Integrand size = 33, antiderivative size = 465 \[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=-\frac {\left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{192 a d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (356 a^2 A b+133 A b^3+128 a^3 B+472 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{192 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (48 a^4 A+120 a^2 A b^2-5 A b^4+160 a^3 b B+40 a b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{64 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a d}+\frac {\left (36 a^2 A+59 A b^2+104 a b B\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{96 d}+\frac {a (11 A b+8 a B) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {a A (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d} \] Output:
-1/192*(284*A*a^2*b+15*A*b^3+128*B*a^3+264*B*a*b^2)*(a+b*cos(d*x+c))^(1/2) *EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/a/d/((a+b*cos(d*x+c ))/(a+b))^(1/2)+1/192*(356*A*a^2*b+133*A*b^3+128*B*a^3+472*B*a*b^2)*((a+b* cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^( 1/2))/d/(a+b*cos(d*x+c))^(1/2)+1/64*(48*A*a^4+120*A*a^2*b^2-5*A*b^4+160*B* a^3*b+40*B*a*b^3)*((a+b*cos(d*x+c))/(a+b))^(1/2)*EllipticPi(sin(1/2*d*x+1/ 2*c),2,2^(1/2)*(b/(a+b))^(1/2))/a/d/(a+b*cos(d*x+c))^(1/2)+1/192*(284*A*a^ 2*b+15*A*b^3+128*B*a^3+264*B*a*b^2)*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a/d+ 1/96*(36*A*a^2+59*A*b^2+104*B*a*b)*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)*tan(d *x+c)/d+1/24*a*(11*A*b+8*B*a)*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^2*tan(d*x+ c)/d+1/4*a*A*(a+b*cos(d*x+c))^(3/2)*sec(d*x+c)^3*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 7.51 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.57 \[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {\frac {2 \left (144 a^3 A b+236 a A b^3+416 a^2 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (288 a^4 A+436 a^2 A b^2-45 A b^4+832 a^3 b B-24 a b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-284 a^2 A b^2-15 A b^4-128 a^3 b B-264 a b^3 B\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{768 a d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{24} \sec ^3(c+d x) \left (17 a A b \sin (c+d x)+8 a^2 B \sin (c+d x)\right )+\frac {1}{96} \sec ^2(c+d x) \left (36 a^2 A \sin (c+d x)+59 A b^2 \sin (c+d x)+104 a b B \sin (c+d x)\right )+\frac {\sec (c+d x) \left (284 a^2 A b \sin (c+d x)+15 A b^3 \sin (c+d x)+128 a^3 B \sin (c+d x)+264 a b^2 B \sin (c+d x)\right )}{192 a}+\frac {1}{4} a^2 A \sec ^3(c+d x) \tan (c+d x)\right )}{d} \] Input:
Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x ]
Output:
((2*(144*a^3*A*b + 236*a*A*b^3 + 416*a^2*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/ (a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(288*a^4*A + 436*a^2*A*b^2 - 45*A*b^4 + 832*a^3*b*B - 24*a*b^3*B)*Sqrt [(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/ Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-284*a^2*A*b^2 - 15*A*b^4 - 128*a^3*b*B - 264*a*b^3*B)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d *x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[ I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[ c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[ c + d*x])^2)))/(768*a*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^3*(17* a*A*b*Sin[c + d*x] + 8*a^2*B*Sin[c + d*x]))/24 + (Sec[c + d*x]^2*(36*a^2*A *Sin[c + d*x] + 59*A*b^2*Sin[c + d*x] + 104*a*b*B*Sin[c + d*x]))/96 + (Sec [c + d*x]*(284*a^2*A*b*Sin[c + d*x] + 15*A*b^3*Sin[c + d*x] + 128*a^3*B*Si n[c + d*x] + 264*a*b^2*B*Sin[c + d*x]))/(192*a) + (a^2*A*Sec[c + d*x]^3*Ta n[c + d*x])/4))/d
Time = 4.23 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.04, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3042, 3468, 27, 3042, 3526, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 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.
