Integrand size = 37, antiderivative size = 429 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {2 a^3 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x)}{3465 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d^2 f}-\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac {2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f} \] Output:
-2/3465*a^3*(15*c^2+10*c*d+7*d^2)*(11*A*d*(c^2-10*c*d+73*d^2)-B*(5*c^3-40* c^2*d+169*c*d^2-710*d^3))*cos(f*x+e)/d^3/f/(a+a*sin(f*x+e))^(1/2)-4/3465*a ^2*(5*c-d)*(11*A*d*(c^2-10*c*d+73*d^2)-B*(5*c^3-40*c^2*d+169*c*d^2-710*d^3 ))*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/d^2/f-2/1155*a*(11*A*d*(c^2-10*c*d+73 *d^2)-B*(5*c^3-40*c^2*d+169*c*d^2-710*d^3))*cos(f*x+e)*(a+a*sin(f*x+e))^(3 /2)/d/f+2/693*a^3*(11*A*(3*c-19*d)*d-B*(15*c^2-65*c*d+194*d^2))*cos(f*x+e) *(c+d*sin(f*x+e))^3/d^3/f/(a+a*sin(f*x+e))^(1/2)+2/99*a^2*(-11*A*d+5*B*c-1 4*B*d)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^3/d^2/f-2/11*a*B *cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3/d/f
Leaf count is larger than twice the leaf count of optimal. \(891\) vs. \(2(429)=858\).
Time = 9.64 (sec) , antiderivative size = 891, normalized size of antiderivative = 2.08 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx =\text {Too large to display} \] Input:
Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f *x])^2,x]
Output:
((a*(1 + Sin[e + f*x]))^(5/2)*(-277200*A*c^2*Cos[(e + f*x)/2] - 207900*B*c ^2*Cos[(e + f*x)/2] - 415800*A*c*d*Cos[(e + f*x)/2] - 360360*B*c*d*Cos[(e + f*x)/2] - 180180*A*d^2*Cos[(e + f*x)/2] - 159390*B*d^2*Cos[(e + f*x)/2] - 46200*A*c^2*Cos[(3*(e + f*x))/2] - 50820*B*c^2*Cos[(3*(e + f*x))/2] - 10 1640*A*c*d*Cos[(3*(e + f*x))/2] - 92400*B*c*d*Cos[(3*(e + f*x))/2] - 46200 *A*d^2*Cos[(3*(e + f*x))/2] - 43890*B*d^2*Cos[(3*(e + f*x))/2] + 5544*A*c^ 2*Cos[(5*(e + f*x))/2] + 13860*B*c^2*Cos[(5*(e + f*x))/2] + 27720*A*c*d*Co s[(5*(e + f*x))/2] + 33264*B*c*d*Cos[(5*(e + f*x))/2] + 16632*A*d^2*Cos[(5 *(e + f*x))/2] + 17325*B*d^2*Cos[(5*(e + f*x))/2] + 1980*B*c^2*Cos[(7*(e + f*x))/2] + 3960*A*c*d*Cos[(7*(e + f*x))/2] + 9900*B*c*d*Cos[(7*(e + f*x)) /2] + 4950*A*d^2*Cos[(7*(e + f*x))/2] + 6435*B*d^2*Cos[(7*(e + f*x))/2] - 1540*B*c*d*Cos[(9*(e + f*x))/2] - 770*A*d^2*Cos[(9*(e + f*x))/2] - 1925*B* d^2*Cos[(9*(e + f*x))/2] - 315*B*d^2*Cos[(11*(e + f*x))/2] + 277200*A*c^2* Sin[(e + f*x)/2] + 207900*B*c^2*Sin[(e + f*x)/2] + 415800*A*c*d*Sin[(e + f *x)/2] + 360360*B*c*d*Sin[(e + f*x)/2] + 180180*A*d^2*Sin[(e + f*x)/2] + 1 59390*B*d^2*Sin[(e + f*x)/2] - 46200*A*c^2*Sin[(3*(e + f*x))/2] - 50820*B* c^2*Sin[(3*(e + f*x))/2] - 101640*A*c*d*Sin[(3*(e + f*x))/2] - 92400*B*c*d *Sin[(3*(e + f*x))/2] - 46200*A*d^2*Sin[(3*(e + f*x))/2] - 43890*B*d^2*Sin [(3*(e + f*x))/2] - 5544*A*c^2*Sin[(5*(e + f*x))/2] - 13860*B*c^2*Sin[(5*( e + f*x))/2] - 27720*A*c*d*Sin[(5*(e + f*x))/2] - 33264*B*c*d*Sin[(5*(e...
