Integrand size = 43, antiderivative size = 622 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=-\frac {2 \left (80 a^5 b B-140 a^3 b^3 B+40 a b^5 B-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-128 a^6 C-3 b^6 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (80 a^4 b B-80 a^2 b^3 B-5 b^5 B-4 a^3 b^2 (10 A-29 C)-128 a^5 C+a b^4 (45 A+17 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b^5 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}-\frac {2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d} \] Output:
-2/15*(80*B*a^5*b-140*B*a^3*b^3+40*B*a*b^5-4*a^4*b^2*(10*A-53*C)+5*a^2*b^4 *(15*A-11*C)-128*a^6*C-3*b^6*(5*A+3*C))*(a+b*cos(d*x+c))^(1/2)*EllipticE(s in(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/b^5/(a^2-b^2)^2/d/((a+b*cos(d*x +c))/(a+b))^(1/2)+2/15*(80*B*a^4*b-80*B*a^2*b^3-5*B*b^5-4*a^3*b^2*(10*A-29 *C)-128*C*a^5+a*b^4*(45*A+17*C))*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJac obiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/b^5/(a^2-b^2)/d/(a+b*cos(d*x+ c))^(1/2)-2/3*(A*b^2-a*(B*b-C*a))*cos(d*x+c)^3*sin(d*x+c)/b/(a^2-b^2)/d/(a +b*cos(d*x+c))^(3/2)+2/3*(6*A*b^4+5*B*a^3*b-9*B*a*b^3-2*a^2*b^2*(A-6*C)-8* a^4*C)*cos(d*x+c)^2*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)+2/ 15*(40*B*a^4*b-65*B*a^2*b^3+5*B*b^5-2*a^3*b^2*(10*A-49*C)+2*a*b^4*(20*A-7* C)-64*C*a^5)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^4/(a^2-b^2)^2/d-2/15*(30* B*a^3*b-50*B*a*b^3-a^2*b^2*(15*A-71*C)+b^4*(35*A-3*C)-48*a^4*C)*cos(d*x+c) *(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^3/(a^2-b^2)^2/d
Time = 7.04 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\frac {2 \left (\frac {a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b^2 \left (-20 a^4 b B+35 a^2 b^3 B+5 b^5 B+2 a^3 b^2 (5 A-22 C)+32 a^5 C-2 a b^4 (15 A+4 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-80 a^5 b B+140 a^3 b^3 B-40 a b^5 B+4 a^4 b^2 (10 A-53 C)+128 a^6 C+3 b^6 (5 A+3 C)+5 a^2 b^4 (-15 A+11 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}+b \left (\frac {10 a^3 \left (A b^2+a (-b B+a C)\right ) \sin (c+d x)}{a^2-b^2}-\frac {10 a^2 \left (-9 A b^4-8 a^3 b B+12 a b^3 B+5 a^2 b^2 (A-3 C)+11 a^4 C\right ) (a+b \cos (c+d x)) \sin (c+d x)}{\left (a^2-b^2\right )^2}+2 (5 b B-14 a C) (a+b \cos (c+d x))^2 \sin (c+d x)+3 b C (a+b \cos (c+d x))^2 \sin (2 (c+d x))\right )}{15 b^5 d (a+b \cos (c+d x))^{3/2}} \] Input:
Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b* Cos[c + d*x])^(5/2),x]
Output:
((2*((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b^2*(-20*a^4*b*B + 35*a^2*b^3*B + 5*b^5*B + 2*a^3*b^2*(5*A - 22*C) + 32*a^5*C - 2*a*b^4*(15*A + 4*C))*Elli pticF[(c + d*x)/2, (2*b)/(a + b)] + (-80*a^5*b*B + 140*a^3*b^3*B - 40*a*b^ 5*B + 4*a^4*b^2*(10*A - 53*C) + 128*a^6*C + 3*b^6*(5*A + 3*C) + 5*a^2*b^4* (-15*A + 11*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*Ellipti cF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)) + b*((10*a^3*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/(a^2 - b^2) - (10*a^2*(-9*A*b^4 - 8*a^3* b*B + 12*a*b^3*B + 5*a^2*b^2*(A - 3*C) + 11*a^4*C)*(a + b*Cos[c + d*x])*Si n[c + d*x])/(a^2 - b^2)^2 + 2*(5*b*B - 14*a*C)*(a + b*Cos[c + d*x])^2*Sin[ c + d*x] + 3*b*C*(a + b*Cos[c + d*x])^2*Sin[2*(c + d*x)]))/(15*b^5*d*(a + b*Cos[c + d*x])^(3/2))
