Integrand size = 33, antiderivative size = 378 \[ \int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=-\frac {\left (16 a^2 A-3 A b^2+6 a b B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{24 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (16 a^2 A-A b^2+18 a b B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{24 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (4 a^2 A b+A b^3+8 a^3 B-2 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{8 a^2 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (16 a^2 A-3 A b^2+6 a b B\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{24 a^2 d}+\frac {(A b+6 a B) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{12 a d}+\frac {A \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{3 d} \] Output:
-1/24*(16*A*a^2-3*A*b^2+6*B*a*b)*(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^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+1 /24*(16*A*a^2-A*b^2+18*B*a*b)*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJacobi AM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/a/d/(a+b*cos(d*x+c))^(1/2)+1/8*( 4*A*a^2*b+A*b^3+8*B*a^3-2*B*a*b^2)*((a+b*cos(d*x+c))/(a+b))^(1/2)*Elliptic Pi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(b/(a+b))^(1/2))/a^2/d/(a+b*cos(d*x+c))^(1 /2)+1/24*(16*A*a^2-3*A*b^2+6*B*a*b)*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a^2/ d+1/12*(A*b+6*B*a)*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)*tan(d*x+c)/a/d+1/3*A* (a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^2*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 7.19 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.68 \[ \int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {\frac {2 \left (4 a A b^2+24 a^2 b B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2 A b+9 A b^3+48 a^3 B-18 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-16 a^2 A b+3 A b^3-6 a b^2 B\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{96 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {\sec ^2(c+d x) (A b \sin (c+d x)+6 a B \sin (c+d x))}{12 a}+\frac {\sec (c+d x) \left (16 a^2 A \sin (c+d x)-3 A b^2 \sin (c+d x)+6 a b B \sin (c+d x)\right )}{24 a^2}+\frac {1}{3} A \sec ^2(c+d x) \tan (c+d x)\right )}{d} \] Input:
Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]
Output:
((2*(4*a*A*b^2 + 24*a^2*b*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*(8*a^2*A*b + 9* A*b^3 + 48*a^3*B - 18*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Elliptic Pi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-16* a^2*A*b + 3*A*b^3 - 6*a*b^2*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*Ar cSinh[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)]*Sqr t[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)))/(96*a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec [c + d*x]^2*(A*b*Sin[c + d*x] + 6*a*B*Sin[c + d*x]))/(12*a) + (Sec[c + d*x ]*(16*a^2*A*Sin[c + d*x] - 3*A*b^2*Sin[c + d*x] + 6*a*b*B*Sin[c + d*x]))/( 24*a^2) + (A*Sec[c + d*x]^2*Tan[c + d*x])/3))/d
Time = 3.25 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.03, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 3478, 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.
| \(\displaystyle \int \sec ^4(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3478 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (3 A b \cos ^2(c+d x)+2 (2 a A+3 b B) \cos (c+d x)+A b+6 a B\right ) \sec ^3(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (3 A b \cos ^2(c+d x)+2 (2 a A+3 b B) \cos (c+d x)+A b+6 a B\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {3 A b \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 (2 a A+3 b B) \sin \left (c+d x+\frac {\pi }{2}\right )+A b+6 a B}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\left (16 A a^2+6 b B a+2 (7 A b+6 a B) \cos (c+d x) a-3 A b^2+b (A b+6 a B) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\left (16 A a^2+6 b B a+2 (7 A b+6 a B) \cos (c+d x) a-3 A b^2+b (A b+6 a B) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {16 A a^2+6 b B a+2 (7 A b+6 a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a-3 A b^2+b (A b+6 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {\left (-b \left (16 A a^2+6 b B a-3 A b^2\right ) \cos ^2(c+d x)+2 a b (A b+6 a B) \cos (c+d x)+3 \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {\left (-b \left (16 A a^2+6 b B a-3 A b^2\right ) \cos ^2(c+d x)+2 a b (A b+6 a B) \cos (c+d x)+3 \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {-b \left (16 A a^2+6 b B a-3 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (A b+6 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {-\left (\left (16 a^2 A+6 a b B-3 A b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx\right )-\frac {\int -\frac {\left (3 b \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )+a b \left (16 A a^2+18 b B a-A b^2\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {\left (3 b \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )+a b \left (16 A a^2+18 b B a-A b^2\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\left (16 a^2 A+6 a b B-3 A b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {3 b \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )+a b \left (16 A a^2+18 b B a-A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\left (16 a^2 A+6 a b B-3 A b^2\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {3 b \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )+a b \left (16 A a^2+18 b B a-A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {3 b \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )+a b \left (16 A a^2+18 b B a-A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {3 b \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )+a b \left (16 A a^2+18 b B a-A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {a b \left (16 a^2 A+18 a b B-A b^2\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx+3 b \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 \left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {a b \left (16 a^2 A+18 a b B-A b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 b \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {a b \left (16 a^2 A+18 a b B-A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+3 b \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {a b \left (16 a^2 A+18 a b B-A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+3 b \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {3 b \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a b \left (16 a^2 A+18 a b B-A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {3 b \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a b \left (16 a^2 A+18 a b B-A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {3 b \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a b \left (16 a^2 A+18 a b B-A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{a d}+\frac {\frac {\frac {2 a b \left (16 a^2 A+18 a b B-A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {6 b \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (16 a^2 A+6 a b B-3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}+\frac {(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
Input:
Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]
Output:
(A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (((A*b + 6*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + (((-2 *(16*a^2*A - 3*A*b^2 + 6*a*b*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*b*(16 *a^2*A - A*b^2 + 18*a*b*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*b*(4*a^2*A*b + A*b^3 + 8*a^3*B - 2*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Elliptic Pi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]))/b)/(2*a) + ((16*a^2*A - 3*A*b^2 + 6*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/( a*d))/(4*a))/6
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*a - A*b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f* x])^n/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[c*(a*A - b*B)*( m + 1) + d*n*(A*b - a*B) + (d*(a*A - b*B)*(m + 1) - c*(A*b - a*B)*(m + 2))* Sin[e + f*x] - d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x]^2, x], x], x] /; Free Q[{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] && LtQ[m, -1] && GtQ[n, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2212\) vs. \(2(362)=724\).
Time = 16.37 (sec) , antiderivative size = 2213, normalized size of antiderivative = 5.85
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2213\) |
| parts | \(\text {Expression too large to display}\) | \(2716\) |
Input:
int((a+cos(d*x+c)*b)^(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)^4,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*b+B*a )*(-1/2*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)^2+3/4*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)-1/8*b/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)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+3/8/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))-3/8*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)*s in(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) )-1/2*(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)*Ellipti cPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))-3/8/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)*EllipticPi(cos(1/2*d*x+1/2*c),2,(- 2*b/(a-b))^(1/2))*b^2)+2*A*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*...
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)^4,x, algorith m="fricas")
Output:
Timed out
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)**4,x)
Output:
Timed out
\[ \int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)^4,x, algorith m="maxima")
Output:
integrate((B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^4, x)
\[ \int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)^4,x, algorith m="giac")
Output:
integrate((B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^4, x)
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,\sqrt {a+b\,\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^4} \,d x \] Input:
int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2))/cos(c + d*x)^4,x)
Output:
int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2))/cos(c + d*x)^4, x)
\[ \int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) a \] Input:
int((a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)^4,x)
Output:
int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**4,x)*b + int(sqrt( cos(c + d*x)*b + a)*sec(c + d*x)**4,x)*a