Integrand size = 33, antiderivative size = 521 \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {\left (15 a^4 A b-29 a^2 A b^3+8 A b^5-35 a^5 B+65 a^3 b^2 B-24 a b^4 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (45 a^5 A b-99 a^3 A b^3+72 a A b^5-105 a^6 B+223 a^4 b^2 B-128 a^2 b^4 B-8 b^6 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{12 b^5 \left (a^2-b^2\right )^2 d}+\frac {a^2 \left (15 a^4 A b-38 a^2 A b^3+35 A b^5-35 a^5 B+86 a^3 b^2 B-63 a b^4 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 (a-b)^2 b^5 (a+b)^3 d}-\frac {\left (15 a^3 A b-33 a A b^3-35 a^4 B+61 a^2 b^2 B-8 b^4 B\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))^2}+\frac {a \left (3 a^2 A b-9 A b^3-7 a^3 B+13 a b^2 B\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} (b+a \sec (c+d x))} \] Output:
1/4*(15*A*a^4*b-29*A*a^2*b^3+8*A*b^5-35*B*a^5+65*B*a^3*b^2-24*B*a*b^4)*cos (d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2)/b^4/( a^2-b^2)^2/d-1/12*(45*A*a^5*b-99*A*a^3*b^3+72*A*a*b^5-105*B*a^6+223*B*a^4* b^2-128*B*a^2*b^4-8*B*b^6)*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c, 2^(1/2))*sec(d*x+c)^(1/2)/b^5/(a^2-b^2)^2/d+1/4*a^2*(15*A*a^4*b-38*A*a^2*b ^3+35*A*b^5-35*B*a^5+86*B*a^3*b^2-63*B*a*b^4)*cos(d*x+c)^(1/2)*EllipticPi( sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))*sec(d*x+c)^(1/2)/(a-b)^2/b^5/(a+b)^3 /d-1/12*(15*A*a^3*b-33*A*a*b^3-35*B*a^4+61*B*a^2*b^2-8*B*b^4)*sin(d*x+c)/b ^3/(a^2-b^2)^2/d/sec(d*x+c)^(1/2)+1/2*a*(A*b-B*a)*sin(d*x+c)/b/(a^2-b^2)/d /sec(d*x+c)^(1/2)/(b+a*sec(d*x+c))^2+1/4*a*(3*A*a^2*b-9*A*b^3-7*B*a^3+13*B *a*b^2)*sin(d*x+c)/b^2/(a^2-b^2)^2/d/sec(d*x+c)^(1/2)/(b+a*sec(d*x+c))
Time = 8.03 (sec) , antiderivative size = 865, normalized size of antiderivative = 1.66 \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {\frac {2 \left (-15 a^4 A b+21 a^2 A b^3-24 A b^5+35 a^5 B-73 a^3 b^2 B+56 a b^4 B\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (-24 a^3 A b^2+96 a A b^4+56 a^4 b B-112 a^2 b^3 B-16 b^5 B\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (-45 a^4 A b+87 a^2 A b^3-24 A b^5+105 a^5 B-195 a^3 b^2 B+72 a b^4 B\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 (2 a-b) b \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 a^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 b^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{48 (a-b)^2 b^3 (a+b)^2 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {a^2 \left (-7 a^2 A b+13 A b^3+11 a^3 B-17 a b^2 B\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2}-\frac {-a^4 A b \sin (c+d x)+a^5 B \sin (c+d x)}{2 b^4 \left (-a^2+b^2\right ) (a+b \cos (c+d x))^2}+\frac {9 a^5 A b \sin (c+d x)-15 a^3 A b^3 \sin (c+d x)-13 a^6 B \sin (c+d x)+19 a^4 b^2 B \sin (c+d x)}{4 b^4 \left (-a^2+b^2\right )^2 (a+b \cos (c+d x))}+\frac {B \sin (2 (c+d x))}{3 b^3}\right )}{d} \] Input:
Integrate[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)) ,x]
Output:
-1/48*((2*(-15*a^4*A*b + 21*a^2*A*b^3 - 24*A*b^5 + 35*a^5*B - 73*a^3*b^2*B + 56*a*b^4*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])* Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-24*a^3*A*b^2 + 96*a*A*b^4 + 56*a^4*b*B - 112*a^2*b^3*B - 16*b^5*B)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], - 1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*C os[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-45*a^4*A*b + 87*a^2*A*b^3 - 24*A*b ^5 + 105*a^5*B - 195*a^3*b^2*B + 72*a*b^4*B)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[ c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)* b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Se c[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*S qrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcS in[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*S in[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/((a - b)^2*b^3*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*((a^2*(-7*a^2*A*b + 13*A*b^3 + 11*a^3*B - 17*a*b^2*B)*Sin[c + d*x])/ (4*b^4*(a^2 - b^2)^2) - (-(a^4*A*b*Sin[c + d*x]) + a^5*B*Sin[c + d*x])/(2* b^4*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (9*a^5*A*b*Sin[c + d*x] - 15...