Time = 3.73 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.01, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 3526, 27, 3042, 3526, 27, 3042, 3528, 27, 3042, 3502, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {2 \int \frac {\cos ^2(c+d x) \left (-\left (\left (8 C a^2-5 b B a+5 A b^2-3 b^2 C\right ) \cos ^2(c+d x)\right )+3 b (b B-a (A+C)) \cos (c+d x)+6 \left (A b^2-a (b B-a C)\right )\right )}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (-\left (\left (8 C a^2-5 b B a+5 A b^2-3 b^2 C\right ) \cos ^2(c+d x)\right )+3 b (b B-a (A+C)) \cos (c+d x)+6 \left (A b^2-a (b B-a C)\right )\right )}{(a+b \cos (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\left (-8 C a^2+5 b B a-5 A b^2+3 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+3 b (b B-a (A+C)) \sin \left (c+d x+\frac {\pi }{2}\right )+6 \left (A b^2-a (b B-a C)\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {\cos (c+d x) \left (-\left (\left (-48 C a^4+30 b B a^3-b^2 (15 A-71 C) a^2-50 b^3 B a+b^4 (35 A-3 C)\right ) \cos ^2(c+d x)\right )+b \left (2 C a^3+b B a^2-2 b^2 (2 A+3 C) a+3 b^3 B\right ) \cos (c+d x)+4 \left (-8 C a^4+5 b B a^3-2 b^2 (A-6 C) a^2-9 b^3 B a+6 A b^4\right )\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\cos (c+d x) \left (-\left (\left (-48 C a^4+30 b B a^3-b^2 (15 A-71 C) a^2-50 b^3 B a+b^4 (35 A-3 C)\right ) \cos ^2(c+d x)\right )+b \left (2 C a^3+b B a^2-2 b^2 (2 A+3 C) a+3 b^3 B\right ) \cos (c+d x)+4 \left (-8 C a^4+5 b B a^3-2 b^2 (A-6 C) a^2-9 b^3 B a+6 A b^4\right )\right )}{\sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (\left (48 C a^4-30 b B a^3+b^2 (15 A-71 C) a^2+50 b^3 B a-b^4 (35 A-3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (2 C a^3+b B a^2-2 b^2 (2 A+3 C) a+3 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+4 \left (-8 C a^4+5 b B a^3-2 b^2 (A-6 C) a^2-9 b^3 B a+6 A b^4\right )\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {-\frac {\frac {2 \int -\frac {-3 \left (-64 C a^5+40 b B a^4-2 b^2 (10 A-49 C) a^3-65 b^3 B a^2+2 b^4 (20 A-7 C) a+5 b^5 B\right ) \cos ^2(c+d x)-b \left (-16 C a^4+10 b B a^3+b^2 (5 A+27 C) a^2-30 b^3 B a+3 b^4 (5 A+3 C)\right ) \cos (c+d x)+2 a \left (-48 C a^4+30 b B a^3-b^2 (15 A-71 C) a^2-50 b^3 B a+b^4 (35 A-3 C)\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {-3 \left (-64 C a^5+40 b B a^4-2 b^2 (10 A-49 C) a^3-65 b^3 B a^2+2 b^4 (20 A-7 C) a+5 b^5 B\right ) \cos ^2(c+d x)-b \left (-16 C a^4+10 b B a^3+b^2 (5 A+27 C) a^2-30 b^3 B a+3 b^4 (5 A+3 C)\right ) \cos (c+d x)+2 a \left (-48 C a^4+30 b B a^3-b^2 (15 A-71 C) a^2-50 b^3 B a+b^4 (35 A-3 C)\right )}{\sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {-3 \left (-64 C a^5+40 b B a^4-2 b^2 (10 A-49 C) a^3-65 b^3 B a^2+2 b^4 (20 A-7 C) a+5 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (-16 C a^4+10 b B a^3+b^2 (5 A+27 C) a^2-30 b^3 B a+3 b^4 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a \left (-48 C a^4+30 b B a^3-b^2 (15 A-71 C) a^2-50 b^3 B a+b^4 (35 A-3 C)\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {2 \int \frac {3 \left (b \left (-32 C a^5+20 b B a^4-2 b^2 (5 A-22 C) a^3-35 b^3 B a^2+2 b^4 (15 A+4 C) a-5 b^5 B\right )+\left (-128 C a^6+80 b B a^5-4 b^2 (10 A-53 C) a^4-140 b^3 B a^3+5 b^4 (15 A-11 C) a^2+40 b^5 B a-3 b^6 (5 A+3 C)\right ) \cos (c+d x)\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {b \left (-32 C a^5+20 b B a^4-2 b^2 (5 A-22 C) a^3-35 b^3 B a^2+2 b^4 (15 A+4 C) a-5 b^5 B\right )+\left (-128 C a^6+80 b B a^5-4 b^2 (10 A-53 C) a^4-140 b^3 B a^3+5 b^4 (15 A-11 C) a^2+40 b^5 B a-3 b^6 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {b \left (-32 C a^5+20 b B a^4-2 b^2 (5 A-22 C) a^3-35 b^3 B a^2+2 b^4 (15 A+4 C) a-5 b^5 B\right )+\left (-128 C a^6+80 b B a^5-4 b^2 (10 A-53 C) a^4-140 b^3 B a^3+5 b^4 (15 A-11 C) a^2+40 b^5 B a-3 b^6 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {\left (-128 a^6 C+80 a^5 b B-4 a^4 b^2 (10 A-53 C)-140 a^3 b^3 B+5 a^2 b^4 (15 A-11 C)+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 C+80 a^4 b