Integrand size = 45, antiderivative size = 707 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=-\frac {(a-b) \sqrt {a+b} \left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{24 a d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (a^2 (48 A-48 B-33 C)-4 b^2 (6 A+3 B+4 C)-2 a b (72 A+27 B+13 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{24 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{8 b d \sqrt {\sec (c+d x)}}-\frac {b (8 a A-2 b B-3 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 d \sqrt {\sec (c+d x)}}-\frac {b (6 A-C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {\left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{24 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)} \sin (c+d x)}{d} \] Output:
-1/24*(a-b)*(a+b)^(1/2)*(54*B*a*b-a^2*(48*A-33*C)+8*b^2*(3*A+2*C))*cos(d*x +c)^(1/2)*csc(d*x+c)*EllipticE((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/sec(d*x+c)^(1/2)-1/24*(a+b)^(1/2)*(a^2*(48*A-48*B-3 3*C)-4*b^2*(6*A+3*B+4*C)-2*a*b*(72*A+27*B+13*C))*cos(d*x+c)^(1/2)*csc(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/sec(d*x+c)^(1/2)-1/8*(a+b)^(1/2)*(30*B*a^2*b+8*B*b^3+5*a^3*C+20*a*b^2*( 2*A+C))*cos(d*x+c)^(1/2)*csc(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)/b/d/sec(d*x+c)^(1/2)-1/4*b*(8*A *a-2*B*b-3*C*a)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/d/sec(d*x+c)^(1/2)-1/3*b *(6*A-C)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+1/24*(54*B*a *b-a^2*(48*A-33*C)+8*b^2*(3*A+2*C))*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(1/2 )*sin(d*x+c)/d+2*A*(a+b*cos(d*x+c))^(5/2)*sec(d*x+c)^(1/2)*sin(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(1920\) vs. \(2(707)=1414\).
Time = 27.56 (sec) , antiderivative size = 1920, normalized size of antiderivative = 2.72 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2)*Sec[c + d*x]^(3/2),x]
Output:
(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((24*a^2*A + b^2*C)*Sin[c + d*x])/12 + (b*(6*b*B + 13*a*C)*Sin[2*(c + d*x)])/24 + (b^2*C*Sin[3*(c + d* x)])/12))/d + (48*a^3*A*Tan[(c + d*x)/2] + 48*a^2*A*b*Tan[(c + d*x)/2] - 2 4*a*A*b^2*Tan[(c + d*x)/2] - 24*A*b^3*Tan[(c + d*x)/2] - 54*a^2*b*B*Tan[(c + d*x)/2] - 54*a*b^2*B*Tan[(c + d*x)/2] - 33*a^3*C*Tan[(c + d*x)/2] - 33* a^2*b*C*Tan[(c + d*x)/2] - 16*a*b^2*C*Tan[(c + d*x)/2] - 16*b^3*C*Tan[(c + d*x)/2] - 96*a^2*A*b*Tan[(c + d*x)/2]^3 + 48*A*b^3*Tan[(c + d*x)/2]^3 + 1 08*a*b^2*B*Tan[(c + d*x)/2]^3 + 66*a^2*b*C*Tan[(c + d*x)/2]^3 + 32*b^3*C*T an[(c + d*x)/2]^3 - 48*a^3*A*Tan[(c + d*x)/2]^5 + 48*a^2*A*b*Tan[(c + d*x) /2]^5 + 24*a*A*b^2*Tan[(c + d*x)/2]^5 - 24*A*b^3*Tan[(c + d*x)/2]^5 + 54*a ^2*b*B*Tan[(c + d*x)/2]^5 - 54*a*b^2*B*Tan[(c + d*x)/2]^5 + 33*a^3*C*Tan[( c + d*x)/2]^5 - 33*a^2*b*C*Tan[(c + d*x)/2]^5 + 16*a*b^2*C*Tan[(c + d*x)/2 ]^5 - 16*b^3*C*Tan[(c + d*x)/2]^5 - 240*a*A*b^2*EllipticPi[-1, ArcSin[Tan[ (c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 180*a^2*b*B*Elli pticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d *x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 48*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]* Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2...
