Integrand size = 43, antiderivative size = 598 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=-\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B+a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{12 a^5 \left (a^2-b^2\right )^2 d}-\frac {b \left (35 A b^6-35 a^5 b B+38 a^3 b^3 B-15 a b^5 B-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^5 (a-b)^2 (a+b)^3 d}+\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))} \] Output:
-1/4*(35*A*b^5-8*a^5*B+29*a^3*b^2*B-15*a*b^4*B+3*a^4*b*(8*A-3*C)-a^2*b^3*( 65*A-3*C))*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+ c)^(1/2)/a^4/(a^2-b^2)^2/d+1/12*(105*A*b^6-72*a^5*b*B+99*a^3*b^3*B-45*a*b^ 5*B+a^4*b^2*(128*A-15*C)-a^2*b^4*(223*A-9*C)+8*a^6*(A+3*C))*cos(d*x+c)^(1/ 2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/a^5/(a^2-b^2)^2 /d-1/4*b*(35*A*b^6-35*a^5*b*B+38*a^3*b^3*B-15*a*b^5*B-a^2*b^4*(86*A-3*C)+3 *a^4*b^2*(21*A-2*C)+15*a^6*C)*cos(d*x+c)^(1/2)*EllipticPi(sin(1/2*d*x+1/2* c),2*a/(a+b),2^(1/2))*sec(d*x+c)^(1/2)/a^5/(a-b)^2/(a+b)^3/d+1/12*(35*A*b^ 4+33*B*a^3*b-15*B*a*b^3+a^4*(8*A-21*C)-a^2*b^2*(61*A-3*C))*sin(d*x+c)/a^3/ (a^2-b^2)^2/d/sec(d*x+c)^(1/2)+1/2*(A*b^2-a*(B*b-C*a))*sin(d*x+c)/a/(a^2-b ^2)/d/sec(d*x+c)^(1/2)/(a+b*sec(d*x+c))^2-1/4*(7*A*b^4+9*B*a^3*b-3*B*a*b^3 -5*a^4*C-a^2*b^2*(13*A+C))*sin(d*x+c)/a^2/(a^2-b^2)^2/d/sec(d*x+c)^(1/2)/( a+b*sec(d*x+c))
Time = 10.39 (sec) , antiderivative size = 1116, normalized size of antiderivative = 1.87 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:
Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]
Output:
((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^ 2)*((2*(-56*a^4*A*b + 73*a^2*A*b^3 - 35*A*b^5 + 24*a^5*B - 21*a^3*b^2*B + 15*a*b^4*B - 15*a^4*b*C - 3*a^2*b^3*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sq rt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1 ])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*C os[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16*a^5*A + 112*a^3*A*b^2 - 56*a*A *b^4 - 96*a^4*b*B + 24*a^2*b^3*B + 48*a^5*C + 24*a^3*b^2*C)*Cos[c + d*x]^2 *EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*S qrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-72*a^4*A*b + 195*a^2*A*b^3 - 105*A*b^5 + 24*a^5*B - 87*a^3* b^2*B + 45*a*b^4*B + 27*a^4*b*C - 9*a^2*b^3*C)*Cos[2*(c + d*x)]*(a + b*Sec [c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Se c[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2 *b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1] *Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), Ar cSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2]) *Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(24*a^3*(a - b)^2*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + ...
Time = 4.24 (sec) , antiderivative size = 595, normalized size of antiderivative = 0.99, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 4588, 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.