B-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+a b^4 (45 A+17 C)-5 b^5 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {\left (-128 a^6 C+80 a^5 b B-4 a^4 b^2 (10 A-53 C)-140 a^3 b^3 B+5 a^2 b^4 (15 A-11 C)+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 C+80 a^4 b B-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+a b^4 (45 A+17 C)-5 b^5 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {\left (-128 a^6 C+80 a^5 b B-4 a^4 b^2 (10 A-53 C)-140 a^3 b^3 B+5 a^2 b^4 (15 A-11 C)+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 C+80 a^4 b B-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+a b^4 (45 A+17 C)-5 b^5 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {\left (-128 a^6 C+80 a^5 b B-4 a^4 b^2 (10 A-53 C)-140 a^3 b^3 B+5 a^2 b^4 (15 A-11 C)+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 C+80 a^4 b B-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+a b^4 (45 A+17 C)-5 b^5 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {2 \left (-128 a^6 C+80 a^5 b B-4 a^4 b^2 (10 A-53 C)-140 a^3 b^3 B+5 a^2 b^4 (15 A-11 C)+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 C+80 a^4 b B-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+a b^4 (45 A+17 C)-5 b^5 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {2 \left (-128 a^6 C+80 a^5 b B-4 a^4 b^2 (10 A-53 C)-140 a^3 b^3 B+5 a^2 b^4 (15 A-11 C)+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 C+80 a^4 b B-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+a b^4 (45 A+17 C)-5 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {2 \left (-128 a^6 C+80 a^5 b B-4 a^4 b^2 (10 A-53 C)-140 a^3 b^3 B+5 a^2 b^4 (15 A-11 C)+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-128 a^5 C+80 a^4 b B-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+a b^4 (45 A+17 C)-5 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {-\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {-\frac {2 \sin (c+d x) \cos (c+d x) \left (-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt {a+b \cos (c+d x)}}{5 b d}-\frac {\frac {\frac {2 \left (-128 a^6 C+80 a^5 b B-4 a^4 b^2 (10 A-53 C)-140 a^3 b^3 B+5 a^2 b^4 (15 A-11 C)+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (-128 a^5 C+80 a^4 b B-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+a b^4 (45 A+17 C)-5 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \sin (c+d x) \left (-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}}{b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]
Output:
(-2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)* d*(a + b*Cos[c + d*x])^(3/2)) - ((-2*(6*A*b^4 + 5*a^3*b*B - 9*a*b^3*B - 2* a^2*b^2*(A - 6*C) - 8*a^4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d *Sqrt[a + b*Cos[c + d*x]]) - ((-2*(30*a^3*b*B - 50*a*b^3*B - a^2*b^2*(15*A - 71*C) + b^4*(35*A - 3*C) - 48*a^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d* x]]*Sin[c + d*x])/(5*b*d) - (((2*(80*a^5*b*B - 140*a^3*b^3*B + 40*a*b^5*B - 4*a^4*b^2*(10*A - 53*C) + 5*a^2*b^4*(15*A - 11*C) - 128*a^6*C - 3*b^6*(5 *A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)]) /(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(80*a^4*b*B - 8 0*a^2*b^3*B - 5*b^5*B - 4*a^3*b^2*(10*A - 29*C) - 128*a^5*C + a*b^4*(45*A + 17*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/( a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]))/b - (2*(40*a^4*b*B - 65*a^2*b^3*B + 5*b^5*B - 2*a^3*b^2*(10*A - 49*C) + 2*a*b^4*(20*A - 7*C) - 64*a^5*C)*Sq rt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d))/(5*b))/(b*(a^2 - b^2)))/(3*b*( a^2 - b^2))
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*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(1783\) vs. \(2(603)=1206\).
