Integrand size = 34, antiderivative size = 280 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {203 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{5/2} d}-\frac {287 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}} \] Output:
203/8*A*arctan(a^(1/2)*tan(d*x+c)/(a-a*sec(d*x+c))^(1/2))/a^(5/2)/d-287/16 *A*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 ^(5/2)/d-1/2*A*cos(d*x+c)^2*sin(d*x+c)/d/(a-a*sec(d*x+c))^(5/2)-19/8*A*cos (d*x+c)^2*sin(d*x+c)/a/d/(a-a*sec(d*x+c))^(3/2)+21/2*A*sin(d*x+c)/a^2/d/(a -a*sec(d*x+c))^(1/2)+119/24*A*cos(d*x+c)*sin(d*x+c)/a^2/d/(a-a*sec(d*x+c)) ^(1/2)+77/24*A*cos(d*x+c)^2*sin(d*x+c)/a^2/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 = 2.50 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.71 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {A \sec ^2(c+d x) \left (-912 (-1+\cos (c+d x)) \cos ^4(c+d x)-192 \cos ^5(c+d x)+6384 (-1+\cos (c+d x))^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4,\frac {3}{2},1+\sec (c+d x)\right )+\frac {287 (-1+\cos (c+d x))^2 \left (27 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right )-24 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right )+\cos (c+d x) \left (21+2 \cos (c+d x)+8 \cos ^2(c+d x)\right ) \sqrt {1+\sec (c+d x)}\right )}{\sqrt {1+\sec (c+d x)}}\right ) \tan (c+d x)}{384 d (a-a \sec (c+d x))^{5/2}} \] Input:
Integrate[(Cos[c + d*x]^3*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(5/2) ,x]
Output:
(A*Sec[c + d*x]^2*(-912*(-1 + Cos[c + d*x])*Cos[c + d*x]^4 - 192*Cos[c + d *x]^5 + 6384*(-1 + Cos[c + d*x])^2*Hypergeometric2F1[1/2, 4, 3/2, 1 + Sec[ c + d*x]] + (287*(-1 + Cos[c + d*x])^2*(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]]))/Sqrt[1 + Sec[ c + d*x]])*Tan[c + d*x])/(384*d*(a - a*Sec[c + d*x])^(5/2))
Time = 2.06 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.09, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.618, Rules used = {3042, 4508, 3042, 4508, 27, 3042, 4510, 25, 3042, 4510, 27, 3042, 4510, 25, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) (A \sec (c+d x)+A)}{(a-a \sec (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A \csc \left (c+d x+\frac {\pi }{2}\right )+A}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (10 a A+9 a \sec (c+d x) A)}{(a-a \sec (c+d x))^{3/2}}dx}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {10 a A+9 a \csc \left (c+d x+\frac {\pi }{2}\right ) A}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\int \frac {7 \cos ^3(c+d x) \left (22 A a^2+19 A \sec (c+d x) a^2\right )}{2 \sqrt {a-a \sec (c+d x)}}dx}{2 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {7 \int \frac {\cos ^3(c+d x) \left (22 A a^2+19 A \sec (c+d x) a^2\right )}{\sqrt {a-a \sec (c+d x)}}dx}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \int \frac {22 A a^2+19 A \csc \left (c+d x+\frac {\pi }{2}\right ) a^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {7 \left (\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}-\frac {\int -\frac {\cos ^2(c+d x) \left (68 A a^3+55 A \sec (c+d x) a^3\right )}{\sqrt {a-a \sec (c+d x)}}dx}{3 a}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\int \frac {\cos ^2(c+d x) \left (68 A a^3+55 A \sec (c+d x) a^3\right )}{\sqrt {a-a \sec (c+d x)}}dx}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\int \frac {68 A a^3+55 A \csc \left (c+d x+\frac {\pi }{2}\right ) a^3}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}-\frac {\int -\frac {6 \cos (c+d x) \left (24 A a^4+17 A \sec (c+d x) a^4\right )}{\sqrt {a-a \sec (c+d x)}}dx}{2 a}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {3 \int \frac {\cos (c+d x) \left (24 A a^4+17 A \sec (c+d x) a^4\right )}{\sqrt {a-a \sec (c+d x)}}dx}{a}+\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {3 \int \frac {24 A a^4+17 A \csc \left (c+d x+\frac {\pi }{2}\right ) a^4}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {3 \left (\frac {24 a^4 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}-\frac {\int -\frac {29 A a^5+12 A \sec (c+d x) a^5}{\sqrt {a-a \sec (c+d x)}}dx}{a}\right )}{a}+\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {3 \left (\frac {\int \frac {29 A a^5+12 A \sec (c+d x) a^5}{\sqrt {a-a \sec (c+d x)}}dx}{a}+\frac {24 a^4 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{a}+\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {3 \left (\frac {\int \frac {29 A a^5+12 A \csc \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {24 