Integrand size = 35, antiderivative size = 427 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\frac {2 \left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d} \] Output:
2/315*(a^2-b^2)*(39*A*a^2*b+8*A*b^3+75*B*a^3-18*B*a*b^2)*((b+a*cos(d*x+c)) /(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(a/(a+b))^(1/2))/a^3/d /cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)+2/315*(147*A*a^4+33*A*a^2*b^2+8*A *b^4+246*B*a^3*b-18*B*a*b^3)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c) ,2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/a^3/d/((b+a*cos(d*x+c))/( a+b))^(1/2)+2/315*(88*A*a^2*b-4*A*b^3+75*B*a^3+9*B*a*b^2)*cos(d*x+c)^(1/2) *(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a^2/d+2/315*(49*A*a^2+3*A*b^2+72*B*a*b) *cos(d*x+c)^(3/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a/d+2/63*(10*A*b+9*B*a )*cos(d*x+c)^(5/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/d+2/9*a*A*cos(d*x+c)^ (7/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/d
Result contains complex when optimal does not.
Time = 17.18 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.26 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\frac {\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {\left (402 a^2 A b-16 A b^3+345 a^3 B+36 a b^2 B\right ) \sin (c+d x)}{630 a^2}+\frac {\left (133 a^2 A+6 A b^2+144 a b B\right ) \sin (2 (c+d x))}{630 a}+\frac {1}{126} (10 A b+9 a B) \sin (3 (c+d x))+\frac {1}{36} a A \sin (4 (c+d x))\right )}{d (b+a \cos (c+d x))}-\frac {2 \cos ^{\frac {3}{2}}(c+d x) \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} (a+b \sec (c+d x))^{3/2} \left (-i (a+b) \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) E\left (i \text {arcsinh}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+i a (a+b) \left (8 A b^3-6 a b^2 (A+3 B)+3 a^3 (49 A+25 B)+3 a^2 b (13 A+57 B)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )^{3/2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 a^3 d (b+a \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x)} \] Input:
Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x ]),x]
Output:
(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(((402*a^2*A*b - 16*A*b^3 + 345*a^3*B + 36*a*b^2*B)*Sin[c + d*x])/(630*a^2) + ((133*a^2*A + 6*A*b^2 + 144*a*b*B)*Sin[2*(c + d*x)])/(630*a) + ((10*A*b + 9*a*B)*Sin[3*(c + d*x)] )/126 + (a*A*Sin[4*(c + d*x)])/36))/(d*(b + a*Cos[c + d*x])) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(3 /2)*((-I)*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a *b^3*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^3*(49*A + 25*B) + 3*a^2*b*(13*A + 57*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (147*a ^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*(b + a*Cos[c + d *x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^3*d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))
Time = 3.99 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.07, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.771, Rules used = {3042, 3434, 3042, 4513, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
\(\Big \downarrow \) 3434 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
\(\Big \downarrow \) 4513 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {2}{9} \int -\frac {3 b (2 a A+3 b B) \sec ^2(c+d x)+\left (7 A a^2+18 b B a+9 A b^2\right ) \sec (c+d x)+a (10 A b+9 a B)}{2 \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {3 b (2 a A+3 b B) \sec ^2(c+d x)+\left (7 A a^2+18 b B a+9 A b^2\right ) \sec (c+d x)+a (10 A b+9 a B)}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {3 b (2 a A+3 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (7 A a^2+18 b B a+9 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (10 A b+9 a B)}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 4592 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \int -\frac {4 a b (10 A b+9 a B) \sec ^2(c+d x)+a \left (45 B a^2+92 A b a+63 b^2 B\right ) \sec (c+d x)+a \left (49 A a^2+72 b B a+3 A b^2\right )}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{7 a}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\int \frac {4 a b (10 A b+9 a B) \sec ^2(c+d x)+a \left (45 B a^2+92 A b a+63 b^2 B\right ) \sec (c+d x)+a \left (49 A a^2+72 b B a+3 A b^2\right )}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\int \frac {4 a b (10 A b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (45 B a^2+92 A b a+63 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (49 A a^2+72 b B a+3 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 4592 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int -\frac {\left (147 A a^2+396 b B a+209 A b^2\right ) \sec (c+d x) a^2+2 b \left (49 A a^2+72 b B a+3 A b^2\right ) \sec ^2(c+d x) a+3 \left (75 B a^3+88 A b a^2+9 b^2 B a-4 A b^3\right ) a}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\int \frac {\left (147 A a^2+396 b B a+209 A b^2\right ) \sec (c+d x) a^2+2 b \left (49 A a^2+72 b B a+3 A b^2\right ) \sec ^2(c+d x) a+3 \left (75 B a^3+88 A b a^2+9 b^2 B a-4 A b^3\right ) a}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\int \frac {\left (147 A a^2+396 b B a+209 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2+2 b \left (49 A a^2+72 b B a+3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 a+3 \left (75 B a^3+88 A b a^2+9 b^2 B a-4 A b^3\right ) a}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 4592 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}-\frac {2 \int -\frac {3 \left (\left (75 B a^3+186 A b a^2+153 b^2 B a+2 A b^3\right ) \sec (c+d x) a^2+\left (147 A a^4+246 b B a^3+33 A b^2 a^2-18 b^3 B a+8 A b^4\right ) a\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\left (75 B a^3+186 A b a^2+153 b^2 B a+2 A b^3\right ) \sec (c+d x) a^2+\left (147 A a^4+246 b B a^3+33 A b^2 a^2-18 b^3 B a+8 A b^4\right ) a}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\left (75 B a^3+186 A b a^2+153 b^2 B a+2 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (147 A a^4+246 b B a^3+33 A b^2 a^2-18 b^3 B a+8 A b^4\right ) a}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 4523 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+\left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 4343 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 4345 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}+\frac {\frac {2 \left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{7 a}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
Input:
Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin [c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((2*(10*A*b + 9*a*B)*Sqrt[a + b*Sec[ c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2 )) + (((2*(a^2 - b^2)*(39*a^2*A*b + 8*A*b^3 + 75*a^3*B - 18*a*b^2*B)*Sqrt[ (b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[S ec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b) ]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec [c + d*x]]))/a + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]))/(5*a))/(7*a))/9)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 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*Csc[e + f*x])^p*(g*Sin[e + f*x])^p Int[(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(g*Csc[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g , m, n, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && !(IntegerQ[m] && I ntegerQ[n])
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[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[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] + Sim p[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[ a*(a*B*n - A*b*(m - n - 1)) + (2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] & & LeQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(2694\) vs. \(2(396)=792\).