Time = 3.89 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.697, Rules used = {3042, 3439, 3042, 4518, 27, 3042, 4588, 27, 3042, 4592, 27, 3042, 4594, 3042, 4274, 3042, 4258, 3042, 3119, 3120, 4336, 3042, 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.
| \(\displaystyle \int \frac {A+B \cos (c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3439 |
| \(\displaystyle \int \frac {A \sec (c+d x)+B}{\sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A \csc \left (c+d x+\frac {\pi }{2}\right )+B}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+b\right )^3}dx\) |
| \(\Big \downarrow \) 4518 |
| \(\displaystyle \frac {\int -\frac {-7 B a^2-5 (A b-a B) \sec ^2(c+d x) a+3 A b a+4 b^2 B+4 b (A b-a B) \sec (c+d x)}{2 \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\int \frac {-7 B a^2-5 (A b-a B) \sec ^2(c+d x) a+3 A b a+4 b^2 B+4 b (A b-a B) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\int \frac {-7 B a^2-5 (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2 a+3 A b a+4 b^2 B+4 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\int \frac {-35 B a^4+15 A b a^3+61 b^2 B a^2-33 A b^3 a-3 \left (-7 B a^3+3 A b a^2+13 b^2 B a-9 A b^3\right ) \sec ^2(c+d x) a-8 b^4 B-4 b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \sec (c+d x)}{2 \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))}dx}{b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\int \frac {-35 B a^4+15 A b a^3+61 b^2 B a^2-33 A b^3 a-3 \left (-7 B a^3+3 A b a^2+13 b^2 B a-9 A b^3\right ) \sec ^2(c+d x) a-8 b^4 B-4 b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))}dx}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\int \frac {-35 B a^4+15 A b a^3+61 b^2 B a^2-33 A b^3 a-3 \left (-7 B a^3+3 A b a^2+13 b^2 B a-9 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 a-8 b^4 B-4 b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {-a \left (-35 B a^4+15 A b a^3+61 b^2 B a^2-33 A b^3 a-8 b^4 B\right ) \sec ^2(c+d x)+4 b \left (-7 B a^4+3 A b a^3+14 b^2 B a^2-12 A b^3 a+2 b^4 B\right ) \sec (c+d x)+3 \left (-35 B a^5+15 A b a^4+65 b^2 B a^3-29 A b^3 a^2-24 b^4 B a+8 A b^5\right )}{2 \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-a \left (-35 B a^4+15 A b a^3+61 b^2 B a^2-33 A b^3 a-8 b^4 B\right ) \sec ^2(c+d x)+4 b \left (-7 B a^4+3 A b a^3+14 b^2 B a^2-12 A b^3 a+2 b^4 B\right ) \sec (c+d x)+3 \left (-35 B a^5+15 A b a^4+65 b^2 B a^3-29 A b^3 a^2-24 b^4 B a+8 A b^5\right )}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-a \left (-35 B a^4+15 A b a^3+61 b^2 B a^2-33 A b^3 a-8 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 b \left (-7 B a^4+3 A b a^3+14 b^2 B a^2-12 A b^3 a+2 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-35 B a^5+15 A b a^4+65 b^2 B a^3-29 A b^3 a^2-24 b^4 B a+8 A b^5\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4594 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)}dx}{b^2}+\frac {\int \frac {3 b \left (-35 B a^5+15 A b a^4+65 b^2 B a^3-29 A b^3 a^2-24 b^4 B a+8 A b^5\right )-\left (-105 B a^6+45 A b a^5+223 b^2 B a^4-99 A b^3 a^3-128 b^4 B a^2+72 A b^5 a-8 b^6 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\int \frac {3 b \left (-35 B a^5+15 A b a^4+65 b^2 B a^3-29 A b^3 a^2-24 b^4 B a+8 A b^5\right )+\left (105 B a^6-45 A b a^5-223 b^2 B a^4+99 A b^3 a^3+128 b^4 B a^2-72 A b^5 a+8 b^6 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {3 b \left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-\left (-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\right ) \int \sqrt {\sec (c+d x)}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {3 b \left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {3 b \left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx-\left (-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {3 b \left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\left (-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {6 b \left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\left (-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {6 b \left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4336 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b^2}+\frac {\frac {6 b \left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b^2}+\frac {\frac {6 b \left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^2}-\frac {\frac {\frac {2 \left (-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right ) \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {\frac {6 a^2 \left (-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b^2 d (a+b)}+\frac {\frac {6 b \left (-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{3 b}}{2 b \left (a^2-b^2\right )}-\frac {a \left (-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)}}{4 b \left (a^2-b^2\right )}\) |
Input:
Int[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]
Output:
