Integrand size = 33, antiderivative size = 268 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {(47 A-38 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 a^{3/2} d}+\frac {(17 A-13 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {7 (3 A-2 B) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A-12 B) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A-3 B) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}} \] Output:
-1/8*(47*A-38*B)*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)*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)*cos(d*x+c)^2*sin(d*x+c)/d/(a+a*sec(d*x+ c))^(3/2)+7/8*(3*A-2*B)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^(1/2)-1/12*(13*A-1 2*B)*cos(d*x+c)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^(1/2)+1/6*(5*A-3*B)*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 = 6.12 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.86 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {B \cos (c+d x) \sin (c+d x)}{2 d (a (1+\sec (c+d x)))^{3/2}}-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a (1+\sec (c+d x)))^{3/2}}-\frac {B (1+\sec (c+d x))^{3/2} \left (\frac {40 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1-\sec (c+d x)\right ) \tan (c+d x)}{d \sqrt {1+\sec (c+d x)}}-\frac {13 \left (7 \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )-4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )-\cos (c+d x) \sqrt {1-\sec (c+d x)}+2 \cos ^2(c+d x) \sqrt {1-\sec (c+d x)}\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}\right )}{16 (a (1+\sec (c+d x)))^{3/2}}-\frac {A (1+\sec (c+d x))^{3/2} \left (\frac {336 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4,\frac {3}{2},1-\sec (c+d x)\right ) \tan (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {17 \left (27 \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )-24 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )-21 \cos (c+d x) \sqrt {1-\sec (c+d x)}+2 \cos ^2(c+d x) \sqrt {1-\sec (c+d x)}-8 \cos ^3(c+d x) \sqrt {1-\sec (c+d x)}\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}\right )}{96 (a (1+\sec (c+d x)))^{3/2}} \] Input:
Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2) ,x]
Output:
-1/2*(B*Cos[c + d*x]*Sin[c + d*x])/(d*(a*(1 + Sec[c + d*x]))^(3/2)) - (A*C os[c + d*x]^2*Sin[c + d*x])/(2*d*(a*(1 + Sec[c + d*x]))^(3/2)) - (B*(1 + S ec[c + d*x])^(3/2)*((40*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*T an[c + d*x])/(d*Sqrt[1 + Sec[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]* Sqrt[1 - Sec[c + d*x]] + 2*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(16*(a*(1 + Sec[ c + d*x]))^(3/2)) - (A*(1 + Sec[c + d*x])^(3/2)*((336*Hypergeometric2F1[1/ 2, 4, 3/2, 1 - Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (1 7*(27*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 24*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] - 21*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 2*Cos[c + d*x] ^2*Sqrt[1 - Sec[c + d*x]] - 8*Cos[c + d*x]^3*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(96*(a*(1 + S ec[c + d*x]))^(3/2))
Time = 1.82 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.08, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.576, Rules used = {3042, 4508, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a \sec (c+d x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (2 a (5 A-3 B)-7 a (A-B) \sec (c+d x))}{2 \sqrt {\sec (c+d x) a+a}}dx}{2 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (2 a (5 A-3 B)-7 a (A-B) \sec (c+d x))}{\sqrt {\sec (c+d x) a+a}}dx}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 a (5 A-3 B)-7 a (A-B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {\int -\frac {\cos ^2(c+d x) \left (2 a^2 (13 A-12 B)-5 a^2 (5 A-3 B) \sec (c+d x)\right )}{\sqrt {\sec (c+d x) a+a}}dx}{3 a}+\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {\cos ^2(c+d x) \left (2 a^2 (13 A-12 B)-5 a^2 (5 A-3 B) \sec (c+d x)\right )}{\sqrt {\sec (c+d x) a+a}}dx}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {2 a^2 (13 A-12 B)-5 a^2 (5 A-3 B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {\int -\frac {3 \cos (c+d x) \left (7 a^3 (3 A-2 B)-a^3 (13 A-12 B) \sec (c+d x)\right )}{\sqrt {\sec (c+d x) a+a}}dx}{2 a}+\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \int \frac {\cos (c+d x) \left (7 a^3 (3 A-2 B)-a^3 (13 A-12 B) \sec (c+d x)\right )}{\sqrt {\sec (c+d x) a+a}}dx}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \int \frac {7 a^3 (3 A-2 B)-a^3 (13 A-12 B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {\int -\frac {a^4 (47 A-38 B)-7 a^4 (3 A-2 B) \sec (c+d x)}{2 \sqrt {\sec (c+d x) a+a}}dx}{a}+\frac {7 a^3 (3 A-2 B) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {7 a^3 (3 A-2 B) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^4 (47 A-38 B)-7 a^4 (3 A-2 B) \sec (c+d x)}{\sqrt {\sec (c+d x) a+a}}dx}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {7 a^3 (3 A-2 B) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^4 (47 A-38 B)-7 a^4 (3 A-2 B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4408 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {7 a^3 (3 A-2 B) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {a^3 (47 A-38 B) \int \sqrt {\sec (c+d x) a+a}dx-4 a^4 (17 A-13 B) \int \frac {\sec (c+d x)}{\sqrt {\sec (c+d x) a+a}}dx}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {7 a^3 (3 A-2 B) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {a^3 (47 A-38 B) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-4 a^4 (17 A-13 B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {7 a^3 (3 A-2 B) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {-4 a^4 (17 A-13 B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {2 a^4 (47 A-38 B) \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{\sec (c+d x) a+a}+a}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {7 a^3 (3 A-2 B) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^{7/2} (47 A-38 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}-4 a^4 (17 A-13 B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {7 a^3 (3 A-2 B) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {8 a^4 (17 A-13 B) \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{\sec (c+d x) a+a}+2 a}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}+\frac {2 a^{7/2} (47 A-38 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {2 a (5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (13 A-12 B) \sin (c+d x) \cos (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {7 a^3 (3 A-2 B) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^{7/2} (47 A-38 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}-\frac {4 \sqrt {2} a^{7/2} (17 A-13 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{d}}{2 a}\right )}{2 a}}{3 a}}{4 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}\) |
Input:
Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]
Output:
-1/2*((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^(3/2)) + ((2*a*(5*A - 3*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d *x]]) - ((a^2*(13*A - 12*B)*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)*ArcTan[(Sqrt[a]*Tan[c + d*x ])/Sqrt[a + a*Sec[c + d*x]]])/d - (4*Sqrt[2]*a^(7/2)*(17*A - 13*B)*ArcTan[ (Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/d)/a + (7*a^3* (3*A - 2*B)*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*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(2*m + 1))), x] - Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Cs c[e + f*x])^n*Simp[b*B*n - a*A*(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[ e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B , 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(473\) vs. \(2(233)=466\).
