Integrand size = 43, antiderivative size = 284 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {(47 A-38 B+24 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 a^{3/2} d}+\frac {(17 A-13 B+9 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(21 A-14 B+12 C) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A-12 B+6 C) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A-3 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}} \] Output:
-1/8*(47*A-38*B+24*C)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^ (3/2)/d+1/4*(17*A-13*B+9*C)*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec (d*x+c))^(1/2))*2^(1/2)/a^(3/2)/d-1/2*(A-B+C)*cos(d*x+c)^2*sin(d*x+c)/d/(a +a*sec(d*x+c))^(3/2)+1/8*(21*A-14*B+12*C)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^ (1/2)-1/12*(13*A-12*B+6*C)*cos(d*x+c)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^(1/2 )+1/6*(5*A-3*B+3*C)*cos(d*x+c)^2*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 5.56 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.71 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {-24 C \left (-2 (3+2 \cos (c+d x)) \sqrt {1-\sec (c+d x)}+12 \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right ) (1+\sec (c+d x))-9 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right ) (1+\sec (c+d x))\right ) \tan (c+d x)-6 B \left (4 \sqrt {1-\sec (c+d x)} \sin (2 (c+d x))-13 \left (7 \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )-4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )+\cos (c+d x) (-1+2 \cos (c+d x)) \sqrt {1-\sec (c+d x)}\right ) (1+\sec (c+d x)) \tan (c+d x)+40 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)} (1+\sec (c+d x)) \tan (c+d x)\right )-A \left (48 \cos ^2(c+d x) \sqrt {1-\sec (c+d x)} \sin (c+d x)+17 \left (27 \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )-24 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )+\cos (c+d x) \left (-21+2 \cos (c+d x)-8 \cos ^2(c+d x)\right ) \sqrt {1-\sec (c+d x)}\right ) (1+\sec (c+d x)) \tan (c+d x)+336 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4,\frac {3}{2},1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)} (1+\sec (c+d x)) \tan (c+d x)\right )}{96 d \sqrt {1-\sec (c+d x)} (a (1+\sec (c+d x)))^{3/2}} \] Input:
Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a* Sec[c + d*x])^(3/2),x]
Output:
(-24*C*(-2*(3 + 2*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]] + 12*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x]) - 9*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*(1 + Sec[c + d*x]))*Tan[c + d*x] - 6*B*(4*Sqrt[1 - Sec[c + d*x]]*Sin[2*(c + d*x)] - 13*(7*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 4*Sqrt[2] *ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(-1 + 2*Cos[c + d* x])*Sqrt[1 - Sec[c + d*x]])*(1 + Sec[c + d*x])*Tan[c + d*x] + 40*Hypergeom etric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*Tan[c + d*x]) - A*(48*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 17*(27*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 24*Sqrt[2]*ArcTanh[Sqrt[ 1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(-21 + 2*Cos[c + d*x] - 8*Cos[c + d*x]^2)*Sqrt[1 - Sec[c + d*x]])*(1 + Sec[c + d*x])*Tan[c + d*x] + 336*Hy pergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*Tan[c + d*x]))/(96*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))
Time = 1.97 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {3042, 4572, 27, 3042, 4510, 25, 3042, 4510, 27, 3042, 4510, 27, 3042, 4408, 3042, 4261, 216, 4282, 216}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a \sec (c+d x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4572 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (2 a (5 A-3 B+3 C)-a (7 A-7 B+3 C) \sec (c+d x))}{2 \sqrt {\sec (c+d x) a+a}}dx}{2 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (2 a (5 A-3 B+3 C)-a (7 A-7 B+3 C) \sec (c+d x))}{\sqrt {\sec (c+d x) a+a}}dx}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 a (5 A-3 B+3 C)-a (7 A-7 B+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {\int -\frac {\cos ^2(c+d x) \left (2 a^2 (13 A-12 B+6 C)-5 a^2 (5 A-3 B+3 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x) a+a}}dx}{3 a}+\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {\cos ^2(c+d x) \left (2 a^2 (13 A-12 B+6 C)-5 a^2 (5 A-3 B+3 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x) a+a}}dx}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {2 a^2 (13 A-12 B+6 C)-5 a^2 (5 A-3 B+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {\int -\frac {3 \cos (c+d x) \left (a^3 (21 A-14 B+12 C)-a^3 (13 A-12 B+6 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x) a+a}}dx}{2 a}+\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \int \frac {\cos (c+d x) \left (a^3 (21 A-14 B+12 C)-a^3 (13 A-12 B+6 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x) a+a}}dx}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \int \frac {a^3 (21 A-14 B+12 C)-a^3 (13 A-12 B+6 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {\int -\frac {a^4 (47 A-38 B+24 C)-a^4 (21 A-14 B+12 C) \sec (c+d x)}{2 \sqrt {\sec (c+d x) a+a}}dx}{a}+\frac {a^3 (21 A-14 B+12 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-14 B+12 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^4 (47 A-38 B+24 C)-a^4 (21 A-14 B+12 C) \sec (c+d x)}{\sqrt {\sec (c+d x) a+a}}dx}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-14 B+12 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^4 (47 A-38 B+24 C)-a^4 (21 A-14 B+12 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4408 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-14 B+12 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {a^3 (47 A-38 B+24 C) \int \sqrt {\sec (c+d x) a+a}dx-4 a^4 (17 A-13 B+9 C) \int \frac {\sec (c+d x)}{\sqrt {\sec (c+d x) a+a}}dx}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-14 B+12 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {a^3 (47 A-38 B+24 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-4 a^4 (17 A-13 B+9 C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-14 B+12 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {-4 a^4 (17 A-13 B+9 C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {2 a^4 (47 A-38 B+24 C) \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{\sec (c+d x) a+a}+a}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-14 B+12 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^{7/2} (47 A-38 B+24 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}-4 a^4 (17 A-13 B+9 C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-14 B+12 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {8 a^4 (17 A-13 B+9 C) \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{\sec (c+d x) a+a}+2 a}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}+\frac {2 a^{7/2} (47 A-38 B+24 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-14 B+12 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^{7/2} (47 A-38 B+24 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}-\frac {4 \sqrt {2} a^{7/2} (17 A-13 B+9 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{d}}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
Input:
Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]
Output:
-1/2*((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^(3/ 2)) + ((2*a*(5*A - 3*B + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a *Sec[c + d*x]]) - ((a^2*(13*A - 12*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(d* Sqrt[a + a*Sec[c + d*x]]) - (3*(-1/2*((2*a^(7/2)*(47*A - 38*B + 24*C)*ArcT an[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (4*Sqrt[2]*a^(7/2 )*(17*A - 13*B + 9*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Se c[c + d*x]])])/d)/a + (a^3*(21*A - 14*B + 12*C)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])))/(2*a))/(3*a))/(4*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c/a Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[(b*c - a*d)/a Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; F reeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d *n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*B* n - A*b*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-(a*A - b*B + a*C))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*B*n - b*C*n - A*b*(2*m + n + 1) - (b*B*(m + n + 1) - a*(A*(m + n + 1) - C*(m - n)))*Csc[ e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Leaf count of result is larger than twice the leaf count of optimal. \(674\) vs. \(2(249)=498\).
