Integrand size = 45, antiderivative size = 622 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {(a-b) \sqrt {a+b} \left (24 a^2 B-12 b^2 B+a b (56 A-27 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{12 a d}-\frac {\sqrt {a+b} \left (a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)-8 a^2 (A-3 B+3 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{12 d}-\frac {\sqrt {a+b} \left (8 A b^2+20 a b B+15 a^2 C+4 b^2 C\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{4 d}-\frac {\left (24 a^2 B-12 b^2 B+a b (56 A-27 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}-\frac {b (8 A b+4 a B-b C) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac {2 (5 A b+3 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)} \] Output:
1/12*(a-b)*(a+b)^(1/2)*(24*B*a^2-12*B*b^2+a*b*(56*A-27*C))*cot(d*x+c)*Elli pticE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^( 1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a/d-1/ 12*(a+b)^(1/2)*(a*b*(56*A-72*B-27*C)-6*b^2*(12*A+2*B+C)-8*a^2*(A-3*B+3*C)) *cot(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2), (-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a- b))^(1/2)/d-1/4*(a+b)^(1/2)*(8*A*b^2+20*B*a*b+15*C*a^2+4*C*b^2)*cot(d*x+c) *EllipticPi((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(a+b)/b,(- (a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b) )^(1/2)/d-1/12*(24*B*a^2-12*B*b^2+a*b*(56*A-27*C))*(a+b*cos(d*x+c))^(1/2)* sin(d*x+c)/d/cos(d*x+c)^(1/2)-1/2*b*(8*A*b+4*B*a-C*b)*cos(d*x+c)^(1/2)*(a+ b*cos(d*x+c))^(1/2)*sin(d*x+c)/d+2/3*(5*A*b+3*B*a)*(a+b*cos(d*x+c))^(3/2)* sin(d*x+c)/d/cos(d*x+c)^(1/2)+2/3*A*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/d/co s(d*x+c)^(3/2)
Result contains complex when optimal does not.
Time = 7.59 (sec) , antiderivative size = 1316, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x] ^2))/Cos[c + d*x]^(5/2),x]
Output:
((-4*a*(8*a^3*A + 16*a*A*b^2 + 48*a^2*b*B + 12*b^3*B + 24*a^3*C + 33*a*b^2 *C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d* x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/ a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/ 2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[ c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-56*a^2*A*b + 24*A*b^3 - 24*a^3 *B + 72*a*b^2*B + 72*a^2*b*C + 12*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^ 2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sq rt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)] *Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d* x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/ a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[( c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[C os[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-56*a*A*b^2 - 24*a^2*b*B + 12 *b^3*B + 27*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*Ellipti cE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x] )*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/...
Time = 3.74 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.02, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.422, Rules used = {3042, 3526, 27, 3042, 3526, 27, 3042, 3528, 27, 3042, 3540, 3042, 3532, 3042, 3288, 3477, 3042, 3295, 3473}
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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {2}{3} \int \frac {(a+b \cos (c+d x))^{3/2} \left (-b (4 A-3 C) \cos ^2(c+d x)+(3 b B+a (A+3 C)) \cos (c+d x)+5 A b+3 a B\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {(a+b \cos (c+d x))^{3/2} \left (-b (4 A-3 C) \cos ^2(c+d x)+(3 b B+a (A+3 C)) \cos (c+d x)+5 A b+3 a B\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-b (4 A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(3 b B+a (A+3 C)) \sin \left (c+d x+\frac {\pi }{2}\right )+5 A b+3 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{3} \left (2 \int \frac {\sqrt {a+b \cos (c+d x)} \left ((A+3 C) a^2+12 b B a+15 A b^2-3 b (8 A b-C b+4 a B) \cos ^2(c+d x)-\left (3 B a^2+8 A b a-6 b C a-3 b^2 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {\sqrt {a+b \cos (c+d x)} \left ((A+3 C) a^2+12 b B a+15 A b^2-3 b (8 A b-C b+4 a B) \cos ^2(c+d x)-\left (3 B a^2+8 A b a-6 b C a-3 b^2 B\right ) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left ((A+3 C) a^2+12 b B a+15 A b^2-3 b (8 A b-C b+4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (-3 B a^2-8 A b a+6 b C a+3 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {-b \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right ) \cos ^2(c+d x)-2 \left (6 B a^3+2 b (7 A-9 C) a^2-18 b^2 B a-3 b^3 (2 A+C)\right ) \cos (c+d x)+a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right )}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \int \frac {-b \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right ) \cos ^2(c+d x)-2 \left (6 B a^3+2 b (7 A-9 C) a^2-18 b^2 B a-3 b^3 (2 A+C)\right ) \cos (c+d x)+a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right )}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \int \frac {-b \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 \left (6 B a^3+2 b (7 A-9 C) a^2-18 b^2 B a-3 b^3 (2 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {\int \frac {3 b^2 \left (15 C a^2+20 b B a+8 A b^2+4 b^2 C\right ) \cos ^2(c+d x)+2 a b \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \cos (c+d x)+a b \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}-\frac {\sin (c+d x) \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {\int \frac {3 b^2 \left (15 C a^2+20 b B a+8 A b^2+4 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a b \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\sin (c+d x) \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {\int \frac {a b \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )+2 a b \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+3 b^2 \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx}{2 b}-\frac {\sin (c+d x) \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {3 b^2 \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {a b \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )+2 a b \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\sin (c+d x) \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3288 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {\int \frac {a b \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )+2 a b \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b \sqrt {a+b} \cot (c+d x) \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {\sin (c+d x) \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {a b \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-a b \left (-8 a^2 (A-3 B+3 C)+a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {6 b \sqrt {a+b} \cot (c+d x) \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {\sin (c+d x) \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {-a b \left (-8 a^2 (A-3 B+3 C)+a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a b \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b \sqrt {a+b} \cot (c+d x) \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {\sin (c+d x) \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {a b \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b \sqrt {a+b} \cot (c+d x) \left (-8 a^2 (A-3 B+3 C)+a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}-\frac {6 b \sqrt {a+b} \cot (c+d x) \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {\sin (c+d x) \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {-\frac {2 b \sqrt {a+b} \cot (c+d x) \left (-8 a^2 (A-3 B+3 C)+a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {2 b (a-b) \sqrt {a+b} \cot (c+d x) \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}-\frac {6 b \sqrt {a+b} \cot (c+d x) \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {\sin (c+d x) \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (4 a B+8 A b-b C) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
Input:
Int[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/C os[c + d*x]^(5/2),x]
Output:
(2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + ( (-3*b*(8*A*b + 4*a*B - b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Si n[c + d*x])/(2*d) + (2*(5*A*b + 3*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (((2*(a - b)*b*Sqrt[a + b]*(24*a^2*B - 12*b ^2*B + a*b*(56*A - 27*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2* b*Sqrt[a + b]*(a*b*(56*A - 72*B - 27*C) - 6*b^2*(12*A + 2*B + C) - 8*a^2*( A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sq rt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d *x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (6*b*Sqrt[a + b]* (8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/ b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d)/(2*b) - ((24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Sqr t[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]))/4)/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c *((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti cPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
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])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*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]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f *x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d) Int[(1/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*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]
Leaf count of result is larger than twice the leaf count of optimal. \(2775\) vs. \(2(559)=1118\).
Time = 66.95 (sec) , antiderivative size = 2776, normalized size of antiderivative = 4.46
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2776\) |
| default | \(\text {Expression too large to display}\) | \(2858\) |
Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2 ),x,method=_RETURNVERBOSE)
Output:
-2/3*A/d*((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+c os(d*x+c)))^(1/2)*b^3*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,(-(a-b)/(a+b))^( 1/2))*(6*cos(d*x+c)^3+12*cos(d*x+c)^2+6*cos(d*x+c))+(cos(d*x+c)/(1+cos(d*x +c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b*Elliptic E(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(-7*cos(d*x+c)^3-14*cos(d*x+ c)^2-7*cos(d*x+c))+(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x +c))/(1+cos(d*x+c)))^(1/2)*a*b^2*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/( a+b))^(1/2))*(-7*cos(d*x+c)^3-14*cos(d*x+c)^2-7*cos(d*x+c))+(cos(d*x+c)/(1 +cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*El lipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(cos(d*x+c)^3+2*cos(d* x+c)^2+cos(d*x+c))+(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x +c))/(1+cos(d*x+c)))^(1/2)*a^2*b*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/( a+b))^(1/2))*(7*cos(d*x+c)^3+14*cos(d*x+c)^2+7*cos(d*x+c))+(cos(d*x+c)/(1+ cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^2*E llipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(9*cos(d*x+c)^3+18*co s(d*x+c)^2+9*cos(d*x+c))+(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(a+b*c os(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^3*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a- b)/(a+b))^(1/2))*(-3*cos(d*x+c)^3-6*cos(d*x+c)^2-3*cos(d*x+c))+(-1-cos(d*x +c))*sin(d*x+c)*a^3+sin(d*x+c)*cos(d*x+c)*(-cos(d*x+c)-8)*b*a^2-7*a*b^2*co s(d*x+c)^2*sin(d*x+c))*(a+b*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2)/(b*cos(d...
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c )^(5/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x +c)**(5/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c )^(5/2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)/cos(d*x + c)^(5/2), x)
\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c )^(5/2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)/cos(d*x + c)^(5/2), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int(((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/c os(c + d*x)^(5/2),x)
Output:
int(((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/c os(c + d*x)^(5/2), x)
\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a^{2} c +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a \,b^{2}+\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{3}+3 \left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{2} b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) b^{2} c +2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}d x \right ) a b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}d x \right ) b^{3} \] Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2 ),x)
Output:
int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d*x),x)*a**2*c + 3*int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d*x),x)*a*b** 2 + int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d*x)**3,x)*a **3 + 3*int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d*x)**2, x)*a**2*b + int(sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x),x )*b**2*c + 2*int(sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)),x)*a*b*c + in t(sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)),x)*b**3