(a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(b + a* Sec[c + d*x])^2) - (-((a*(3*a^2*A*b - 9*A*b^3 - 7*a^3*B + 13*a*b^2*B)*Sin[ c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x]))) + (-1 /3*(((6*b*(15*a^4*A*b - 29*a^2*A*b^3 + 8*A*b^5 - 35*a^5*B + 65*a^3*b^2*B - 24*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x ]])/d - (2*(45*a^5*A*b - 99*a^3*A*b^3 + 72*a*A*b^5 - 105*a^6*B + 223*a^4*b ^2*B - 128*a^2*b^4*B - 8*b^6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d)/b^2 + (6*a^2*(15*a^4*A*b - 38*a^2*A*b^3 + 35*A*b ^5 - 35*a^5*B + 86*a^3*b^2*B - 63*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[( 2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d))/b + (2* (15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*Sin[c + d*x] )/(3*b*d*Sqrt[Sec[c + d*x]]))/(2*b*(a^2 - b^2)))/(4*b*(a^2 - b^2))
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[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)* (x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim p[g^(m + n) Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(d + c*Csc[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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]] Int[ 1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[b*(A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(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)*(d*Csc[e + f*x])^n*Simp[A*(a^2*(m + 1) - b^2*(m + n + 1)) + a*b*B*n - a*(A*b - a*B)*(m + 1)*Csc[e + f*x] + b*(A*b - a*B)*(m + n + 2) *Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A* b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && IL tQ[n, 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*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc [e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f *x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x ] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[n, 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 + 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_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2) Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Simp[1/a^2 Int[(a*A - (A*b - a *B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2194\) vs. \(2(496)=992\).
Time = 13.55 (sec) , antiderivative size = 2195, normalized size of antiderivative = 4.21
Input:
int((A+B*cos(d*x+c))/(a+cos(d*x+c)*b)^3/sec(d*x+c)^(7/2),x,method=_RETURNV ERBOSE)
Output:
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2/3/b^5/(-2*s in(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(-4*B*cos(1/2*d*x+1/2*c)*s in(1/2*d*x+1/2*c)^4*b^2+9*A*a*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d* x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+3*A*(sin(1/2*d*x +1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/ 2*c),2^(1/2))*b^2+2*B*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*b^2-18*a^2*B *(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(c os(1/2*d*x+1/2*c),2^(1/2))-B*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d *x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-9*B*(sin(1/2*d* x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1 /2*c),2^(1/2))*a*b)+2*a^4*(A*b-B*a)/b^5*(-1/2*b^2/a/(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*b*cos(1/2*d *x+1/2*c)^2+a-b)^2-3/4*b^2*(3*a^2-b^2)/a^2/(a^2-b^2)^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*b*cos(1/2*d*x+1/2* c)^2+a-b)-7/8/(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)*El lipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1/4/(a+b)/(a^2-b^2)/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+si n(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b+3/8/(a+b )/(a^2-b^2)/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1...
Timed out. \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c))/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x, algorith m="fricas")
Output:
Timed out
Timed out. \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((A+B*cos(d*x+c))/(a+b*cos(d*x+c))**3/sec(d*x+c)**(7/2),x)
Output:
Timed out
\[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((A+B*cos(d*x+c))/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x, algorith m="maxima")
Output:
integrate((B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)^3*sec(d*x + c)^(7/2)) , x)
\[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((A+B*cos(d*x+c))/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x, algorith m="giac")
Output:
integrate((B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)^3*sec(d*x + c)^(7/2)) , x)
Timed out. \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {A+B\,\cos \left (c+d\,x\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:
int((A + B*cos(c + d*x))/((1/cos(c + d*x))^(7/2)*(a + b*cos(c + d*x))^3),x )
Output:
int((A + B*cos(c + d*x))/((1/cos(c + d*x))^(7/2)*(a + b*cos(c + d*x))^3), x)
\[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {\sqrt {\sec \left (d x +c \right )}}{\cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4} b^{2}+2 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4} a b +\sec \left (d x +c \right )^{4} a^{2}}d x \] Input:
int((A+B*cos(d*x+c))/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x)
Output:
int(sqrt(sec(c + d*x))/(cos(c + d*x)**2*sec(c + d*x)**4*b**2 + 2*cos(c + d *x)*sec(c + d*x)**4*a*b + sec(c + d*x)**4*a**2),x)