Integrand size = 41, antiderivative size = 306 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {2 \left (15 A b^2+20 a b B+3 a^2 C+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (5 a^2 b B-5 b^3 B+3 a^3 C-3 a b^2 (5 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 a^2 A \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {2 (5 b B+3 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \] Output:
2/15*(15*A*b^2+20*B*a*b+3*C*a^2+9*C*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))/b/d/((a+b*cos(d*x+c))/(a+b))^( 1/2)-2/15*(5*B*a^2*b-5*B*b^3+3*a^3*C-3*a*b^2*(5*A+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))/b/d/(a+b *cos(d*x+c))^(1/2)+2*a^2*A*((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)+2/15*(5* B*b+3*C*a)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/d+2/5*C*(a+b*cos(d*x+c))^(3/2 )*sin(d*x+c)/d
Result contains complex when optimal does not.
Time = 10.12 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.49 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {\frac {4 \left (15 a^2 B+5 b^2 B+6 a b (5 A+2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (20 a b B+3 a^2 (10 A+C)+3 b^2 (5 A+3 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 i \left (15 A b^2+20 a b B+3 a^2 C+9 b^2 C\right ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {\frac {b (1+\cos (c+d x))}{-a+b}} \csc (c+d x) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right )}{a b^2 \sqrt {-\frac {1}{a+b}}}+4 \sqrt {a+b \cos (c+d x)} (5 b B+6 a C+3 b C \cos (c+d x)) \sin (c+d x)}{30 d} \] 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],x]
Output:
((4*(15*a^2*B + 5*b^2*B + 6*a*b*(5*A + 2*C))*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 *(20*a*b*B + 3*a^2*(10*A + C) + 3*b^2*(5*A + 3*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)*(15*A*b^2 + 20*a*b*B + 3*a^2*C + 9*b^2*C)*Sqrt[-((b*(-1 + C os[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[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*A rcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/ (a*b^2*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*(5*b*B + 6*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x])/(30*d)
Time = 2.60 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.03, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 3528, 27, 3042, 3528, 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.
| \(\displaystyle \int \sec (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left ((5 b B+3 a C) \cos ^2(c+d x)+(5 A b+3 C b+5 a B) \cos (c+d x)+5 a A\right ) \sec (c+d x)dx+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \sqrt {a+b \cos (c+d x)} \left ((5 b B+3 a C) \cos ^2(c+d x)+(5 A b+3 C b+5 a B) \cos (c+d x)+5 a A\right ) \sec (c+d x)dx+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left ((5 b B+3 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(5 A b+3 C b+5 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {\left (15 A a^2+\left (3 C a^2+20 b B a+15 A b^2+9 b^2 C\right ) \cos ^2(c+d x)+\left (15 B a^2+30 A b a+12 b C a+5 b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {\left (15 A a^2+\left (3 C a^2+20 b B a+15 A b^2+9 b^2 C\right ) \cos ^2(c+d x)+\left (15 B a^2+30 A b a+12 b C a+5 b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {15 A a^2+\left (3 C a^2+20 b B a+15 A b^2+9 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (15 B a^2+30 A b a+12 b C a+5 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\int -\frac {\left (15 a^2 A b-\left (3 C a^3+5 b B a^2-3 b^2 (5 A+C) a-5 b^3 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}+\frac {\int \frac {\left (15 a^2 A b-\left (3 C a^3+5 b B a^2-3 b^2 (5 A+C) a-5 b^3 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {\int \frac {15 A b a^2+\left (-3 C a^3-5 b B a^2+3 b^2 (5 A+C) a+5 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {15 A b a^2+\left (-3 C a^3-5 b B a^2+3 b^2 (5 A+C) a+5 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {15 A b a^2+\left (-3 C a^3-5 b B a^2+3 b^2 (5 A+C) a+5 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\int \frac {15 A b a^2+\left (-3 C a^3-5 b B a^2+3 b^2 (5 A+C) a+5 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {15 a^2 A b \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\left (3 a^3 C+5 a^2 b B-3 a b^2 (5 A+C)-5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}+\frac {2 \left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {15 a^2 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (3 a^3 C+5 a^2 b B-3 a b^2 (5 A+C)-5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {15 a^2 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\left (3 a^3 C+5 a^2 b B-3 a b^2 (5 A+C)-5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {15 a^2 