Time = 1.93 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.86, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {3042, 3455, 27, 3042, 3455, 27, 3042, 3460, 3042, 3240, 27, 3042, 3230, 3042, 3125}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2dx\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^2 (a (11 A d+3 B (c+2 d))-a (5 B c-11 A d-14 B d) \sin (e+f x))dx}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^2 (a (11 A d+3 B (c+2 d))-a (5 B c-11 A d-14 B d) \sin (e+f x))dx}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^2 (a (11 A d+3 B (c+2 d))-a (5 B c-11 A d-14 B d) \sin (e+f x))dx}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2 \left (a^2 \left (11 A d (c+15 d)-B \left (5 c^2-11 d c-138 d^2\right )\right )-a^2 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 d c+194 d^2\right )\right ) \sin (e+f x)\right )dx}{9 d}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2 \left (a^2 \left (11 A d (c+15 d)-B \left (5 c^2-11 d c-138 d^2\right )\right )-a^2 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 d c+194 d^2\right )\right ) \sin (e+f x)\right )dx}{9 d}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2 \left (a^2 \left (11 A d (c+15 d)-B \left (5 c^2-11 d c-138 d^2\right )\right )-a^2 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 d c+194 d^2\right )\right ) \sin (e+f x)\right )dx}{9 d}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2dx}{7 d}+\frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}}{9 d}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2dx}{7 d}+\frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}}{9 d}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 3240 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \left (\frac {2 \int \frac {1}{2} \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}+\frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}}{9 d}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \left (\frac {\int \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}+\frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}}{9 d}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \left (\frac {\int \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}+\frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}}{9 d}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \left (\frac {\frac {1}{3} a \left (15 c^2+10 c d+7 d^2\right ) \int \sqrt {\sin (e+f x) a+a}dx-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}+\frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}}{9 d}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \left (\frac {\frac {1}{3} a \left (15 c^2+10 c d+7 d^2\right ) \int \sqrt {\sin (e+f x) a+a}dx-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}+\frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}}{9 d}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}+\frac {\frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}+\frac {3 a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \left (\frac {-\frac {2 a^2 \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}}{9 d}}{11 d}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f}\) |
Input:
Int[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2 ,x]
Output:
(-2*a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3)/(1 1*d*f) + ((2*a^2*(5*B*c - 11*A*d - 14*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3)/(9*d*f) + ((2*a^3*(11*A*(3*c - 19*d)*d - B* (15*c^2 - 65*c*d + 194*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(7*d*f*S qrt[a + a*Sin[e + f*x]]) + (3*a^2*(11*A*d*(c^2 - 10*c*d + 73*d^2) - B*(5*c ^3 - 40*c^2*d + 169*c*d^2 - 710*d^3))*((-2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*a*f) + ((-2*a^2*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/( 3*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a*(5*c - d)*d*Cos[e + f*x]*Sqrt[a + a*S in[e + f*x]])/(3*f))/(5*a)))/(7*d))/(9*d))/(11*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^2, x_Symbol] :> Simp[(-d^2)*Cos[e + f*x]*((a + b*Sin[e + f*x])^ (m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^ m*Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && !LtQ[m, -1]
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)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[n, -1]