| \(\displaystyle \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\int \frac {-\left ((4 A-3 C) a^2\right )-3 b B a+4 (A b+C b-a B) \sec (c+d x) a+7 A b^2-5 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\int \frac {-\left ((4 A-3 C) a^2\right )-3 b B a+4 (A b+C b-a B) \sec (c+d x) a+7 A b^2-5 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}dx}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\int \frac {-\left ((4 A-3 C) a^2\right )-3 b B a+4 (A b+C b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a+7 A b^2-5 \left (A b^2-a (b B-a C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\int \frac {(8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+4 \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \sec (c+d x) a+35 A b^4-3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right ) \sec ^2(c+d x)}{2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}dx}{a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\int \frac {(8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+4 \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \sec (c+d x) a+35 A b^4-3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}dx}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\int \frac {(8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+4 \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+35 A b^4-3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {-b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \sec ^2(c+d x)+4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \sec (c+d x)+3 \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )}{2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \sec ^2(c+d x)+4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \sec (c+d x)+3 \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4594 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)}dx}{a^2}+\frac {\int \frac {3 a \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )-\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\int \frac {3 a \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )+\left (-8 (A+3 C) a^6+72 b B a^5-b^2 (128 A-15 C) a^4-99 b^3 B a^3+b^4 (223 A-9 C) a^2+45 b^5 B a-105 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {3 a \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-\left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right ) \int \sqrt {\sec (c+d x)}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 a \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {3 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\cos (c+d x)}dx-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {3 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\frac {6 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\frac {6 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right )}{d}}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4336 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a^2}+\frac {\frac {6 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right )}{d}}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2}+\frac {\frac {6 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right )}{d}}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {6 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a^2 d (a+b)}+\frac {\frac {6 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right )}{d}}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
Input:
Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + b*Sec [c + d*x])^3),x]
Output:
((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x ]]*(a + b*Sec[c + d*x])^2) - (((7*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])) - (-1/3*(((6*a*(35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15 *a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sqrt[Cos[c + d*x]]* EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d - (2*(105*A*b^6 - 72*a^5*b *B + 99*a^3*b^3*B - 45*a*b^5*B + a^4*b^2*(128*A - 15*C) - a^2*b^4*(223*A - 9*C) + 8*a^6*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt [Sec[c + d*x]])/d)/a^2 + (6*b*(35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a *b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Sqrt[Co s[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/ (a^2*(a + b)*d))/a + (2*(35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 2 1*C) - a^2*b^2*(61*A - 3*C))*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]))/(2* a*(a^2 - b^2)))/(4*a*(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[(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[((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. \(2288\) vs. \(2(573)=1146\).
Time = 10.66 (sec) , antiderivative size = 2289, normalized size of antiderivative = 3.83
Input:
int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c))^3,x, method=_RETURNVERBOSE)
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/a^5*(4*A*c os(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*a^2-2*A*cos(1/2*d*x+1/2*c)*sin(1/2* d*x+1/2*c)^2*a^2+A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/ 2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a^2+18*A*EllipticF(cos(1/2*d* x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^ (1/2)*b^2+9*A*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)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a*b-9*B*EllipticF(cos(1/2*d*x+1/2* c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)* a*b-3*B*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)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a^2+3*C*EllipticF(cos(1/2*d*x+1/2*c),2^( 1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*a^2)/( -2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2/a^5*b^2*(5*A*b^2-4*B *a*b+3*C*a^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*(s in(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+sin(1/2*d*x...
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c) )^3,x, algorithm="fricas")
Output:
integral((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sqrt(sec(d*x + c))/(b^3*s ec(d*x + c)^5 + 3*a*b^2*sec(d*x + c)^4 + 3*a^2*b*sec(d*x + c)^3 + a^3*sec( d*x + c)^2), x)
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(3/2)/(a+b*sec(d*x+ c))**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c) )^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c) )^3,x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)/((b*sec(d*x + c) + a)^3* sec(d*x + c)^(3/2)), x)
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/((a + b/cos(c + d*x))^3*(1/cos (c + d*x))^(3/2)),x)
Output:
int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/((a + b/cos(c + d*x))^3*(1/cos (c + d*x))^(3/2)), x)
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5} b^{3}+3 \sec \left (d x +c \right )^{4} a \,b^{2}+3 \sec \left (d x +c \right )^{3} a^{2} b +\sec \left (d x +c \right )^{2} a^{3}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4} b^{3}+3 \sec \left (d x +c \right )^{3} a \,b^{2}+3 \sec \left (d x +c \right )^{2} a^{2} b +\sec \left (d x +c \right ) a^{3}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3} b^{3}+3 \sec \left (d x +c \right )^{2} a \,b^{2}+3 \sec \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c \] Input:
int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c))^3,x)
Output:
int(sqrt(sec(c + d*x))/(sec(c + d*x)**5*b**3 + 3*sec(c + d*x)**4*a*b**2 + 3*sec(c + d*x)**3*a**2*b + sec(c + d*x)**2*a**3),x)*a + int(sqrt(sec(c + d *x))/(sec(c + d*x)**4*b**3 + 3*sec(c + d*x)**3*a*b**2 + 3*sec(c + d*x)**2* a**2*b + sec(c + d*x)*a**3),x)*b + int(sqrt(sec(c + d*x))/(sec(c + d*x)**3 *b**3 + 3*sec(c + d*x)**2*a*b**2 + 3*sec(c + d*x)*a**2*b + a**3),x)*c