Integrand size = 37, antiderivative size = 308 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx=-\frac {(c-d) \left (B \left (5 c^2+62 c d-163 d^2\right )+3 A \left (c^2+6 c d+25 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}+\frac {d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \cos (e+f x)}{24 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {d^2 (9 A c+15 B c+39 A d-95 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{48 a^3 f}-\frac {(3 A c+5 B c+9 A d-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a+a \sin (e+f x))^{5/2}} \] Output:
-1/32*(c-d)*(B*(5*c^2+62*c*d-163*d^2)+3*A*(c^2+6*c*d+25*d^2))*arctanh(1/2* a^(1/2)*cos(f*x+e)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))*2^(1/2)/a^(5/2)/f+1/24* d*(A*(9*c^2+36*c*d-93*d^2)+B*(15*c^2-228*c*d+197*d^2))*cos(f*x+e)/a^2/f/(a +a*sin(f*x+e))^(1/2)+1/48*d^2*(9*A*c+39*A*d+15*B*c-95*B*d)*cos(f*x+e)*(a+a *sin(f*x+e))^(1/2)/a^3/f-1/16*(3*A*c+9*A*d+5*B*c-17*B*d)*cos(f*x+e)*(c+d*s in(f*x+e))^2/a/f/(a+a*sin(f*x+e))^(3/2)-1/4*(A-B)*cos(f*x+e)*(c+d*sin(f*x+ e))^3/f/(a+a*sin(f*x+e))^(5/2)
Result contains complex when optimal does not.
Time = 8.32 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (24 (A-B) (c-d)^3 \sin \left (\frac {1}{2} (e+f x)\right )-12 (A-B) (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+6 (c-d)^2 (B (5 c-29 d)+3 A (c+7 d)) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-3 (c-d)^2 (B (5 c-29 d)+3 A (c+7 d)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+(3+3 i) (-1)^{3/4} (c-d) \left (B \left (5 c^2+62 c d-163 d^2\right )+3 A \left (c^2+6 c d+25 d^2\right )\right ) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-16 B d^3 \cos \left (\frac {3}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+(24+24 i) d^2 (-6 B c-2 A d+5 B d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+i \sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+(24+24 i) d^2 (6 B c+2 A d-5 B d) \left (i \cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-16 B d^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {3}{2} (e+f x)\right )\right )}{48 f (a (1+\sin (e+f x)))^{5/2}} \] Input:
Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x ])^(5/2),x]
Output:
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(24*(A - B)*(c - d)^3*Sin[(e + f*x) /2] - 12*(A - B)*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 6*(c - d)^2*(B*(5*c - 29*d) + 3*A*(c + 7*d))*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 3*(c - d)^2*(B*(5*c - 29*d) + 3*A*(c + 7*d))*(Cos[( e + f*x)/2] + Sin[(e + f*x)/2])^3 + (3 + 3*I)*(-1)^(3/4)*(c - d)*(B*(5*c^2 + 62*c*d - 163*d^2) + 3*A*(c^2 + 6*c*d + 25*d^2))*ArcTanh[(1/2 + I/2)*(-1 )^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 16*B*d^3*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + ( 24 + 24*I)*d^2*(-6*B*c - 2*A*d + 5*B*d)*(Cos[(e + f*x)/2] + I*Sin[(e + f*x )/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 + (24 + 24*I)*d^2*(6*B*c + 2 *A*d - 5*B*d)*(I*Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 16*B*d^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sin [(3*(e + f*x))/2]))/(48*f*(a*(1 + Sin[e + f*x]))^(5/2))
Time = 1.97 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.05, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {3042, 3456, 27, 3042, 3456, 27, 3042, 3447, 3042, 3502, 27, 3042, 3230, 3042, 3128, 219}
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.