Time = 39.15 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.60
| method | result | size |
| default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (315 B \cos \left (f x +e \right )^{4} \sin \left (f x +e \right ) d^{2}+\left (385 A \,d^{2}+770 B c d +1120 B \,d^{2}\right ) \cos \left (f x +e \right )^{4}+\left (-990 A c d -1430 A \,d^{2}-495 B \,c^{2}-2860 B c d -2405 B \,d^{2}\right ) \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )+\left (-693 A \,c^{2}-3960 A c d -3179 A \,d^{2}-1980 B \,c^{2}-6358 B c d -4370 B \,d^{2}\right ) \cos \left (f x +e \right )^{2}+\left (3234 A \,c^{2}+8580 A c d +4642 A \,d^{2}+4290 B \,c^{2}+9284 B c d +4930 B \,d^{2}\right ) \sin \left (f x +e \right )+10626 A \,c^{2}+19140 A c d +9218 A \,d^{2}+9570 B \,c^{2}+18436 B c d +8930 B \,d^{2}\right )}{3465 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(257\) |
| parts | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) d \left (A d +2 B c \right ) \left (35 \sin \left (f x +e \right )^{4}+130 \sin \left (f x +e \right )^{3}+219 \sin \left (f x +e \right )^{2}+292 \sin \left (f x +e \right )+584\right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) c \left (2 A d +B c \right ) \left (3 \sin \left (f x +e \right )^{3}+12 \sin \left (f x +e \right )^{2}+23 \sin \left (f x +e \right )+46\right )}{21 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 A \,c^{2} \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (3 \sin \left (f x +e \right )^{2}+14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 B \,d^{2} \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (63 \sin \left (f x +e \right )^{5}+224 \sin \left (f x +e \right )^{4}+355 \sin \left (f x +e \right )^{3}+426 \sin \left (f x +e \right )^{2}+568 \sin \left (f x +e \right )+1136\right )}{693 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(344\) |
Input:
int((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x,method=_R ETURNVERBOSE)
Output:
2/3465*(1+sin(f*x+e))*a^3*(sin(f*x+e)-1)*(315*B*cos(f*x+e)^4*sin(f*x+e)*d^ 2+(385*A*d^2+770*B*c*d+1120*B*d^2)*cos(f*x+e)^4+(-990*A*c*d-1430*A*d^2-495 *B*c^2-2860*B*c*d-2405*B*d^2)*cos(f*x+e)^2*sin(f*x+e)+(-693*A*c^2-3960*A*c *d-3179*A*d^2-1980*B*c^2-6358*B*c*d-4370*B*d^2)*cos(f*x+e)^2+(3234*A*c^2+8 580*A*c*d+4642*A*d^2+4290*B*c^2+9284*B*c*d+4930*B*d^2)*sin(f*x+e)+10626*A* c^2+19140*A*c*d+9218*A*d^2+9570*B*c^2+18436*B*c*d+8930*B*d^2)/cos(f*x+e)/( a+a*sin(f*x+e))^(1/2)/f
Time = 0.11 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.38 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx =\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, al gorithm="fricas")
Output:
-2/3465*(315*B*a^2*d^2*cos(f*x + e)^6 + 35*(22*B*a^2*c*d + (11*A + 32*B)*a ^2*d^2)*cos(f*x + e)^5 + 1056*(7*A + 5*B)*a^2*c^2 + 704*(15*A + 13*B)*a^2* c*d + 32*(143*A + 125*B)*a^2*d^2 - 5*(99*B*a^2*c^2 + 22*(9*A + 19*B)*a^2*c *d + (209*A + 320*B)*a^2*d^2)*cos(f*x + e)^4 - (99*(7*A + 20*B)*a^2*c^2 + 22*(180*A + 289*B)*a^2*c*d + (3179*A + 4370*B)*a^2*d^2)*cos(f*x + e)^3 + ( 33*(77*A + 85*B)*a^2*c^2 + 22*(255*A + 263*B)*a^2*c*d + (2893*A + 2965*B)* a^2*d^2)*cos(f*x + e)^2 + 2*(33*(161*A + 145*B)*a^2*c^2 + 22*(435*A + 419* B)*a^2*c*d + (4609*A + 4465*B)*a^2*d^2)*cos(f*x + e) + (315*B*a^2*d^2*cos( f*x + e)^5 - 1056*(7*A + 5*B)*a^2*c^2 - 704*(15*A + 13*B)*a^2*c*d - 32*(14 3*A + 125*B)*a^2*d^2 - 35*(22*B*a^2*c*d + (11*A + 23*B)*a^2*d^2)*cos(f*x + e)^4 - 5*(99*B*a^2*c^2 + 22*(9*A + 26*B)*a^2*c*d + 13*(22*A + 37*B)*a^2*d ^2)*cos(f*x + e)^3 + 3*(33*(7*A + 15*B)*a^2*c^2 + 22*(45*A + 53*B)*a^2*c*d + (583*A + 655*B)*a^2*d^2)*cos(f*x + e)^2 + 2*(33*(49*A + 65*B)*a^2*c^2 + 22*(195*A + 211*B)*a^2*c*d + (2321*A + 2465*B)*a^2*d^2)*cos(f*x + e))*sin (f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)
Timed out. \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**2,x)
Output:
Timed out
\[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{2} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, al gorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c)^2, x)
Time = 0.35 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.59 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, al gorithm="giac")
Output:
