Integrand size = 43, antiderivative size = 540 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a b d}+\frac {\sqrt {a+b} \left (3 A b^2+4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a d}+\frac {\sqrt {a+b} \left (A b^3-8 a^3 B-6 a b^2 B-12 a^2 b (A+2 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a^2 d}+\frac {\left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac {(A b+2 a B) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d} \] Output:
1/24*(a-b)*(a+b)^(1/2)*(3*A*b^2+30*B*a*b+8*a^2*(2*A+3*C))*cot(d*x+c)*Ellip ticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x +c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a/b/d+1/24*(a+b)^(1/2)*( 3*A*b^2+4*a^2*(4*A+3*B+6*C)+2*a*b*(7*A+15*B+24*C))*cot(d*x+c)*EllipticF((a +b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a +b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a/d+1/8*(a+b)^(1/2)*(A*b^3-8*B* a^3-6*B*a*b^2-12*a^2*b*(A+2*C))*cot(d*x+c)*EllipticPi((a+b*sec(d*x+c))^(1/ 2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2) *(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/d+1/24*(3*A*b^2+30*B*a*b+8*a^2*(2*A+3 *C))*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a/d+1/4*(A*b+2*B*a)*cos(d*x+c)*(a+b *sec(d*x+c))^(1/2)*sin(d*x+c)/d+1/3*A*cos(d*x+c)^2*(a+b*sec(d*x+c))^(3/2)* sin(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(5040\) vs. \(2(540)=1080\).
Time = 25.05 (sec) , antiderivative size = 5040, normalized size of antiderivative = 9.33 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
Result too large to show
Time = 2.57 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.395, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4592, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}
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.
| \(\displaystyle \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{2} \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (A+6 C) \sec ^2(c+d x)+2 (2 a A+3 b B+3 a C) \sec (c+d x)+3 (A b+2 a B)\right )dx+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (A+6 C) \sec ^2(c+d x)+2 (2 a A+3 b B+3 a C) \sec (c+d x)+3 (A b+2 a B)\right )dx+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (A+6 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 (2 a A+3 b B+3 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 (A b+2 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {\cos (c+d x) \left (8 (2 A+3 C) a^2+30 b B a+3 A b^2+b (7 A b+24 C b+6 a B) \sec ^2(c+d x)+2 \left (6 B a^2+b (13 A+24 C) a+12 b^2 B\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \frac {\cos (c+d x) \left (8 (2 A+3 C) a^2+30 b B a+3 A b^2+b (7 A b+24 C b+6 a B) \sec ^2(c+d x)+2 \left (6 B a^2+b (13 A+24 C) a+12 b^2 B\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \frac {8 (2 A+3 C) a^2+30 b B a+3 A b^2+b (7 A b+24 C b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (6 B a^2+b (13 A+24 C) a+12 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (8 (2 A+3 C) a^2+30 b B a+3 A b^2\right ) \sec ^2(c+d x)-2 a b (7 A b+24 C b+6 a B) \sec (c+d x)+3 \left (-8 B a^3-12 b (A+2 C) a^2-6 b^2 B a+A b^3\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (8 (2 A+3 C) a^2+30 b B a+3 A b^2\right ) \sec ^2(c+d x)-2 a b (7 A b+24 C b+6 a B) \sec (c+d x)+3 \left (-8 B a^3-12 b (A+2 C) a^2-6 b^2 B a+A b^3\right )}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (8 (2 A+3 C) a^2+30 b B a+3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a b (7 A b+24 C b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-8 B a^3-12 b (A+2 C) a^2-6 b^2 B a+A b^3\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {3 \left (-8 B a^3-12 b (A+2 C) a^2-6 b^2 B a+A b^3\right )+\left (-2 a b (7 A b+24 C b+6 a B)-b \left (8 (2 A+3 C) a^2+30 b B a+3 A b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {3 \left (-8 B a^3-12 b (A+2 C) a^2-6 b^2 B a+A b^3\right )+\left (-2 a b (7 A b+24 C b+6 a B)-b \left (8 (2 A+3 C) a^2+30 b B a+3 A b^2\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)+3 A b^2\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+3 \left (-8 a^3 B-12 a^2 b (A+2 C)-6 a b^2 B+A b^3\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-b \left (4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)+3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (-8 a^3 B-12 a^2 b (A+2 C)-6 a b^2 B+A b^3\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-b \left (4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)+3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-8 a^3 B-12 a^2 b (A+2 C)-6 a b^2 B+A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \cot (c+d x) \left (4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)+3 A b^2\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-8 a^3 B-12 a^2 b (A+2 C)-6 a b^2 B+A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-\frac {2 \sqrt {a+b} \cot (c+d x) \left (4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)+3 A b^2\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-8 a^3 B-12 a^2 b (A+2 C)-6 a b^2 B+A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