| \(\displaystyle \int \sec ^5(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (b (3 a A+8 b B) \cos ^2(c+d x)+2 \left (3 A a^2+8 b B a+4 A b^2\right ) \cos (c+d x)+a (11 A b+8 a B)\right ) \sec ^4(c+d x)dx+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \sqrt {a+b \cos (c+d x)} \left (b (3 a A+8 b B) \cos ^2(c+d x)+2 \left (3 A a^2+8 b B a+4 A b^2\right ) \cos (c+d x)+a (11 A b+8 a B)\right ) \sec ^4(c+d x)dx+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b (3 a A+8 b B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (3 A a^2+8 b B a+4 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (11 A b+8 a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int \frac {\left (3 b \left (8 B a^2+17 A b a+16 b^2 B\right ) \cos ^2(c+d x)+2 \left (16 B a^3+49 A b a^2+72 b^2 B a+24 A b^3\right ) \cos (c+d x)+a \left (36 A a^2+104 b B a+59 A b^2\right )\right ) \sec ^3(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {\left (3 b \left (8 B a^2+17 A b a+16 b^2 B\right ) \cos ^2(c+d x)+2 \left (16 B a^3+49 A b a^2+72 b^2 B a+24 A b^3\right ) \cos (c+d x)+a \left (36 A a^2+104 b B a+59 A b^2\right )\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {3 b \left (8 B a^2+17 A b a+16 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (16 B a^3+49 A b a^2+72 b^2 B a+24 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (36 A a^2+104 b B a+59 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {\left (a b \left (36 A a^2+104 b B a+59 A b^2\right ) \cos ^2(c+d x)+2 a \left (36 A a^3+152 b B a^2+161 A b^2 a+96 b^3 B\right ) \cos (c+d x)+a \left (128 B a^3+284 A b a^2+264 b^2 B a+15 A b^3\right )\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {\left (a b \left (36 A a^2+104 b B a+59 A b^2\right ) \cos ^2(c+d x)+2 a \left (36 A a^3+152 b B a^2+161 A b^2 a+96 b^3 B\right ) \cos (c+d x)+a \left (128 B a^3+284 A b a^2+264 b^2 B a+15 A b^3\right )\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {a b \left (36 A a^2+104 b B a+59 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (36 A a^3+152 b B a^2+161 A b^2 a+96 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (128 B a^3+284 A b a^2+264 b^2 B a+15 A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\int \frac {\left (2 b \left (36 A a^2+104 b B a+59 A b^2\right ) \cos (c+d x) a^2-b \left (128 B a^3+284 A b a^2+264 b^2 B a+15 A b^3\right ) \cos ^2(c+d x) a+3 \left (48 A a^4+160 b B a^3+120 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\int \frac {\left (2 b \left (36 A a^2+104 b B a+59 A b^2\right ) \cos (c+d x) a^2-b \left (128 B a^3+284 A b a^2+264 b^2 B a+15 A b^3\right ) \cos ^2(c+d x) a+3 \left (48 A a^4+160 b B a^3+120 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\int \frac {2 b \left (36 A a^2+104 b B a+59 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2-b \left (128 B a^3+284 A b a^2+264 b^2 B a+15 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+3 \left (48 A a^4+160 b B a^3+120 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {-a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \int \sqrt {a+b \cos (c+d x)}dx-\frac {\int -\frac {\left (b \left (128 B a^3+356 A b a^2+472 b^2 B a+133 A b^3\right ) \cos (c+d x) a^2+3 b \left (48 A a^4+160 b B a^3+120 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {\int \frac {\left (b \left (128 B a^3+356 A b a^2+472 b^2 B a+133 A b^3\right ) \cos (c+d x) a^2+3 b \left (48 A a^4+160 b B a^3+120 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \int \sqrt {a+b \cos (c+d x)}dx}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (128 B a^3+356 A b a^2+472 b^2 B a+133 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (48 A a^4+160 b B a^3+120 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (128 B a^3+356 A b a^2+472 b^2 B a+133 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (48 A a^4+160 b B a^3+120 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (128 B a^3+356 A b a^2+472 b^2 B a+133 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (48 A a^4+160 b B a^3+120 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (128 B a^3+356 A b a^2+472 b^2 B a+133 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (48 A a^4+160 b B a^3+120 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {a^2 b \left (128 a^3 B+356 a^2 A b+472 a b^2 B+133 A b^3\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx+3 a b \left (48 a^4 A+160 a^3 b B+120 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {a^2 b \left (128 a^3 B+356 a^2 A b+472 a b^2 B+133 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 a b \left (48 a^4 A+160 a^3 b B+120 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {\frac {a^2 b \left (128 a^3 B+356 a^2 A b+472 a b^2 B+133 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+3 a b \left (48 a^4 A+160 a^3 b B+120 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {\frac {a^2 b \left (128 a^3 B+356 a^2 A b+472 a b^2 B+133 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+3 a b \left (48 a^4 A+160 a^3 b B+120 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {3 a b \left (48 a^4 A+160 a^3 b B+120 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 b \left (128 a^3 B+356 a^2 A b+472 a b^2 B+133 