| \(\displaystyle \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a \sin (e+f x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a \sin (e+f x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^2 (a (3 A c+5 B c+6 A d-6 B d)-a (3 A-11 B) d \sin (e+f x))}{2 (\sin (e+f x) a+a)^{3/2}}dx}{4 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^2 (a (3 A c+5 B c+6 A d-6 B d)-a (3 A-11 B) d \sin (e+f x))}{(\sin (e+f x) a+a)^{3/2}}dx}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^2 (a (3 A c+5 B c+6 A d-6 B d)-a (3 A-11 B) d \sin (e+f x))}{(\sin (e+f x) a+a)^{3/2}}dx}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x)) \left (a^2 \left (B \left (5 c^2+47 d c-68 d^2\right )+3 A \left (c^2+3 d c+12 d^2\right )\right )-a^2 d (9 A c+15 B c+39 A d-95 B d) \sin (e+f x)\right )}{2 \sqrt {\sin (e+f x) a+a}}dx}{2 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x)) \left (a^2 \left (B \left (5 c^2+47 d c-68 d^2\right )+3 A \left (c^2+3 d c+12 d^2\right )\right )-a^2 d (9 A c+15 B c+39 A d-95 B d) \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a}}dx}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(c+d \sin (e+f x)) \left (a^2 \left (B \left (5 c^2+47 d c-68 d^2\right )+3 A \left (c^2+3 d c+12 d^2\right )\right )-a^2 d (9 A c+15 B c+39 A d-95 B d) \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a}}dx}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {\int \frac {-d^2 (9 A c+15 B c+39 A d-95 B d) \sin ^2(e+f x) a^2+c \left (B \left (5 c^2+47 d c-68 d^2\right )+3 A \left (c^2+3 d c+12 d^2\right )\right ) a^2+\left (a^2 d \left (B \left (5 c^2+47 d c-68 d^2\right )+3 A \left (c^2+3 d c+12 d^2\right )\right )-a^2 c d (9 A c+15 B c+39 A d-95 B d)\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a}}dx}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-d^2 (9 A c+15 B c+39 A d-95 B d) \sin (e+f x)^2 a^2+c \left (B \left (5 c^2+47 d c-68 d^2\right )+3 A \left (c^2+3 d c+12 d^2\right )\right ) a^2+\left (a^2 d \left (B \left (5 c^2+47 d c-68 d^2\right )+3 A \left (c^2+3 d c+12 d^2\right )\right )-a^2 c d (9 A c+15 B c+39 A d-95 B d)\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a}}dx}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {a^3 \left (3 A \left (3 c^3+9 d c^2+33 d^2 c-13 d^3\right )+B \left (15 c^3+141 d c^2-219 d^2 c+95 d^3\right )\right )-2 a^3 d \left (A \left (9 c^2+36 d c-93 d^2\right )+B \left (15 c^2-228 d c+197 d^2\right )\right ) \sin (e+f x)}{2 \sqrt {\sin (e+f x) a+a}}dx}{3 a}+\frac {2 a d^2 (9 A c+39 A d+15 B c-95 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a^3 \left (3 A \left (3 c^3+9 d c^2+33 d^2 c-13 d^3\right )+B \left (15 c^3+141 d c^2-219 d^2 c+95 d^3\right )\right )-2 a^3 d \left (A \left (9 c^2+36 d c-93 d^2\right )+B \left (15 c^2-228 d c+197 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a}}dx}{3 a}+\frac {2 a d^2 (9 A c+39 A d+15 B c-95 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a^3 \left (3 A \left (3 c^3+9 d c^2+33 d^2 c-13 d^3\right )+B \left (15 c^3+141 d c^2-219 d^2 c+95 d^3\right )\right )-2 a^3 d \left (A \left (9 c^2+36 d c-93 d^2\right )+B \left (15 c^2-228 d c+197 d^2\right )\right ) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a}}dx}{3 a}+\frac {2 a d^2 (9 A c+39 A d+15 B c-95 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {\frac {3 a^3 (c-d) \left (3 A \left (c^2+6 c d+25 d^2\right )+B \left (5 c^2+62 c d-163 d^2\right )\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx+\frac {4 a^3 d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{3 a}+\frac {2 a d^2 (9 A c+39 A d+15 B c-95 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 a^3 (c-d) \left (3 A \left (c^2+6 c d+25 d^2\right )+B \left (5 c^2+62 c d-163 d^2\right )\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx+\frac {4 a^3 d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}}{3 a}+\frac {2 a d^2 (9 A c+39 A d+15 B c-95 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\frac {\frac {\frac {4 