Integrand size = 43, antiderivative size = 307 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=-\frac {(5 A b+4 a B-8 b C) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (4 a^2 B+8 b^2 B+a b (7 A+8 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (3 A b^2+12 a b B+4 a^2 (A+2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 d \sqrt {a+b \cos (c+d x)}}+\frac {(3 A b+4 a B) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^{3/2} \sec (c+d x) \tan (c+d x)}{2 d} \] Output:
-1/4*(5*A*b+4*B*a-8*C*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))/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+1/4*(4*B*a^2+ 8*B*b^2+a*b*(7*A+8*C))*((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)+1/4*(3*A*b^2+1 2*B*a*b+4*a^2*(A+2*C))*((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))/d/(a+b*cos(d*x+c))^(1/2)+1/4*(3*A*b+4 *B*a)*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/d+1/2*A*(a+b*cos(d*x+c))^(3/2)*sec (d*x+c)*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 7.36 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.89 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {\frac {2 \left (4 a A b+16 b^2 B+32 a b C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2 A+A b^2+20 a b B+16 a^2 C+8 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-5 A b^2-4 a b B+8 b^2 C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{16 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} \sec (c+d x) (5 A b \sin (c+d x)+4 a B \sin (c+d x))+\frac {1}{2} a A \sec (c+d x) \tan (c+d x)\right )}{d} \] Input:
Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2)*Sec[c + d*x]^3,x]
Output:
((2*(4*a*A*b + 16*b^2*B + 32*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Ell ipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(8*a^2*A + A*b^2 + 20*a*b*B + 16*a^2*C + 8*b^2*C)*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)*(-5*A*b^2 - 4*a*b*B + 8*b^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sq rt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*Elliptic E[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)))/(16*d) + (Sqrt[a + b*Cos[c + d*x]]*((S ec[c + d*x]*(5*A*b*Sin[c + d*x] + 4*a*B*Sin[c + d*x]))/4 + (a*A*Sec[c + d* x]*Tan[c + d*x])/2))/d
Time = 2.70 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.03, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 3526, 27, 3042, 3526, 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.
| \(\displaystyle \int \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (-b (A-4 C) \cos ^2(c+d x)+2 (2 b B+a (A+2 C)) \cos (c+d x)+3 A b+4 a B\right ) \sec ^2(c+d x)dx+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \sqrt {a+b \cos (c+d x)} \left (-b (A-4 C) \cos ^2(c+d x)+2 (2 b B+a (A+2 C)) \cos (c+d x)+3 A b+4 a B\right ) \sec ^2(c+d x)dx+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (-b (A-4 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 (2 b B+a (A+2 C)) \sin \left (c+d x+\frac {\pi }{2}\right )+3 A b+4 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{4} \left (\int \frac {\left (4 (A+2 C) a^2+12 b B a+3 A b^2-b (5 A b-8 C b+4 a B) \cos ^2(c+d x)+2 b (4 b B+a (A+8 C)) \cos (c+d x)\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {\left (4 (A+2 C) a^2+12 b B a+3 A b^2-b (5 A b-8 C b+4 a B) \cos ^2(c+d x)+2 b (4 b B+a (A+8 C)) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {4 (A+2 C) a^2+12 b B a+3 A b^2-b (5 A b-8 C b+4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (4 b B+a (A+8 C)) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\frac {\int -\frac {\left (b \left (4 (A+2 C) a^2+12 b B a+3 A b^2\right )+b \left (4 B a^2+b (7 A+8 C) a+8 b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\left ((4 a B+5 A b-8 b C) \int \sqrt {a+b \cos (c+d x)}dx\right )\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {\int \frac {\left (b \left (4 (A+2 C) a^2+12 b B a+3 A b^2\right )+b \left (4 B a^2+b (7 A+8 C) a+8 b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-(4 a B+5 A b-8 b C) \int \sqrt {a+b \cos (c+d x)}dx\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {\int \frac {b \left (4 (A+2 C) a^2+12 b B a+3 A b^2\right )+b \left (4 B a^2+b (7 A+8 C) a+8 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-(4 a B+5 A b-8 b C) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {\int \frac {b \left (4 (A+2 C) a^2+12 b B a+3 A b^2\right )+b \left (4 B a^2+b (7 A+8 C) a+8 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {(4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {\int \frac {b \left (4 (A+2 C) a^2+12 b B a+3 A b^2\right )+b \left (4 B a^2+b (7 A+8 C) a+8 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {(4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {\int \frac {b \left (4 (A+2 C) a^2+12 b B a+3 A b^2\right )+b \left (4 B a^2+b (7 A+8 C) a+8 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 (4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {b \left (4 a^2 B+a b (7 A+8 C)+8 b^2 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx+b \left (4 a^2 (A+2 C)+12 a b B+3 A b^2\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 (4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {b \left (4 a^2 B+a b (7 A+8 C)+8 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (4 a^2 (A+2 C)+12 a b B+3 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 (4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {b \left (4 a^2 (A+2 C)+12 a b B+3 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {b \left (4 a^2 B+a b (7 A+8 C)+8 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 (4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {b \left (4 a^2 (A+2 C)+12 a b B+3 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {b \left (4 a^2 B+a b (7 A+8 C)+8 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 (4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {b \left (4 a^2 (A+2 C)+12 a b B+3 A b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \left (4 a^2 B+a b (7 A+8 C)+8 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 (4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {\frac {b \left (4 a^2 (A+2 C)+12 a b B+3 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (4 a^2 B+a b (7 A+8 C)+8 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 (4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {\frac {b \left (4 a^2 (A+2 C)+12 a b B+3 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (4 a^2 B+a b (7 A+8 C)+8 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 (4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {\frac {2 b \left (4 a^2 B+a b (7 A+8 C)+8 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (4 a^2 (A+2 C)+12 a b B+3 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 (4 a B+5 A b-8 b C) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {(4 a B+3 A b) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec [c + d*x]^3,x]
Output:
(A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (((-2*(5* A*b + 4*a*B - 8*b*C)*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*b*(4*a^2*B + 8*b^2 *B + a*b*(7*A + 8*C))*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]]) + (2*b*(3*A*b^2 + 12*a* b*B + 4*a^2*(A + 2*C))*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 + ((3*A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d)/4
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_)]), 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_)] + (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. \(1742\) vs. \(2(295)=590\).
Time = 26.79 (sec) , antiderivative size = 1743, normalized size of antiderivative = 5.68
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1743\) |
| parts | \(\text {Expression too large to display}\) | \(1971\) |
Input:
int((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3,x, method=_RETURNVERBOSE)
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*B*b^2*(s in(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))-2*(A*b^2+2*B*a*b+C*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))+2*A*a^2*(-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)/(-1+2*cos(1/2*d*x+1/2*c)^2)^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)/(-1+2*cos(1/2*d*x+1/2*c)^2)-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*(si n(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))-1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*co s(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...
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^3,x, algorithm="fricas")
Output:
integral((C*b*cos(d*x + c)^3 + (C*a + B*b)*cos(d*x + c)^2 + A*a + (B*a + A *b)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^3, x)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x +c)**3,x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^3,x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/ 2)*sec(d*x + c)^3, x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^3,x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/ 2)*sec(d*x + c)^3, x)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^3} \,d x \] Input:
int(((a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/c os(c + d*x)^3,x)
Output:
int(((a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/c os(c + d*x)^3, x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3}d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}d x \right ) b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3}d x \right ) a c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3}d x \right ) b^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}d x \right ) a^{2} \] Input:
int((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3,x)
Output:
2*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**3,x)*a*b + int(s qrt(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**3,x)*b*c + int(sqrt( cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**3,x)*a*c + int(sqrt(cos( c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**3,x)*b**2 + int(sqrt(cos(c + d*x)*b + a)*sec(c + d*x)**3,x)*a**2