Integrand size = 33, antiderivative size = 420 \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\frac {\left (3 a^3 A b-9 a A b^3-15 a^4 B+29 a^2 b^2 B-8 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^3 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 A b-7 A b^3-5 a^3 B+11 a b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 b^2 \left (a^2-b^2\right )^2 d}+\frac {\left (3 a^4 A b-6 a^2 A b^3+15 A b^5-15 a^5 B+38 a^3 b^2 B-35 a b^4 B\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{4 (a-b)^2 b^3 (a+b)^3 d}-\frac {\left (3 a^3 A b-9 a A b^3-15 a^4 B+29 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{4 b^3 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2}+\frac {a \left (a^2 A b-7 A b^3-5 a^3 B+11 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \]
1/4*(3*A*a^3*b-9*A*a*b^3-15*B*a^4+29*B*a^2*b^2-8*B*b^4)*(cos(1/2*d*x+1/2*c )^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/b^3/(a ^2-b^2)^2/d+1/4*(A*a^2*b-7*A*b^3-5*B*a^3+11*B*a*b^2)*(cos(1/2*d*x+1/2*c)^2 )^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/b^2/(a^2- b^2)^2/d+1/4*(3*A*a^4*b-6*A*a^2*b^3+15*A*b^5-15*B*a^5+38*B*a^3*b^2-35*B*a* b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d* x+1/2*c),2*a/(a+b),2^(1/2))/(a-b)^2/b^3/(a+b)^3/d-1/4*(3*A*a^3*b-9*A*a*b^3 -15*B*a^4+29*B*a^2*b^2-8*B*b^4)*sin(d*x+c)/b^3/(a^2-b^2)^2/d/cos(d*x+c)^(1 /2)+1/2*a*(A*b-B*a)*sin(d*x+c)/b/(a^2-b^2)/d/(b+a*cos(d*x+c))^2/cos(d*x+c) ^(1/2)+1/4*a*(A*a^2*b-7*A*b^3-5*B*a^3+11*B*a*b^2)*sin(d*x+c)/b^2/(a^2-b^2) ^2/d/(b+a*cos(d*x+c))/cos(d*x+c)^(1/2)
Time = 5.97 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.09 \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\frac {-\frac {\frac {\left (-9 a^4 A b+19 a^2 A b^3-16 A b^5+45 a^5 B-95 a^3 b^2 B+56 a b^4 B\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {8 b \left (-a^3 A b+4 a A b^3+5 a^4 B-10 a^2 b^2 B+2 b^4 B\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-b \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a (a+b)}+\frac {\left (-3 a^3 A b+9 a A b^3+15 a^4 B-29 a^2 b^2 B+8 b^4 B\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b \sqrt {\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}+\frac {\sqrt {\cos (c+d x)} \left (2 a b \left (-5 a^3 A b+11 a A b^3+25 a^4 B-47 a^2 b^2 B+16 b^4 B\right ) \sin (c+d x)+a^2 \left (-3 a^3 A b+9 a A b^3+15 a^4 B-29 a^2 b^2 B+8 b^4 B\right ) \sin (2 (c+d x))+16 \left (-a^2 b+b^3\right )^2 B \tan (c+d x)\right )}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}}{8 b^3 d} \]
(-((((-9*a^4*A*b + 19*a^2*A*b^3 - 16*A*b^5 + 45*a^5*B - 95*a^3*b^2*B + 56* a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(-(a^3* A*b) + 4*a*A*b^3 + 5*a^4*B - 10*a^2*b^2*B + 2*b^4*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + ((-3*a^3*A*b + 9*a*A*b^3 + 15*a^4*B - 29*a^2*b^2*B + 8*b^4*B)*(-2*a*b*El lipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqr t[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2)) + (Sqrt[Cos[c + d*x]]*(2*a*b*(-5*a^3*A*b + 11*a*A*b^3 + 25*a^4*B - 47*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x] + a^2*(-3*a^3*A*b + 9*a*A*b^3 + 15*a ^4*B - 29*a^2*b^2*B + 8*b^4*B)*Sin[2*(c + d*x)] + 16*(-(a^2*b) + b^3)^2*B* Tan[c + d*x]))/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2))/(8*b^3*d)
Time = 3.12 (sec) , antiderivative size = 413, normalized size of antiderivative = 0.98, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {3042, 3433, 3042, 3479, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3119, 3481, 3042, 3120, 3284}
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.