Time = 20.71 (sec) , antiderivative size = 2695, normalized size of antiderivative = 6.31
Input:
int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x,method=_RET URNVERBOSE)
Output:
-2/315/d*((-4*cos(d*x+c)+4)*sin(d*x+c)*A*((a-b)/(a+b))^(1/2)*a*b^4+18*B*(( a-b)/(a+b))^(1/2)*a*b^4*sin(d*x+c)-8*A*((a-b)/(a+b))^(1/2)*b^5*sin(d*x+c)+ (-18*cos(d*x+c)^2-36*cos(d*x+c)-18)*B*(1/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(b +a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b^3*EllipticF(((a-b)/(a+b))^(1/2) *(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(147*cos(d*x+c)^2+294*cos(d *x+c)+147)*A*(1/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x +c)))^(1/2)*a^4*b*EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(- (a+b)/(a-b))^(1/2))+(-33*cos(d*x+c)^2-66*cos(d*x+c)-33)*A*(1/(1+cos(d*x+c) ))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*b^2*EllipticE (((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(33*cos (d*x+c)^2+66*cos(d*x+c)+33)*A*(1/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(b+a*cos(d *x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b^3*EllipticE(((a-b)/(a+b))^(1/2)*(csc(d* x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(-8*cos(d*x+c)^2-16*cos(d*x+c)-8)*A *(1/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)* a*b^4*EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b)) ^(1/2))+(-246*cos(d*x+c)^2-492*cos(d*x+c)-246)*B*(1/(1+cos(d*x+c)))^(1/2)* (1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^4*b*EllipticE(((a-b)/(a+ b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(246*cos(d*x+c)^2+ 492*cos(d*x+c)+246)*B*(1/(1+cos(d*x+c)))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/( 1+cos(d*x+c)))^(1/2)*a^3*b^2*EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.54 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x, algo rithm="fricas")
Output:
2/945*(3*(35*A*a^5*cos(d*x + c)^3 + 75*B*a^5 + 88*A*a^4*b + 9*B*a^3*b^2 - 4*A*a^2*b^3 + 5*(9*B*a^5 + 10*A*a^4*b)*cos(d*x + c)^2 + (49*A*a^5 + 72*B*a ^4*b + 3*A*a^3*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))* sqrt(cos(d*x + c))*sin(d*x + c) - sqrt(1/2)*(225*I*B*a^5 + 264*I*A*a^4*b - 33*I*B*a^3*b^2 - 60*I*A*a^2*b^3 + 36*I*B*a*b^4 - 16*I*A*b^5)*sqrt(a)*weie rstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3* (3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a) - sqrt(1/2)*(-225*I*B*a^5 - 264*I*A*a^4*b + 33*I*B*a^3*b^2 + 60*I*A*a^2*b^3 - 36*I*B*a*b^4 + 16*I*A *b^5)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqr t(1/2)*(-147*I*A*a^5 - 246*I*B*a^4*b - 33*I*A*a^3*b^2 + 18*I*B*a^2*b^3 - 8 *I*A*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2* b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2* b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a)) - 3* sqrt(1/2)*(147*I*A*a^5 + 246*I*B*a^4*b + 33*I*A*a^3*b^2 - 18*I*B*a^2*b^3 + 8*I*A*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^ 2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^ 2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a)))/( a^4*d)
Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(9/2)*(a+b*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c)),x)
Output:
Timed out
\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x, algo rithm="maxima")
Output:
integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/2)*cos(d*x + c)^(9/ 2), x)
\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x, algo rithm="giac")
Output:
integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/2)*cos(d*x + c)^(9/ 2), x)
Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^{9/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \] Input:
int(cos(c + d*x)^(9/2)*(A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(3/2),x)
Output:
int(cos(c + d*x)^(9/2)*(A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(3/2), x)
\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) b^{2}+2 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) a b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a^{2} \] Input:
int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d* x)**2,x)*b**2 + 2*int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x),x)*a*b + int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d* x))*cos(c + d*x)**4,x)*a**2