Input:
Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[ c + d*x]^2),x]
Output:
(A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((3*(A* b + 2*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (-1 /2*((-2*(a - b)*Sqrt[a + b]*(3*A*b^2 + 30*a*b*B + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/ (a - b))])/(b*d) - (2*Sqrt[a + b]*(3*A*b^2 + 4*a^2*(4*A + 3*B + 6*C) + 2*a *b*(7*A + 15*B + 24*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d *x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*S qrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (6*Sqrt[a + b]*(A*b^3 - 8*a^3* B - 6*a*b^2*B - 12*a^2*b*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, Arc Sin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - S ec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d))/a + ((3*A*b^2 + 30*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d))/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[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) *((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt [a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[d Int[Csc[e + f*x]/Sqrt[a + b*Csc[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[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C )*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C Int[Csc[e + f*x]*(( 1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A , B, C}, x] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(2328\) vs. \(2(495)=990\).
Time = 15.98 (sec) , antiderivative size = 2329, normalized size of antiderivative = 4.31
Input:
int(cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
Output:
1/24/d/a*(sin(d*x+c)*cos(d*x+c)^2*(8*cos(d*x+c)^2+8*cos(d*x+c)+16)*a^3*A+( -16*cos(d*x+c)^2-32*cos(d*x+c)-16)*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c) +1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^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)*C*(1/(a+b )*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2) *a^3*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+sin(d*x+c)*cos( d*x+c)*(22*cos(d*x+c)^2+22*cos(d*x+c)+16)*A*a^2*b+sin(d*x+c)*cos(d*x+c)*(1 7*cos(d*x+c)+14)*a*A*b^2+sin(d*x+c)*cos(d*x+c)*(42*cos(d*x+c)+12)*B*a^2*b+ (-48*cos(d*x+c)^2-96*cos(d*x+c)-48)*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c )+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^3*EllipticPi(-csc(d*x+c)+c ot(d*x+c),-1,((a-b)/(a+b))^(1/2))+(24*cos(d*x+c)^2+48*cos(d*x+c)+24)*B*(1/ (a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^( 1/2)*a^3*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(-3*cos(d*x +c)^2-6*cos(d*x+c)-3)*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(c os(d*x+c)/(cos(d*x+c)+1))^(1/2)*a*b^2*EllipticE(-csc(d*x+c)+cot(d*x+c),((a -b)/(a+b))^(1/2))+24*C*a^2*b*cos(d*x+c)*sin(d*x+c)+30*B*a*b^2*cos(d*x+c)*s in(d*x+c)+(-72*cos(d*x+c)^2-144*cos(d*x+c)-72)*A*(1/(a+b)*(b+a*cos(d*x+c)) /(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^2*b*EllipticPi( -csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+(-144*cos(d*x+c)^2-288*cos( d*x+c)-144)*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+...
Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*(a+b*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+ c)**2),x)
Output:
Timed out
\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{3} \,d x } \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/ 2)*cos(d*x + c)^3, x)
\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{3} \,d x } \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/ 2)*cos(d*x + c)^3, x)
Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^3\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \] Input:
int(cos(c + d*x)^3*(a + b/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/cos( c + d*x)^2),x)
Output:
int(cos(c + d*x)^3*(a + b/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/cos( c + d*x)^2), x)
\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}d x \right ) b c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{2}d x \right ) a c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{2}d x \right ) b^{2}+2 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )d x \right ) a b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) a^{2} \] Input:
int(cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**3,x)*b*c + int( sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**2,x)*a*c + int(sqrt (sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**2,x)*b**2 + 2*int(sqrt( sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x),x)*a*b + int(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3,x)*a**2