| \(\displaystyle \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3433 |
| \(\displaystyle \int \frac {A \cos (c+d x)+B}{\cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A \sin \left (c+d x+\frac {\pi }{2}\right )+B}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^3}dx\) |
| \(\Big \downarrow \) 3479 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\int \frac {-5 B a^2-3 (A b-a B) \cos ^2(c+d x) a+A b a+4 b^2 B+4 b (A b-a B) \cos (c+d x)}{2 \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\int \frac {-5 B a^2-3 (A b-a B) \cos ^2(c+d x) a+A b a+4 b^2 B+4 b (A b-a B) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\int \frac {-5 B a^2-3 (A b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+A b a+4 b^2 B+4 b (A b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {-\frac {\int -\frac {-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-\left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \cos ^2(c+d x) a-8 b^4 B-4 b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \cos (c+d x)}{2 \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))}dx}{b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\int \frac {-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-\left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \cos ^2(c+d x) a-8 b^4 B-4 b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))}dx}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\int \frac {-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-\left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a-8 b^4 B-4 b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \int -\frac {-15 B a^5+3 A b a^4+33 b^2 B a^3-5 A b^3 a^2+\left (-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-8 b^4 B\right ) \cos ^2(c+d x) a-24 b^4 B a+8 A b^5+4 b \left (-5 B a^4+A b a^3+10 b^2 B a^2-4 A b^3 a-2 b^4 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{b}+\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-15 B a^5+3 A b a^4+33 b^2 B a^3-5 A b^3 a^2+\left (-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-8 b^4 B\right ) \cos ^2(c+d x) a-24 b^4 B a+8 A b^5+4 b \left (-5 B a^4+A b a^3+10 b^2 B a^2-4 A b^3 a-2 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-15 B a^5+3 A b a^4+33 b^2 B a^3-5 A b^3 a^2+\left (-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-8 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a-24 b^4 B a+8 A b^5+4 b \left (-5 B a^4+A b a^3+10 b^2 B a^2-4 A b^3 a-2 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {b \left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \cos (c+d x) a^2+\left (-15 B a^5+3 A b a^4+33 b^2 B a^3-5 A b^3 a^2-24 b^4 B a+8 A b^5\right ) a}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}}{b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx+\frac {\int \frac {b \left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \cos (c+d x) a^2+\left (-15 B a^5+3 A b a^4+33 b^2 B a^3-5 A b^3 a^2-24 b^4 B a+8 A b^5\right ) a}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}}{b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {b \left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (-15 B a^5+3 A b a^4+33 b^2 B a^3-5 A b^3 a^2-24 b^4 B a+8 A b^5\right ) a}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {b \left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (-15 B a^5+3 A b a^4+33 b^2 B a^3-5 A b^3 a^2-24 b^4 B a+8 A b^5\right ) a}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}+\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {a b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+a \left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}+\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {a b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}+\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {a \left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {2 a b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a}+\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {\frac {2 a b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {2 a \left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}}{a}}{b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
(a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(b + a* Cos[c + d*x])^2) - (-((a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x]))) + (-((( 2*(3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*EllipticE[(c + d*x)/2, 2])/d + ((2*a*b*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Elli pticF[(c + d*x)/2, 2])/d + (2*a*(3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a ^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2 ])/((a + b)*d))/a)/b) + (2*(3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2* B - 8*b^4*B)*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]))/(2*b*(a^2 - b^2)))/(4 *b*(a^2 - b^2))
3.6.92.3.1 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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]* (d_.) + (c_))^(n_.)*((g_.)*sin[(e_.) + (f_.)*(x_)])^(p_.), x_Symbol] :> Sim p[g^(m + n) Int[(g*Sin[e + f*x])^(p - m - n)*(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(A*b^2 - a*b*B))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin [e + f*x])^(1 + n)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m + n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B) *(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n }, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Rat ionalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(I ntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0]) ))
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, 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_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{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. \(1996\) vs. \(2(480)=960\).
Time = 36.68 (sec) , antiderivative size = 1997, normalized size of antiderivative = 4.75
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B/b^3/sin(1/ 2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2 *d*x+1/2*c)^2)^(1/2)*(2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-(sin(1/2*d *x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/ 2*c)^2-1)^(1/2))-2*B*a/b^2*(a^2/b/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2 *d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*a*cos(1/2*d*x+1/2*c)^2-a+b)-1 /2/(a+b)/b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/ (-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x +1/2*c),2^(1/2))+1/2*a/b/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/ 2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/ 2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1/2*a/b/(a^2-b^2)*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+s in(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-1/2/b/(a^ 2-b^2)/(a^2-a*b)*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2 +1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi( cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/2))+3/2*b/(a^2-b^2)/(a^2-a*b)*a*(sin(1/2 *d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2 *c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a-b), 2^(1/2)))+2*B*a^2/b^3/(a^2-a*b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d *x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/...
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]