Time = 4.05 (sec) , antiderivative size = 676, normalized size of antiderivative = 0.96, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.489, Rules used = {3042, 4709, 3042, 3526, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3540, 25, 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.
| \(\displaystyle \int \sec ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{3/2} (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^{5/2} \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (2 \int \frac {(a+b \cos (c+d x))^{3/2} \left (-b (6 A-C) \cos ^2(c+d x)+(b B-a (A-C)) \cos (c+d x)+5 A b+a B\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\int \frac {(a+b \cos (c+d x))^{3/2} \left (-b (6 A-C) \cos ^2(c+d x)+(b B-a (A-C)) \cos (c+d x)+5 A b+a B\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-b (6 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(b B-a (A-C)) \sin \left (c+d x+\frac {\pi }{2}\right )+5 A b+a B\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \int \frac {\sqrt {a+b \cos (c+d x)} \left (-3 b (8 a A-2 b B-3 a C) \cos ^2(c+d x)+2 \left (-3 (A-C) a^2+6 b B a+b^2 (3 A+2 C)\right ) \cos (c+d x)+a (24 A b+C b+6 a B)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \int \frac {\sqrt {a+b \cos (c+d x)} \left (-3 b (8 a A-2 b B-3 a C) \cos ^2(c+d x)+2 \left (-3 (A-C) a^2+6 b B a+b^2 (3 A+2 C)\right ) \cos (c+d x)+a (24 A b+C b+6 a B)\right )}{\sqrt {\cos (c+d x)}}dx-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (-3 b (8 a A-2 b B-3 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (-3 (A-C) a^2+6 b B a+b^2 (3 A+2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (24 A b+C b+6 a B)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {b \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right ) \cos ^2(c+d x)+2 \left (-12 (A-C) a^3+36 b B a^2+b^2 (36 A+19 C) a+6 b^3 B\right ) \cos (c+d x)+a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right )}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {b \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right ) \cos ^2(c+d x)+2 \left (-12 (A-C) a^3+36 b B a^2+b^2 (36 A+19 C) a+6 b^3 B\right ) \cos (c+d x)+a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right )}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {b \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (-12 (A-C) a^3+36 b B a^2+b^2 (36 A+19 C) a+6 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\int -\frac {-3 b \left (5 C a^3+30 b B a^2+20 b^2 (2 A+C) a+8 b^3 B\right ) \cos ^2(c+d x)-2 a b \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \cos (c+d x)+a b \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}+\frac {\sin (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-3 b \left (5 C a^3+30 b B a^2+20 b^2 (2 A+C) a+8 b^3 B\right ) \cos ^2(c+d x)-2 a b \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \cos (c+d x)+a b \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-3 b \left (5 C a^3+30 b B a^2+20 b^2 (2 A+C) a+8 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a b \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a b \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )-2 a b \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-3 b \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a b \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )-2 a b \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-3 b \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3288 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a b \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )-2 a b \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {a+b} \cot (c+d x) \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {a b \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+a b \left (a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {6 \sqrt {a+b} \cot (c+d x) \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {a b \left (a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a b \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {a+b} \cot (c+d x) \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {a b \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sqrt {a+b} \cot (c+d x) \left (a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {6 \sqrt {a+b} \cot (c+d x) \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {\frac {2 b \sqrt {a+b} \cot (c+d x) \left (a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {2 b (a-b) \sqrt {a+b} \cot (c+d x) \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}+\frac {6 \sqrt {a+b} \cot (c+d x) \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}\right )-\frac {3 b \sin (c+d x) \sqrt {\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {b (6 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt {\cos (c+d x)}}\right )\) |
Input:
Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec [c + d*x]^(3/2),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1/3*(b*(6*A - C)*Sqrt[Cos[c + d*x] ]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/d + (2*A*(a + b*Cos[c + d*x])^( 5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ((-3*b*(8*a*A - 2*b*B - 3*a*C) *Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (-1/2*( (2*(a - b)*b*Sqrt[a + b]*(54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*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^2*(48*A - 48*B - 33*C) - 4*b^2*(6*A + 3*B + 4*C) - 2*a*b*(72*A + 27*B + 13*C))*Co t[c + d*x]*EllipticF[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)]*Sqr t[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (6*Sqrt[a + b]*(30*a^2*b*B + 8*b^3* B + 5*a^3*C + 20*a*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSi n[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)/b + ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]))/4)/6)
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]
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(2901\) vs. \(2(634)=1268\).
