Integrand size = 42, antiderivative size = 375 \[ \int (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=-\frac {\left (16 a^2 B+3 b^2 B+30 a b C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{24 a d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (16 a^2 B+17 b^2 B+42 a b C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{24 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (12 a^2 b B-b^3 B+8 a^3 C+6 a b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{8 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (16 a^2 B+3 b^2 B+30 a b C\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{24 a d}+\frac {(7 b B+6 a C) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{12 d}+\frac {a B \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{3 d} \] Output:
-1/24*(16*B*a^2+3*B*b^2+30*C*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/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+1/ 24*(16*B*a^2+17*B*b^2+42*C*a*b)*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJaco biAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/d/(a+b*cos(d*x+c))^(1/2)+1/8*( 12*B*a^2*b-B*b^3+8*C*a^3+6*C*a*b^2)*((a+b*cos(d*x+c))/(a+b))^(1/2)*Ellipti cPi(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)+1/24*(16*B*a^2+3*B*b^2+30*C*a*b)*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a/d+ 1/12*(7*B*b+6*C*a)*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)*tan(d*x+c)/d+1/3*a*B* (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 = 6.95 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.69 \[ \int (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\frac {2 \left (28 a b^2 B+24 a^2 b C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (56 a^2 b B-9 b^3 B+48 a^3 C+6 a b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-16 a^2 b B-3 b^3 B-30 a b^2 C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{96 a d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{12} \sec ^2(c+d x) (7 b B \sin (c+d x)+6 a C \sin (c+d x))+\frac {\sec (c+d x) \left (16 a^2 B \sin (c+d x)+3 b^2 B \sin (c+d x)+30 a b C \sin (c+d x)\right )}{24 a}+\frac {1}{3} a B \sec ^2(c+d x) \tan (c+d x)\right )}{d} \] Input:
Integrate[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*S ec[c + d*x]^5,x]
Output:
((2*(28*a*b^2*B + 24*a^2*b*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*(56*a^2*b*B - 9*b^3*B + 48*a^3*C + 6*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Ellipti cPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-16 *a^2*b*B - 3*b^3*B - 30*a*b^2*C)*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)]*S qrt[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*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec [c + d*x]^2*(7*b*B*Sin[c + d*x] + 6*a*C*Sin[c + d*x]))/12 + (Sec[c + d*x]* (16*a^2*B*Sin[c + d*x] + 3*b^2*B*Sin[c + d*x] + 30*a*b*C*Sin[c + d*x]))/(2 4*a) + (a*B*Sec[c + d*x]^2*Tan[c + d*x])/3))/d
Time = 3.48 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.04, number of steps used = 26, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3508, 3042, 3468, 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.
| \(\displaystyle \int \sec ^5(c+d x) (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \sec ^4(c+d x) (a+b \cos (c+d x))^{3/2} (B+C \cos (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (3 b (a B+2 b C) \cos ^2(c+d x)+2 \left (2 B a^2+6 b C a+3 b^2 B\right ) \cos (c+d x)+a (7 b B+6 a C)\right ) \sec ^3(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (3 b (a B+2 b C) \cos ^2(c+d x)+2 \left (2 B a^2+6 b C a+3 b^2 B\right ) \cos (c+d x)+a (7 b B+6 a C)\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {3 b (a B+2 b C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (2 B a^2+6 b C a+3 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (7 b B+6 a C)}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\left (a b (7 b B+6 a C) \cos ^2(c+d x)+2 a \left (6 C a^2+13 b B a+12 b^2 C\right ) \cos (c+d x)+a \left (16 B a^2+30 b C a+3 b^2 B\right )\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\left (a b (7 b B+6 a C) \cos ^2(c+d x)+2 a \left (6 C a^2+13 b B a+12 b^2 C\right ) \cos (c+d x)+a \left (16 B a^2+30 b C a+3 b^2 B\right )\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {a b (7 b B+6 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (6 C a^2+13 b B a+12 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (16 B a^2+30 b C a+3 b^2 B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {\left (2 b (7 b B+6 a C) \cos (c+d x) a^2-b \left (16 B a^2+30 b C a+3 b^2 B\right ) \cos ^2(c+d x) a+3 \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {\left (2 b (7 b B+6 a C) \cos (c+d x) a^2-b \left (16 B a^2+30 b C a+3 b^2 B\right ) \cos ^2(c+d x) a+3 \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {2 b (7 b B+6 a C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2-b \left (16 B a^2+30 b C a+3 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+3 \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {-a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \int \sqrt {a+b \cos (c+d x)}dx-\frac {\int -\frac {\left (b \left (16 B a^2+42 b C a+17 b^2 B\right ) \cos (c+d x) a^2+3 b \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {\left (b \left (16 B a^2+42 b C a+17 b^2 B\right ) \cos (c+d x) a^2+3 b \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \int \sqrt {a+b \cos (c+d x)}dx}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 B a^2+42 b C a+17 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 B a^2+42 b C a+17 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 B a^2+42 b C a+17 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 B a^2+42 b C a+17 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {a^2 b \left (16 a^2 B+42 a b C+17 b^2 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx+3 a b \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {a^2 b \left (16 a^2 B+42 a b C+17 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 a b \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {a^2 b \left (16 a^2 B+42 a b C+17 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+3 a b \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {a^2 b \left (16 a^2 B+42 a b C+17 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+3 a b \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {3 a b \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 b \left (16 a^2 B+42 a b C+17 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {3 a b \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a^2 b \left (16 a^2 B+42 a b C+17 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {3 a b \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a^2 b \left (16 a^2 B+42 a b C+17 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}+\frac {\frac {\frac {2 a^2 b \left (16 a^2 B+42 a b C+17 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {6 a b \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (16 a^2 B+30 a b C+3 b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}+\frac {(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]
Output:
(a*B*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (((7*b* B + 6*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d) + ((( -2*a*(16*a^2*B + 3*b^2*B + 30*a*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*a^ 2*b*(16*a^2*B + 17*b^2*B + 42*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*El lipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (6*a*b *(12*a^2*b*B - b^3*B + 8*a^3*C + 6*a*b^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*a) + ((16*a^2*B + 3*b^2*B + 30*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*T an[c + d*x])/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*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[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 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. \(2326\) vs. \(2(359)=718\).
Time = 7.43 (sec) , antiderivative size = 2327, normalized size of antiderivative = 6.21
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2327\) |
| parts | \(\text {Expression too large to display}\) | \(2727\) |
Input:
int((a+b*cos(d*x+c))^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x,me thod=_RETURNVERBOSE)
Output:
-(-(-2*cos(1/2*d*x+1/2*c)^2*b-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B*a^2*(- 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)/(-1+2*cos(1/2*d*x+1/2*c)^2)^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)/(-1+2*cos(1/2*d *x+1/2*c)^2)^2-1/24*(16*a^2+15*b^2)/a^3*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)+ 5/48*b^2/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+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*cos(1/2*d*x+1/2*c)^2*b+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*cos(1/2*d*x+1/2*c)^2*b +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*cos(1/2*d*x+1/2*c)^2*b+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*cos (1/2*d*x+1/2*c)^2*b+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*cos(1/2*d*x+1/2*c)^2*b+a-b)/(...
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^ 5,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c )**5,x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{5} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^ 5,x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*(b*cos(d*x + c) + a)^(3/2)*s ec(d*x + c)^5, x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{5} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^ 5,x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*(b*cos(d*x + c) + a)^(3/2)*s ec(d*x + c)^5, x)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^5} \,d x \] Input:
int(((B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2))/cos(c + d*x)^5,x)
Output:
int(((B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2))/cos(c + d*x)^5, x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{5}d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{5}d x \right ) b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{5}d x \right ) a c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{5}d x \right ) b^{2} \] Input:
int((a+b*cos(d*x+c))^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x)
Output:
int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**5,x)*a*b + int(sqr t(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**5,x)*b*c + int(sqrt(co s(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**5,x)*a*c + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**5,x)*b**2