a^3 d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}-\frac {6 a^3 (c-d) \left (3 A \left (c^2+6 c d+25 d^2\right )+B \left (5 c^2+62 c d-163 d^2\right )\right ) \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}}{3 a}+\frac {2 a d^2 (9 A c+39 A d+15 B c-95 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\frac {4 a^3 d \left (A \left (9 c^2+36 c d-93 d^2\right )+B \left (15 c^2-228 c d+197 d^2\right )\right ) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}-\frac {3 \sqrt {2} a^{5/2} (c-d) \left (3 A \left (c^2+6 c d+25 d^2\right )+B \left (5 c^2+62 c d-163 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f}}{3 a}+\frac {2 a d^2 (9 A c+39 A d+15 B c-95 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{4 a^2}-\frac {a (3 A c+9 A d+5 B c-17 B d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
Input:
Int[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^(5/ 2),x]
Output:
-1/4*((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(f*(a + a*Sin[e + f*x]) ^(5/2)) + (-1/2*(a*(3*A*c + 5*B*c + 9*A*d - 17*B*d)*Cos[e + f*x]*(c + d*Si n[e + f*x])^2)/(f*(a + a*Sin[e + f*x])^(3/2)) + ((2*a*d^2*(9*A*c + 15*B*c + 39*A*d - 95*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) + ((-3*Sqr t[2]*a^(5/2)*(c - d)*(B*(5*c^2 + 62*c*d - 163*d^2) + 3*A*(c^2 + 6*c*d + 25 *d^2))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])]) /f + (4*a^3*d*(A*(9*c^2 + 36*c*d - 93*d^2) + B*(15*c^2 - 228*c*d + 197*d^2 ))*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]))/(3*a))/(4*a^2))/(8*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[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_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 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[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] & & NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[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[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1156\) vs. \(2(281)=562\).
Time = 2.31 (sec) , antiderivative size = 1157, normalized size of antiderivative = 3.76
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1157\) |
| default | \(\text {Expression too large to display}\) | \(1438\) |
Input:
int((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(5/2),x,method=_R ETURNVERBOSE)
Output:
-1/32*A*c^3/a^(9/2)*(-3*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2) /a^(1/2))*a^2*cos(f*x+e)^2+6*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^ (1/2)/a^(1/2))*a^2*sin(f*x+e)+6*a^(3/2)*(a-a*sin(f*x+e))^(1/2)*sin(f*x+e)+ 6*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2+14*(a-a* sin(f*x+e))^(1/2)*a^(3/2))*(-a*(sin(f*x+e)-1))^(1/2)/(1+sin(f*x+e))/cos(f* x+e)/(a+a*sin(f*x+e))^(1/2)/f-1/96*B*d^3/a^(9/2)*(-489*2^(1/2)*arctanh(1/2 *(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*cos(f*x+e)^2+384*a^(3/2)*(a-a *sin(f*x+e))^(1/2)*cos(f*x+e)^2+64*a^(1/2)*(a-a*sin(f*x+e))^(3/2)*cos(f*x+ e)^2+978*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*s in(f*x+e)-768*a^(3/2)*(a-a*sin(f*x+e))^(1/2)*sin(f*x+e)-128*(a-a*sin(f*x+e ))^(3/2)*a^(1/2)*sin(f*x+e)+978*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2) *2^(1/2)/a^(1/2))*a^2-1092*(a-a*sin(f*x+e))^(1/2)*a^(3/2)+46*(a-a*sin(f*x+ e))^(3/2)*a^(1/2))*(-a*(sin(f*x+e)-1))^(1/2)/(1+sin(f*x+e))/cos(f*x+e)/(a+ a*sin(f*x+e))^(1/2)/f-1/32*c^2*(3*A*d+B*c)/a^(11/2)*(-5*2^(1/2)*arctanh(1/ 2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^3*cos(f*x+e)^2+10*2^(1/2)*arct anh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^3+12*a^(5/2)* (a-a*sin(f*x+e))^(1/2)-10*(a-a*sin(f*x+e))^(3/2)*a^(3/2)+10*2^(1/2)*arctan h(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^3)*(-a*(sin(f*x+e)-1))^(1/ 2)/(1+sin(f*x+e))/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f+1/32*d^2*(A*d+3*B*c) /a^(9/2)*(-75*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2...
Leaf count of result is larger than twice the leaf count of optimal. 980 vs. \(2 (282) = 564\).