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {\frac {3 a b \left (48 a^4 A+160 a^3 b B+120 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a^2 b \left (128 a^3 B+356 a^2 A b+472 a b^2 B+133 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\frac {\frac {3 a b \left (48 a^4 A+160 a^3 b B+120 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a^2 b \left (128 a^3 B+356 a^2 A b+472 a b^2 B+133 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (36 a^2 A+104 a b B+59 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {\frac {\left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}+\frac {\frac {\frac {2 a^2 b \left (128 a^3 B+356 a^2 A b+472 a b^2 B+133 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {6 a b \left (48 a^4 A+160 a^3 b B+120 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]
Output:
(a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((a*( 11*A*b + 8*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d ) + (((36*a^2*A + 59*A*b^2 + 104*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d *x]*Tan[c + d*x])/(2*d) + (((-2*a*(284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 26 4*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)]) /(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((2*a^2*b*(356*a^2*A*b + 133*A*b ^3 + 128*a^3*B + 472*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF [(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (6*a*b*(48*a^ 4*A + 120*a^2*A*b^2 - 5*A*b^4 + 160*a^3*b*B + 40*a*b^3*B)*Sqrt[(a + b*Cos[ c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]))/b)/(2*a) + ((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a* b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d)/(4*a))/6)/8
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[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[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(b*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2 , 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(3547\) vs. \(2(445)=890\).
Time = 537.63 (sec) , antiderivative size = 3548, normalized size of antiderivative = 7.63
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3548\) |
| parts | \(\text {Expression too large to display}\) | \(4064\) |
Input:
int((a+cos(d*x+c)*b)^(5/2)*(A+B*cos(d*x+c))*sec(d*x+c)^5,x,method=_RETURNV ERBOSE)
Output:
-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*a^2*(3*A *b+B*a)*(-1/3*cos(1/2*d*x+1/2*c)/a*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/ 2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2-1)^3+5/12*b/a^2*cos(1/2*d*x+ 1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos (1/2*d*x+1/2*c)^2-1)^2-1/24*(16*a^2+15*b^2)/a^3*cos(1/2*d*x+1/2*c)*(-2*b*s in(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c )^2-1)+5/48*b^2/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^ 2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2) ^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+1/3*(sin(1/2*d*x+1 /2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2* d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c ),(-2*b/(a-b))^(1/2))-1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1 /2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2 *c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+1/3/a*(sin(1 /2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b* sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*b*EllipticE(cos(1/2 *d*x+1/2*c),(-2*b/(a-b))^(1/2))-5/16*b^2/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)* ((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a +b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^ (1/2))+5/16/a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2...
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c))*sec(d*x+c)^5,x, algorith m="fricas")
Output:
Timed out
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c))*sec(d*x+c)**5,x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{5} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c))*sec(d*x+c)^5,x, algorith m="maxima")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)*sec(d*x + c)^5, x)
\[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{5} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c))*sec(d*x+c)^5,x, algorith m="giac")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)*sec(d*x + c)^5, x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^5} \,d x \] Input:
int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2))/cos(c + d*x)^5,x)
Output:
int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2))/cos(c + d*x)^5, x)
\[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{5}d x \right ) a^{2} b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{5}d x \right ) b^{3}+3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{5}d x \right ) a \,b^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) a^{3} \] Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c))*sec(d*x+c)^5,x)
Output:
3*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**5,x)*a**2*b + in t(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**5,x)*b**3 + 3*int (sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**5,x)*a*b**2 + int( sqrt(cos(c + d*x)*b + a)*sec(c + d*x)**5,x)*a**3