Time = 33.79 (sec) , antiderivative size = 2902, normalized size of antiderivative = 4.10
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2902\) |
| parts | \(\text {Expression too large to display}\) | \(2990\) |
Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(3/2 ),x,method=_RETURNVERBOSE)
Output:
-1/24/d*((48*cos(d*x+c)^2+96*cos(d*x+c)+48)*B*(1/(a+b)*(a+b*cos(d*x+c))/(1 +cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^3*EllipticPi(-csc( d*x+c)+cot(d*x+c),-1,(-(a-b)/(a+b))^(1/2))+(24*cos(d*x+c)^2+48*cos(d*x+c)+ 24)*A*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d *x+c)))^(1/2)*b^3*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))+( 16*cos(d*x+c)^2+32*cos(d*x+c)+16)*C*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c )))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^3*EllipticE(-csc(d*x+c)+cot( d*x+c),(-(a-b)/(a+b))^(1/2))+(-24*cos(d*x+c)^2-48*cos(d*x+c)-24)*B*(1/(a+b )*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2) *b^3*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))+(24*cos(d*x+c) ^2+48*cos(d*x+c)+24)*A*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(co s(d*x+c)/(1+cos(d*x+c)))^(1/2)*a*b^2*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a -b)/(a+b))^(1/2))-48*A*a^2*b*cos(d*x+c)*sin(d*x+c)+(30*cos(d*x+c)^2+60*cos (d*x+c)+30)*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*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,(-(a-b)/(a+ b))^(1/2))+sin(d*x+c)*cos(d*x+c)*(-66*cos(d*x+c)-12)*B*a*b^2+sin(d*x+c)*co s(d*x+c)*(-59*cos(d*x+c)-26)*a^2*b*C+sin(d*x+c)*cos(d*x+c)*(-34*cos(d*x+c) ^2-34*cos(d*x+c)-16)*C*a*b^2+(48*cos(d*x+c)^2+96*cos(d*x+c)+48)*A*(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*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))+(48*cos(d*x+...
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(3/2),x, algorithm="fricas")
Output:
integral((C*b^2*cos(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^3 + A*a^2 + (C*a^2 + 2*B*a*b + A*b^2)*cos(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c ))*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(3/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x +c)**(3/2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(3/2),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)*sec(d*x + c)^(3/2), x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(3/2),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)*sec(d*x + c)^(3/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int((1/cos(c + d*x))^(3/2)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
int((1/cos(c + d*x))^(3/2)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) b^{2} c +2 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )d x \right ) a b c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )d x \right ) b^{3}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )d x \right ) a^{2} c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )d x \right ) a \,b^{2}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )d x \right ) a^{3} \] Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(3/2 ),x)
Output:
3*int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x ),x)*a**2*b + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x) **4*sec(c + d*x),x)*b**2*c + 2*int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x),x)*a*b*c + int(sqrt(sec(c + d*x))*sqrt(c os(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x),x)*b**3 + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x),x)*a**2*c + 3 *int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d *x),x)*a*b**2 + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*sec(c + d* x),x)*a**3