Time = 58.90 (sec) , antiderivative size = 675, normalized size of antiderivative = 2.38
| method | result | size |
| default | \(\frac {\left (\left (141 \cos \left (d x +c \right )^{2}+282 \cos \left (d x +c \right )+141\right ) A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\csc \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot \left (d x +c \right )^{2}-1}}\right )+\left (-114 \cos \left (d x +c \right )^{2}-228 \cos \left (d x +c \right )-114\right ) B \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\csc \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot \left (d x +c \right )^{2}-1}}\right )+\left (72 \cos \left (d x +c \right )^{2}+144 \cos \left (d x +c \right )+72\right ) C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\csc \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot \left (d x +c \right )^{2}-1}}\right )+\left (102 \cos \left (d x +c \right )^{2}+204 \cos \left (d x +c \right )+102\right ) \sqrt {2}\, A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\left (-78 \cos \left (d x +c \right )^{2}-156 \cos \left (d x +c \right )-78\right ) \sqrt {2}\, B \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\left (54 \cos \left (d x +c \right )^{2}+108 \cos \left (d x +c \right )+54\right ) \sqrt {2}\, C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (8 \cos \left (d x +c \right )^{3}-6 \cos \left (d x +c \right )^{2}+37 \cos \left (d x +c \right )+63\right ) A +\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (12 \cos \left (d x +c \right )^{2}-18 \cos \left (d x +c \right )-42\right ) B +\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (24 \cos \left (d x +c \right )+36\right ) C \right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{24 d \,a^{2} \left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right )}\) | \(675\) |
Input:
int(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(3/2),x, method=_RETURNVERBOSE)
Output:
1/24/d/a^2*((141*cos(d*x+c)^2+282*cos(d*x+c)+141)*A*(-cos(d*x+c)/(cos(d*x+ c)+1))^(1/2)*arctanh(2^(1/2)*(-csc(d*x+c)+cot(d*x+c))/(csc(d*x+c)^2-2*csc( d*x+c)*cot(d*x+c)+cot(d*x+c)^2-1)^(1/2))+(-114*cos(d*x+c)^2-228*cos(d*x+c) -114)*B*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(2^(1/2)*(-csc(d*x+c)+co t(d*x+c))/(csc(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c)+cot(d*x+c)^2-1)^(1/2))+(72 *cos(d*x+c)^2+144*cos(d*x+c)+72)*C*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arct anh(2^(1/2)*(-csc(d*x+c)+cot(d*x+c))/(csc(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c) +cot(d*x+c)^2-1)^(1/2))+(102*cos(d*x+c)^2+204*cos(d*x+c)+102)*2^(1/2)*A*(- cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-c ot(d*x+c)+csc(d*x+c))+(-78*cos(d*x+c)^2-156*cos(d*x+c)-78)*2^(1/2)*B*(-cos (d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cot( d*x+c)+csc(d*x+c))+(54*cos(d*x+c)^2+108*cos(d*x+c)+54)*2^(1/2)*C*(-cos(d*x +c)/(cos(d*x+c)+1))^(1/2)*ln((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cot(d*x+ c)+csc(d*x+c))+sin(d*x+c)*cos(d*x+c)*(8*cos(d*x+c)^3-6*cos(d*x+c)^2+37*cos (d*x+c)+63)*A+sin(d*x+c)*cos(d*x+c)*(12*cos(d*x+c)^2-18*cos(d*x+c)-42)*B+s in(d*x+c)*cos(d*x+c)*(24*cos(d*x+c)+36)*C)*(a*(1+sec(d*x+c)))^(1/2)/(cos(d *x+c)^2+2*cos(d*x+c)+1)
Time = 22.56 (sec) , antiderivative size = 724, normalized size of antiderivative = 2.55 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(3 /2),x, algorithm="fricas")
Output:
[-1/48*(6*sqrt(2)*((17*A - 13*B + 9*C)*cos(d*x + c)^2 + 2*(17*A - 13*B + 9 *C)*cos(d*x + c) + 17*A - 13*B + 9*C)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqr t((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d *x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 3*((47*A - 38*B + 24*C)*cos(d*x + c)^2 + 2*(47*A - 38*B + 24*C)*cos(d*x + c) + 47*A - 38*B + 24*C)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sq rt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d* x + c) - a)/(cos(d*x + c) + 1)) - 2*(8*A*cos(d*x + c)^4 - 6*(A - 2*B)*cos( d*x + c)^3 + (37*A - 18*B + 24*C)*cos(d*x + c)^2 + 3*(21*A - 14*B + 12*C)* cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^2*d *cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d), -1/24*(6*sqrt(2)*((17*A - 13*B + 9*C)*cos(d*x + c)^2 + 2*(17*A - 13*B + 9*C)*cos(d*x + c) + 17*A - 13*B + 9*C)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) *cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - 3*((47*A - 38*B + 24*C)*cos(d*x + c)^2 + 2*(47*A - 38*B + 24*C)*cos(d*x + c) + 47*A - 38*B + 24*C)*sqrt(a)*a rctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d* x + c))) - (8*A*cos(d*x + c)^4 - 6*(A - 2*B)*cos(d*x + c)^3 + (37*A - 18*B + 24*C)*cos(d*x + c)^2 + 3*(21*A - 14*B + 12*C)*cos(d*x + c))*sqrt((a*cos (d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d *cos(d*x + c) + a^2*d)]
\[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cos(d*x+c)**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))* *(3/2),x)
Output:
Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*cos(c + d*x)**3/(a*(sec( c + d*x) + 1))**(3/2), x)
\[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(3 /2),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*cos(d*x + c)^3/(a*sec(d* x + c) + a)^(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 1056 vs. \(2 (249) = 498\).
Time = 1.19 (sec) , antiderivative size = 1056, normalized size of antiderivative = 3.72 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(3 /2),x, algorithm="giac")
Output:
-1/48*(6*sqrt(2)*(17*A - 13*B + 9*C)*log((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2)/(sqrt(-a)*a*sgn(cos(d*x + c))) - 3 *(47*A - 38*B + 24*C)*log(abs(-1947111321950560360698936123457536*(sqrt(-a )*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - 38942226 43901120721397872246915072*sqrt(2)*abs(a) + 584133396585168108209680837037 2608*a)/abs(-1947111321950560360698936123457536*(sqrt(-a)*tan(1/2*d*x + 1/ 2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 + 38942226439011207213978722 46915072*sqrt(2)*abs(a) + 5841333965851681082096808370372608*a))/(sqrt(-a) *abs(a)*sgn(cos(d*x + c))) - 12*(sqrt(2)*A*a*sgn(cos(d*x + c)) - sqrt(2)*B *a*sgn(cos(d*x + c)) + sqrt(2)*C*a*sgn(cos(d*x + c)))*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*tan(1/2*d*x + 1/2*c)/a^3 - 4*sqrt(2)*(339*(sqrt(-a)*tan(1/ 2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^10*A - 174*(sqrt(-a) *tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^10*B + 72*(sq rt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^10*C - 3165*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)) ^8*A*a + 1842*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c )^2 + a))^8*B*a - 888*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^8*C*a + 9198*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*ta n(1/2*d*x + 1/2*c)^2 + a))^6*A*a^2 - 5292*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^6*B*a^2 + 3024*(sqrt(-a)*tan(1/2*...
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int((cos(c + d*x)^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + a/cos(c + d*x))^(3/2),x)
Output:
int((cos(c + d*x)^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + a/cos(c + d*x))^(3/2), x)
\[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{2}+2 \sec \left (d x +c \right )+1}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )}{\sec \left (d x +c \right )^{2}+2 \sec \left (d x +c \right )+1}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{2}+2 \sec \left (d x +c \right )+1}d x \right ) a \right )}{a^{2}} \] Input:
int(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(3/2),x)
Output:
(sqrt(a)*(int((sqrt(sec(c + d*x) + 1)*cos(c + d*x)**3*sec(c + d*x)**2)/(se c(c + d*x)**2 + 2*sec(c + d*x) + 1),x)*c + int((sqrt(sec(c + d*x) + 1)*cos (c + d*x)**3*sec(c + d*x))/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1),x)*b + i nt((sqrt(sec(c + d*x) + 1)*cos(c + d*x)**3)/(sec(c + d*x)**2 + 2*sec(c + d *x) + 1),x)*a))/a**2