1/55440*sqrt(2)*(315*B*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-11 /4*pi + 11/2*f*x + 11/2*e) + 6930*(40*A*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 30*B*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 60*A*a^2*c* d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 52*B*a^2*c*d*sgn(cos(-1/4*pi + 1/2 *f*x + 1/2*e)) + 26*A*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 23*B*a ^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 2310*(20*A*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 22*B*a^2*c^2*s gn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 44*A*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f* x + 1/2*e)) + 40*B*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 20*A*a^2* d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 19*B*a^2*d^2*sgn(cos(-1/4*pi + 1 /2*f*x + 1/2*e)))*sin(-3/4*pi + 3/2*f*x + 3/2*e) + 693*(8*A*a^2*c^2*sgn(co s(-1/4*pi + 1/2*f*x + 1/2*e)) + 20*B*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1 /2*e)) + 40*A*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 48*B*a^2*c*d*s gn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 24*A*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f* x + 1/2*e)) + 25*B*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-5/4*p i + 5/2*f*x + 5/2*e) + 495*(4*B*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e) ) + 8*A*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 20*B*a^2*c*d*sgn(cos (-1/4*pi + 1/2*f*x + 1/2*e)) + 10*A*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/ 2*e)) + 13*B*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-7/4*pi + 7/ 2*f*x + 7/2*e) + 385*(4*B*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) +...
Timed out. \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2 \,d x \] Input:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^2 ,x)
Output:
int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^2 , x)
\[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\sqrt {a}\, a^{2} \left (\left (\int \sqrt {\sin \left (f x +e \right )+1}d x \right ) a \,c^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{5}d x \right ) b \,d^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) a \,d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) b c d +2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) b \,d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) a c d +2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) a \,d^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) b \,c^{2}+4 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) b c d +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) b \,d^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) a \,c^{2}+4 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) a c d +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) a \,d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) b \,c^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) b c d +2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) a \,c^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) a c d +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) b \,c^{2}\right ) \] Input:
int((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x)
Output:
sqrt(a)*a**2*(int(sqrt(sin(e + f*x) + 1),x)*a*c**2 + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**5,x)*b*d**2 + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)* *4,x)*a*d**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**4,x)*b*c*d + 2*i nt(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**4,x)*b*d**2 + 2*int(sqrt(sin(e + f *x) + 1)*sin(e + f*x)**3,x)*a*c*d + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f *x)**3,x)*a*d**2 + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**3,x)*b*c**2 + 4*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**3,x)*b*c*d + int(sqrt(sin(e + f *x) + 1)*sin(e + f*x)**3,x)*b*d**2 + int(sqrt(sin(e + f*x) + 1)*sin(e + f* x)**2,x)*a*c**2 + 4*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**2,x)*a*c*d + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**2,x)*a*d**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**2,x)*b*c**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**2,x)*b*c*d + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x),x)*a*c**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x),x)*a*c*d + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x),x)*b*c**2)