Time = 11.51 (sec) , antiderivative size = 1784, normalized size of antiderivative = 2.87
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1784\) |
| parts | \(\text {Expression too large to display}\) | \(3896\) |
Input:
int(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2),x, method=_RETURNVERBOSE)
Output:
-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*a^2/b^5 *(3*A*b^2-4*B*a*b+5*C*a^2)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2*b- a-b)/(a^2-b^2)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2 )*(2*b*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+(sin(1/2*d*x+1/2*c)^2)^(1/2 )*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d* x+1/2*c),(-2*b/(a-b))^(1/2))*a-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*si n(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b /(a-b))^(1/2)))-2*(2*A*a*b^2+A*b^3-3*B*a^2*b-2*B*a*b^2-B*b^3+4*C*a^3+3*C*a ^2*b+2*C*a*b^2+C*b^3)/b^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1 /2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2 *c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+16*C/b^2*(-1 /10/b*cos(1/2*d*x+1/2*c)^3*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/ 2*c)^2)^(1/2)-1/60/b^2*(-4*a+12*b)*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/ 2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+1/60/b^2*(-4*a+12*b)*(a-b)*(sin(1 /2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b* sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d *x+1/2*c),(-2*b/(a-b))^(1/2))-1/60*(4*a^2-15*a*b+27*b^2)/b^3*(a-b)*(sin(1/ 2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*s in(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d *x+1/2*c),(-2*b/(a-b))^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))...
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 1742, normalized size of antiderivative = 2.80 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5 /2),x, algorithm="fricas")
Output:
-2/45*(sqrt(1/2)*(-256*I*C*a^9 + 160*I*B*a^8*b - 40*I*(2*A - 13*C)*a^7*b^2 - 340*I*B*a^6*b^3 + 2*I*(90*A - 121*C)*a^5*b^4 + 185*I*B*a^4*b^5 - 6*I*(2 0*A + 7*C)*a^3*b^6 + 15*I*B*a^2*b^7 + (-256*I*C*a^7*b^2 + 160*I*B*a^6*b^3 - 40*I*(2*A - 13*C)*a^5*b^4 - 340*I*B*a^4*b^5 + 2*I*(90*A - 121*C)*a^3*b^6 + 185*I*B*a^2*b^7 - 6*I*(20*A + 7*C)*a*b^8 + 15*I*B*b^9)*cos(d*x + c)^2 + 2*(-256*I*C*a^8*b + 160*I*B*a^7*b^2 - 40*I*(2*A - 13*C)*a^6*b^3 - 340*I*B *a^5*b^4 + 2*I*(90*A - 121*C)*a^4*b^5 + 185*I*B*a^3*b^6 - 6*I*(20*A + 7*C) *a^2*b^7 + 15*I*B*a*b^8)*cos(d*x + c))*sqrt(b)*weierstrassPInverse(4/3*(4* a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I *b*sin(d*x + c) + 2*a)/b) + sqrt(1/2)*(256*I*C*a^9 - 160*I*B*a^8*b + 40*I* (2*A - 13*C)*a^7*b^2 + 340*I*B*a^6*b^3 - 2*I*(90*A - 121*C)*a^5*b^4 - 185* I*B*a^4*b^5 + 6*I*(20*A + 7*C)*a^3*b^6 - 15*I*B*a^2*b^7 + (256*I*C*a^7*b^2 - 160*I*B*a^6*b^3 + 40*I*(2*A - 13*C)*a^5*b^4 + 340*I*B*a^4*b^5 - 2*I*(90 *A - 121*C)*a^3*b^6 - 185*I*B*a^2*b^7 + 6*I*(20*A + 7*C)*a*b^8 - 15*I*B*b^ 9)*cos(d*x + c)^2 + 2*(256*I*C*a^8*b - 160*I*B*a^7*b^2 + 40*I*(2*A - 13*C) *a^6*b^3 + 340*I*B*a^5*b^4 - 2*I*(90*A - 121*C)*a^4*b^5 - 185*I*B*a^3*b^6 + 6*I*(20*A + 7*C)*a^2*b^7 - 15*I*B*a*b^8)*cos(d*x + c))*sqrt(b)*weierstra ssPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b* cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3*sqrt(1/2)*(-128*I*C*a^8*b + 80*I*B*a^7*b^2 - 4*I*(10*A - 53*C)*a^6*b^3 - 140*I*B*a^5*b^4 + 5*I*(1...
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))* *(5/2),x)
Output:
Timed out
\[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5 /2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^3/(b*cos(d* x + c) + a)^(5/2), x)
\[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5 /2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^3/(b*cos(d* x + c) + a)^(5/2), x)
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:
int((cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(5/2),x)
Output:
int((cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(5/2), x)
\[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{5}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) a \] Input:
int(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2),x)
Output:
int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**5)/(cos(c + d*x)**3*b**3 + 3*c os(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*c + int((sqrt(cos (c + d*x)*b + a)*cos(c + d*x)**4)/(cos(c + d*x)**3*b**3 + 3*cos(c + d*x)** 2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*b + int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3)/(cos(c + d*x)**3*b**3 + 3*cos(c + d*x)**2*a*b**2 + 3* cos(c + d*x)*a**2*b + a**3),x)*a