Time = 0.14 (sec) , antiderivative size = 980, normalized size of antiderivative = 3.18 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(5/2),x, al gorithm="fricas")
Output:
-1/192*(3*sqrt(2)*(4*(3*A + 5*B)*c^3 + 12*(5*A + 19*B)*c^2*d + 12*(19*A - 75*B)*c*d^2 - 4*(75*A - 163*B)*d^3 - ((3*A + 5*B)*c^3 + 3*(5*A + 19*B)*c^2 *d + 3*(19*A - 75*B)*c*d^2 - (75*A - 163*B)*d^3)*cos(f*x + e)^3 - 3*((3*A + 5*B)*c^3 + 3*(5*A + 19*B)*c^2*d + 3*(19*A - 75*B)*c*d^2 - (75*A - 163*B) *d^3)*cos(f*x + e)^2 + 2*((3*A + 5*B)*c^3 + 3*(5*A + 19*B)*c^2*d + 3*(19*A - 75*B)*c*d^2 - (75*A - 163*B)*d^3)*cos(f*x + e) + (4*(3*A + 5*B)*c^3 + 1 2*(5*A + 19*B)*c^2*d + 12*(19*A - 75*B)*c*d^2 - 4*(75*A - 163*B)*d^3 - ((3 *A + 5*B)*c^3 + 3*(5*A + 19*B)*c^2*d + 3*(19*A - 75*B)*c*d^2 - (75*A - 163 *B)*d^3)*cos(f*x + e)^2 + 2*((3*A + 5*B)*c^3 + 3*(5*A + 19*B)*c^2*d + 3*(1 9*A - 75*B)*c*d^2 - (75*A - 163*B)*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a )*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos (f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)* sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - co s(f*x + e) - 2)) + 4*(32*B*d^3*cos(f*x + e)^4 - 12*(A - B)*c^3 + 36*(A - B )*c^2*d - 36*(A - B)*c*d^2 + 12*(A - B)*d^3 + 32*(9*B*c*d^2 + (3*A - 5*B)* d^3)*cos(f*x + e)^3 - 3*((3*A + 5*B)*c^3 + 3*(5*A - 13*B)*c^2*d - 3*(13*A - 53*B)*c*d^2 + (53*A - 93*B)*d^3)*cos(f*x + e)^2 - 3*((7*A + B)*c^3 + 3*( A - 9*B)*c^2*d - 27*(A - 9*B)*c*d^2 + 9*(9*A - 17*B)*d^3)*cos(f*x + e) + ( 32*B*d^3*cos(f*x + e)^3 + 12*(A - B)*c^3 - 36*(A - B)*c^2*d + 36*(A - B)*c *d^2 - 12*(A - B)*d^3 - 96*(3*B*c*d^2 + (A - 2*B)*d^3)*cos(f*x + e)^2 -...
Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))**3/(a+a*sin(f*x+e))**(5/2),x)
Output:
Timed out
\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(5/2),x, al gorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(d*sin(f*x + e) + c)^3/(a*sin(f*x + e) + a) ^(5/2), x)
Exception generated. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(5/2),x, al gorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \] Input:
int(((A + B*sin(e + f*x))*(c + d*sin(e + f*x))^3)/(a + a*sin(e + f*x))^(5/ 2),x)
Output:
int(((A + B*sin(e + f*x))*(c + d*sin(e + f*x))^3)/(a + a*sin(e + f*x))^(5/ 2), x)
\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) a \,c^{3}+\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) b \,d^{3}+\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) a \,d^{3}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) b c \,d^{2}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) a c \,d^{2}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) b \,c^{2} d +3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) a \,c^{2} d +\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) b \,c^{3}\right )}{a^{3}} \] Input:
int((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(5/2),x)
Output:
(sqrt(a)*(int(sqrt(sin(e + f*x) + 1)/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x)*a*c**3 + int((sqrt(sin(e + f*x) + 1)*sin(e + f*x) **4)/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x)*b*d**3 + int((sqrt(sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x)*a*d**3 + 3*int((sqrt(sin(e + f*x) + 1)* sin(e + f*x)**3)/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1 ),x)*b*c*d**2 + 3*int((sqrt(sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f* x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x)*a*c*d**2 + 3*int((sqrt( sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x)*b*c**2*d + 3*int((sqrt(sin(e + f*x) + 1)*sin(e + f* x))/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x)*a*c**2*d + int((sqrt(sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x)*b*c**3))/a**3