))
\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\sqrt {c}\, c^{2} \left (\left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) b +\left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) a -2 \left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) b -2 \left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) a +\left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) b +\left (\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {-\sin \left (f x +e \right )+1}d x \right ) a \right ) \] Input:
int((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x)
Output:
sqrt(c)*c**2*(int((sin(e + f*x)*a + a)**m*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3,x)*b + int((sin(e + f*x)*a + a)**m*sqrt( - sin(e + f*x) + 1)*sin (e + f*x)**2,x)*a - 2*int((sin(e + f*x)*a + a)**m*sqrt( - sin(e + f*x) + 1 )*sin(e + f*x)**2,x)*b - 2*int((sin(e + f*x)*a + a)**m*sqrt( - sin(e + f*x ) + 1)*sin(e + f*x),x)*a + int((sin(e + f*x)*a + a)**m*sqrt( - sin(e + f*x ) + 1)*sin(e + f*x),x)*b + int((sin(e + f*x)*a + a)**m*sqrt( - sin(e + f*x ) + 1),x)*a)