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\left (3 a^3 C+5 a^2 b B-3 a b^2 (5 A+C)-5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {15 a^2 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (3 a^3 C+5 a^2 b B-3 a b^2 (5 A+C)-5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\frac {15 a^2 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 \left (3 a^3 C+5 a^2 b B-3 a b^2 (5 A+C)-5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\frac {15 a^2 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 \left (3 a^3 C+5 a^2 b B-3 a b^2 (5 A+C)-5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\frac {30 a^2 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (3 a^3 C+5 a^2 b B-3 a b^2 (5 A+C)-5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}\right )+\frac {2 (3 a C+5 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
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],x]
Output:
(2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (((2*(15*A*b^2 + 20* a*b*B + 3*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((-2*(5*a^2*b* B - 5*b^3*B + 3*a^3*C - 3*a*b^2*(5*A + 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]]) + (30*a^2*A*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)/3 + (2*(5*b*B + 3*a*C)*Sq rt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d))/5
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)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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. \(1329\) vs. \(2(296)=592\).
Time = 29.75 (sec) , antiderivative size = 1330, normalized size of antiderivative = 4.35
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1330\) |
| parts | \(\text {Expression too large to display}\) | \(1364\) |
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),x,me thod=_RETURNVERBOSE)
Output:
-2/15*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(24*C*co s(1/2*d*x+1/2*c)^7*b^3+20*B*cos(1/2*d*x+1/2*c)^5*b^3-48*C*cos(1/2*d*x+1/2* c)^5*b^3-30*B*cos(1/2*d*x+1/2*c)^3*b^3+30*C*cos(1/2*d*x+1/2*c)^3*b^3+10*B* cos(1/2*d*x+1/2*c)*b^3-6*C*cos(1/2*d*x+1/2*c)*b^3-15*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)*EllipticE(cos(1/2*d *x+1/2*c),(-2*b/(a-b))^(1/2))*b^3+5*B*b^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)*EllipticF(cos(1/2*d*x+1/2*c),(-2 *b/(a-b))^(1/2))-3*C*(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)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+ 3*C*(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)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3-9*C*(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)*EllipticE(cos (1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3+36*C*cos(1/2*d*x+1/2*c)^5*a*b^2+10 *B*cos(1/2*d*x+1/2*c)^3*a*b^2+12*C*cos(1/2*d*x+1/2*c)^3*a^2*b-54*C*cos(1/2 *d*x+1/2*c)^3*a*b^2-10*B*cos(1/2*d*x+1/2*c)*a*b^2-12*C*cos(1/2*d*x+1/2*c)* a^2*b+18*C*cos(1/2*d*x+1/2*c)*a*b^2-20*B*(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)*EllipticE(cos(1/2*d*x+1/2*c),(-2* b/(a-b))^(1/2))*a*b^2+3*C*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)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2 ))*b^2-3*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)...
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right ) \,d x } \] 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 ),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), x)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Timed out} \] 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),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right ) \,d x } \] 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 ),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), x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right ) \,d x } \] 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 ),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), x)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\cos \left (c+d\,x\right )} \,d x \] 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),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), x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )d x \right ) b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )d x \right ) a c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )d x \right ) b^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )d x \right ) a^{2} \] 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),x)
Output:
2*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x),x)*a*b + int(sqrt (cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x),x)*b*c + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x),x)*a*c + int(sqrt(cos(c + d*x)* b + a)*cos(c + d*x)**2*sec(c + d*x),x)*b**2 + int(sqrt(cos(c + d*x)*b + a) *sec(c + d*x),x)*a**2