Time = 2.05 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.77
| 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 )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\csc \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 )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\csc \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 )}{1+\cos \left (d x +c \right )}}\, \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-\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 )}{1+\cos \left (d x +c \right )}}\, \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-\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 \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 )}\) | \(474\) |
Input:
int(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^(3/2),x,method=_RETURNV ERBOSE)
Output:
1/24/d/a^2*((141*cos(d*x+c)^2+282*cos(d*x+c)+141)*A*(-cos(d*x+c)/(1+cos(d* x+c)))^(1/2)*arctanh(2^(1/2)*(-csc(d*x+c)+cot(d*x+c))/(cot(d*x+c)^2-2*csc( d*x+c)*cot(d*x+c)+csc(d*x+c)^2-1)^(1/2))+(-114*cos(d*x+c)^2-228*cos(d*x+c) -114)*B*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(2^(1/2)*(-csc(d*x+c)+co t(d*x+c))/(cot(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c)+csc(d*x+c)^2-1)^(1/2))+(10 2*cos(d*x+c)^2+204*cos(d*x+c)+102)*2^(1/2)*A*(-cos(d*x+c)/(1+cos(d*x+c)))^ (1/2)*ln((-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-cot(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)/(1+cos(d*x+c)))^(1/ 2)*ln((-2*cos(d*x+c)/(1+cos(d*x+c)))^(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)*(a*(1+sec(d*x+c)))^(1/2 )/(cos(d*x+c)^2+2*cos(d*x+c)+1)
Time = 3.03 (sec) , antiderivative size = 675, normalized size of antiderivative = 2.52 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^(3/2),x, algorith m="fricas")
Output:
[1/48*(6*sqrt(2)*((17*A - 13*B)*cos(d*x + c)^2 + 2*(17*A - 13*B)*cos(d*x + c) + 17*A - 13*B)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((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*co s(d*x + c) + a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 3*((47*A - 38*B)* cos(d*x + c)^2 + 2*(47*A - 38*B)*cos(d*x + c) + 47*A - 38*B)*sqrt(-a)*log( (2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*c os(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)*cos(d*x + c) ^2 + 21*(3*A - 2*B)*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)*cos(d*x + c)^2 + 2*(17*A - 13*B)*cos(d*x + c) + 17*A - 13*B)*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)*cos(d*x + c)^2 + 2*(47*A - 38*B)*cos(d*x + c) + 47*A - 38*B)*sqrt(a)*arctan(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)*cos(d*x + c)^2 + 21*(3*A - 2*B)*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)]
\[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cos(d*x+c)**3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))**(3/2),x)
Output:
Integral((A + B*sec(c + d*x))*cos(c + d*x)**3/(a*(sec(c + d*x) + 1))**(3/2 ), x)
\[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^(3/2),x, algorith m="maxima")
Output:
integrate((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. 816 vs. \(2 (233) = 466\).
Time = 1.16 (sec) , antiderivative size = 816, normalized size of antiderivative = 3.04 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^(3/2),x, algorith m="giac")
Output:
-1/48*(6*sqrt(2)*(17*A - 13*B)*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)*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 - 389422264390112072139 7872246915072*sqrt(2)*abs(a) + 5841333965851681082096808370372608*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 + 3894222643901120721397872246915072*sqrt (2)*abs(a) + 5841333965851681082096808370372608*a))/(sqrt(-a)*abs(a)*sgn(c os(d*x + c))) - 12*(sqrt(2)*A*a*sgn(cos(d*x + c)) - sqrt(2)*B*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 - 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 + 9198*(sqrt(-a)*tan(1/2*d*x + 1/2* c) - sqrt(-a*tan(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 - 4938*(sqrt( -a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4*A*a^3 + 2820*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)) ^4*B*a^3 + 975*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1...
Timed out. \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int((cos(c + d*x)^3*(A + B/cos(c + d*x)))/(a + a/cos(c + d*x))^(3/2),x)
Output:
int((cos(c + d*x)^3*(A + B/cos(c + d*x)))/(a + a/cos(c + d*x))^(3/2), x)
\[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )}{\sec \left (d x +c \right )^{2}+2 \sec \left (d x +c \right )+1}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{2}+2 \sec \left (d x +c \right )+1}d x \right ) a \right )}{a^{2}} \] Input:
int(cos(d*x+c)^3*(A+B*sec(d*x+c))/(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))/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1),x)*b + int((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