Integrand size = 35, antiderivative size = 645 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=-\frac {\left (105 A b^4+a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{12 a^4 \sqrt {a+b} \left (a^2-b^2\right ) d}+\frac {\left (35 a A b^3+105 A b^4+6 a^4 (A-8 C)-3 a^2 b^2 (45 A-8 C)-a^3 (27 A b-8 b C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{12 a^4 \sqrt {a+b} \left (a^2-b^2\right ) d}-\frac {\sqrt {a+b} \left (35 A b^2+4 a^2 (A+2 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{4 a^5 d}-\frac {7 A b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac {A \cos (c+d x) \sin (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}+\frac {b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {b^2 \left (105 A b^4+a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)\right ) \tan (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}} \]
-7/4*A*b*sin(d*x+c)/a^2/d/(a+b*sec(d*x+c))^(3/2)+1/2*A*cos(d*x+c)*sin(d*x+ c)/a/d/(a+b*sec(d*x+c))^(3/2)-1/12*(105*A*b^4+a^4*(33*A-56*C)-2*a^2*b^2*(8 5*A-12*C))*cot(d*x+c)*EllipticE((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^4/(a^2-b^2)/d/(a+b)^(1/2)+1/12*(35*a*A*b^3+105*A*b^4+6*a^4*(A-8*C)-3* a^2*b^2*(45*A-8*C)-a^3*(27*A*b-8*C*b))*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^4/(a^2-b^2)/d/(a+b)^(1/2)-1/4*(35*A*b^2+4 *a^2*(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))*(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^5/d+1/12*b^2*(35*A*b^2-a^2*(27*A-8*C))*tan(d* x+c)/a^3/(a^2-b^2)/d/(a+b*sec(d*x+c))^(3/2)-1/12*b^2*(105*A*b^4+a^4*(33*A- 56*C)-2*a^2*b^2*(85*A-12*C))*tan(d*x+c)/a^4/(a^2-b^2)^2/d/(a+b*sec(d*x+c)) ^(1/2)
Time = 18.46 (sec) , antiderivative size = 799, normalized size of antiderivative = 1.24 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\frac {1}{2} \left (\frac {(b+a \cos (c+d x))^3 \sec ^3(c+d x) \left (\frac {4 b \left (-13 a^2 A b^2+9 A b^4-7 a^4 C+3 a^2 b^2 C\right ) \sin (c+d x)}{3 a^4 \left (-a^2+b^2\right )^2}-\frac {4 \left (A b^5 \sin (c+d x)+a^2 b^3 C \sin (c+d x)\right )}{3 a^4 \left (a^2-b^2\right ) (b+a \cos (c+d x))^2}-\frac {8 \left (-7 a^2 A b^4 \sin (c+d x)+5 A b^6 \sin (c+d x)-4 a^4 b^2 C \sin (c+d x)+2 a^2 b^4 C \sin (c+d x)\right )}{3 a^4 \left (a^2-b^2\right )^2 (b+a \cos (c+d x))}+\frac {A \sin (2 (c+d x))}{2 a^3}\right )}{d (a+b \sec (c+d x))^{5/2}}-\frac {(b+a \cos (c+d x))^2 \sec (c+d x) \left (a b (a+b) \left (105 A b^4+a^4 (33 A-56 C)+2 a^2 b^2 (-85 A+12 C)\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-b (a+b) \left (210 a A b^4-105 A b^5+2 a^2 b^3 (29 A-12 C)+12 a^3 b^2 (-19 A+4 C)-6 a^5 (A+12 C)+a^4 b (39 A+16 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+3 (a-b)^2 (a+b)^2 \left (35 A b^2+4 a^2 (A+2 C)\right ) \left ((a-b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )-2 a \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+a b \left (105 A b^4+a^4 (33 A-56 C)+2 a^2 b^2 (-85 A+12 C)\right ) (b+a \cos (c+d x)) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sec (c+d x) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^5 \left (a^2-b^2\right )^2 d \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} (a+b \sec (c+d x))^{5/2}}\right ) \]
(((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((4*b*(-13*a^2*A*b^2 + 9*A*b^4 - 7 *a^4*C + 3*a^2*b^2*C)*Sin[c + d*x])/(3*a^4*(-a^2 + b^2)^2) - (4*(A*b^5*Sin [c + d*x] + a^2*b^3*C*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)*(b + a*Cos[c + d*x ])^2) - (8*(-7*a^2*A*b^4*Sin[c + d*x] + 5*A*b^6*Sin[c + d*x] - 4*a^4*b^2*C *Sin[c + d*x] + 2*a^2*b^4*C*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(b + a*Cos [c + d*x])) + (A*Sin[2*(c + d*x)])/(2*a^3)))/(d*(a + b*Sec[c + d*x])^(5/2) ) - ((b + a*Cos[c + d*x])^2*Sec[c + d*x]*(a*b*(a + b)*(105*A*b^4 + a^4*(33 *A - 56*C) + 2*a^2*b^2*(-85*A + 12*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d *x)/2]^2)/(a + b)] - b*(a + b)*(210*a*A*b^4 - 105*A*b^5 + 2*a^2*b^3*(29*A - 12*C) + 12*a^3*b^2*(-19*A + 4*C) - 6*a^5*(A + 12*C) + a^4*b*(39*A + 16*C ))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 *Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(a - b)^2*(a + b)^2*(35*A*b^2 + 4*a^2*(A + 2*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x )/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x) /2]^2)/(a + b)] + a*b*(105*A*b^4 + a^4*(33*A - 56*C) + 2*a^2*b^2*(-85*A + 12*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(6*a^5*(a^2 - b^2)^2*d*(Cos[c + d*x]*Sec[(c + d* x)/2]^2)^(3/2)*(a + b*Sec[c + d*x])^(5/2)))/2
Time = 3.15 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.05, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4593, 27, 3042, 4592, 27, 3042, 4548, 27, 3042, 4548, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4593 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {\cos (c+d x) \left (-5 A b \sec ^2(c+d x)-2 a (A+2 C) \sec (c+d x)+7 A b\right )}{2 (a+b \sec (c+d x))^{5/2}}dx}{2 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {\cos (c+d x) \left (-5 A b \sec ^2(c+d x)-2 a (A+2 C) \sec (c+d x)+7 A b\right )}{(a+b \sec (c+d x))^{5/2}}dx}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {-5 A b \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a (A+2 C) \csc \left (c+d x+\frac {\pi }{2}\right )+7 A b}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{4 a}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {4 (A+2 C) a^2+10 A b \sec (c+d x) a+35 A b^2-21 A b^2 \sec ^2(c+d x)}{2 (a+b \sec (c+d x))^{5/2}}dx}{a}}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {4 (A+2 C) a^2+10 A b \sec (c+d x) a+35 A b^2-21 A b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}}dx}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {4 (A+2 C) a^2+10 A b \csc \left (c+d x+\frac {\pi }{2}\right ) a+35 A b^2-21 A b^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}-\frac {2 \int -\frac {b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \sec ^2(c+d x)-6 a b \left (7 A b^2-a^2 (3 A-4 C)\right ) \sec (c+d x)+3 \left (a^2-b^2\right ) \left (4 (A+2 C) a^2+35 A b^2\right )}{2 (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\int \frac {b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \sec ^2(c+d x)-6 a b \left (7 A b^2-a^2 (3 A-4 C)\right ) \sec (c+d x)+3 \left (a^2-b^2\right ) \left (4 (A+2 C) a^2+35 A b^2\right )}{(a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\int \frac {b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-6 a b \left (7 A b^2-a^2 (3 A-4 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (a^2-b^2\right ) \left (4 (A+2 C) a^2+35 A b^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {-\frac {2 \int -\frac {3 \left (4 (A+2 C) a^2+35 A b^2\right ) \left (a^2-b^2\right )^2+b^2 \left ((33 A-56 C) a^4-2 b^2 (85 A-12 C) a^2+105 A b^4\right ) \sec ^2(c+d x)+2 a b \left (3 (A-8 C) a^4-2 b^2 (27 A-4 C) a^2+35 A b^4\right ) \sec (c+d x)}{2 \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {3 \left (4 (A+2 C) a^2+35 A b^2\right ) \left (a^2-b^2\right )^2+b^2 \left ((33 A-56 C) a^4-2 b^2 (85 A-12 C) a^2+105 A b^4\right ) \sec ^2(c+d x)+2 a b \left (3 (A-8 C) a^4-2 b^2 (27 A-4 C) a^2+35 A b^4\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {\int \frac {3 \left (4 (A+2 C) a^2+35 A b^2\right ) \left (a^2-b^2\right )^2+b^2 \left ((33 A-56 C) a^4-2 b^2 (85 A-12 C) a^2+105 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (3 (A-8 C) a^4-2 b^2 (27 A-4 C) a^2+35 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {3 \left (4 (A+2 C) a^2+35 A b^2\right ) \left (a^2-b^2\right )^2+\left (2 a b \left (3 (A-8 C) a^4-2 b^2 (27 A-4 C) a^2+35 A b^4\right )-b^2 \left ((33 A-56 C) a^4-2 b^2 (85 A-12 C) a^2+105 A b^4\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {3 \left (4 (A+2 C) a^2+35 A b^2\right ) \left (a^2-b^2\right )^2+\left (2 a b \left (3 (A-8 C) a^4-2 b^2 (27 A-4 C) a^2+35 A b^4\right )-b^2 \left ((33 A-56 C) a^4-2 b^2 (85 A-12 C) a^2+105 A b^4\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {3 \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)+35 A b^2\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx+b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b (a-b) \left (6 a^4 (A-8 C)-a^3 (27 A b-8 b C)-3 a^2 b^2 (45 A-8 C)+35 a A b^3+105 A b^4\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {3 \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)+35 A b^2\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b (a-b) \left (6 a^4 (A-8 C)-a^3 (27 A b-8 b C)-3 a^2 b^2 (45 A-8 C)+35 a A b^3+105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b (a-b) \left (6 a^4 (A-8 C)-a^3 (27 A b-8 b C)-3 a^2 b^2 (45 A-8 C)+35 a A b^3+105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)+35 A b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}-\frac {2 b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {\frac {b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)+35 A b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \left (6 a^4 (A-8 C)-a^3 (27 A b-8 b C)-3 a^2 b^2 (45 A-8 C)+35 a A b^3+105 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{a \left (a^2-b^2\right )}-\frac {2 b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {7 A b \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 b^2 \left (35 A b^2-a^2 (27 A-8 C)\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac {\frac {-\frac {6 \sqrt {a+b} \left (a^2-b^2\right )^2 \left (4 a^2 (A+2 C)+35 A b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}-\frac {2 (a-b) \sqrt {a+b} \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{d}+\frac {2 (a-b) \sqrt {a+b} \left (6 a^4 (A-8 C)-a^3 (27 A b-8 b C)-3 a^2 b^2 (45 A-8 C)+35 a A b^3+105 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{a \left (a^2-b^2\right )}-\frac {2 b^2 \left (a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}}{2 a}}{4 a}\) |
(A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*(a + b*Sec[c + d*x])^(3/2)) - ((7*A*b *Sin[c + d*x])/(a*d*(a + b*Sec[c + d*x])^(3/2)) - ((2*b^2*(35*A*b^2 - a^2* (27*A - 8*C))*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (((-2*(a - b)*Sqrt[a + b]*(105*A*b^4 + a^4*(33*A - 56*C) - 2*a^2*b^2*(8 5*A - 12*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))])/d + (2*(a - b)*Sqrt[a + b]*(35*a*A*b^3 + 105* A*b^4 + 6*a^4*(A - 8*C) - 3*a^2*b^2*(45*A - 8*C) - a^3*(27*A*b - 8*b*C))*C ot[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)]*Sqrt[-((b*(1 + Sec[c + d*x ]))/(a - b))])/d - (6*Sqrt[a + b]*(a^2 - b^2)^2*(35*A*b^2 + 4*a^2*(A + 2*C ))*Cot[c + d*x]*EllipticPi[(a + b)/a, 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))])/(a*d))/(a*(a^2 - b^2)) - (2*b^2*(105*A*b^4 + a^4*(33*A - 56*C) - 2*a^2*b^2*(85*A - 12*C))*Tan[c + d*x])/(a*(a^2 - b^2 )*d*Sqrt[a + b*Sec[c + d*x]]))/(3*a*(a^2 - b^2)))/(2*a))/(4*a)
3.8.53.3.1 Defintions of rubi rules used
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_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^( m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x ] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -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]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Co t[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*(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, 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. \(12227\) vs. \(2(596)=1192\).
Time = 7.11 (sec) , antiderivative size = 12228, normalized size of antiderivative = 18.96
\[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
integral((C*cos(d*x + c)^2*sec(d*x + c)^2 + A*cos(d*x + c)^2)*sqrt(b*sec(d *x + c) + a)/(b^3*sec(d*x + c)^3 + 3*a*b^2*sec(d*x + c)^2 + 3*a^2*b*sec(d* x + c) + a^3), x)
\[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]