a^4 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{a}+\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4408 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {3 \left (\frac {41 a^5 A \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}}dx+29 a^4 A \int \sqrt {a-a \sec (c+d x)}dx}{a}+\frac {24 a^4 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{a}+\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {3 \left (\frac {41 a^5 A \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+29 a^4 A \int \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {24 a^4 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{a}+\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {3 \left (\frac {41 a^5 A \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {58 a^5 A \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{a-a \sec (c+d x)}+a}d\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}}{d}}{a}+\frac {24 a^4 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{a}+\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {3 \left (\frac {41 a^5 A \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {58 a^{9/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}}{a}+\frac {24 a^4 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{a}+\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {\frac {7 \left (\frac {\frac {3 \left (\frac {\frac {58 a^{9/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}-\frac {82 a^5 A \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{a-a \sec (c+d x)}+2 a}d\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}}{d}}{a}+\frac {24 a^4 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{a}+\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{3 a}+\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {7 \left (\frac {22 a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}+\frac {\frac {34 a^3 A \sin (c+d x) \cos (c+d x)}{d \sqrt {a-a \sec (c+d x)}}+\frac {3 \left (\frac {\frac {58 a^{9/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}-\frac {41 \sqrt {2} a^{9/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{d}}{a}+\frac {24 a^4 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{a}}{3 a}\right )}{4 a^2}-\frac {19 a A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{3/2}}}{4 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}\) |
Input:
Int[(Cos[c + d*x]^3*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(5/2),x]
Output:
-1/2*(A*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a - a*Sec[c + d*x])^(5/2)) + ((-1 9*a*A*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a - a*Sec[c + d*x])^(3/2)) + (7*( (22*a^2*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a - a*Sec[c + d*x]]) + (( 34*a^3*A*Cos[c + d*x]*Sin[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]]) + (3*(((5 8*a^(9/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/d - ( 41*Sqrt[2]*a^(9/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec [c + d*x]])])/d)/a + (24*a^4*A*Sin[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]])) )/a)/(3*a)))/(4*a^2))/(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. \(1398\) vs. \(2(241)=482\).
Time = 2.62 (sec) , antiderivative size = 1399, normalized size of antiderivative = 5.00
Input:
int(cos(d*x+c)^3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(5/2),x,method=_RETURNV ERBOSE)
Output:
A*(1/32/d/(-a/(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/((2*c os(1/2*d*x+1/2*c)^2-1)/(cos(1/2*d*x+1/2*c)+1)^2)^(1/2)/a^2*((32*cos(1/2*d* x+1/2*c)^4-102*cos(1/2*d*x+1/2*c)^2+66)*2^(1/2)*((2*cos(1/2*d*x+1/2*c)^2-1 )/(cos(1/2*d*x+1/2*c)+1)^2)^(1/2)*cot(1/2*d*x+1/2*c)*csc(1/2*d*x+1/2*c)^2+ 2^(1/2)*arctanh((2*cos(1/2*d*x+1/2*c)-1)/(cos(1/2*d*x+1/2*c)+1)/((2*cos(1/ 2*d*x+1/2*c)^2-1)/(cos(1/2*d*x+1/2*c)+1)^2)^(1/2))*(79*cot(1/2*d*x+1/2*c)- 79*csc(1/2*d*x+1/2*c))+2^(1/2)*ln(2*(((2*cos(1/2*d*x+1/2*c)^2-1)/(cos(1/2* d*x+1/2*c)+1)^2)^(1/2)*cos(1/2*d*x+1/2*c)+((2*cos(1/2*d*x+1/2*c)^2-1)/(cos (1/2*d*x+1/2*c)+1)^2)^(1/2)-2*cos(1/2*d*x+1/2*c)-1)/(cos(1/2*d*x+1/2*c)+1) )*(-79*cot(1/2*d*x+1/2*c)+79*csc(1/2*d*x+1/2*c))+arctanh(2^(1/2)*cos(1/2*d *x+1/2*c)/(cos(1/2*d*x+1/2*c)+1)/((2*cos(1/2*d*x+1/2*c)^2-1)/(cos(1/2*d*x+ 1/2*c)+1)^2)^(1/2))*(-224*cot(1/2*d*x+1/2*c)+224*csc(1/2*d*x+1/2*c)))-1/8/ d/(-a/(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/((2*cos(1/2*d *x+1/2*c)^2-1)/(cos(1/2*d*x+1/2*c)+1)^2)^(1/2)/a^2*(2^(1/2)*arctanh((2*cos (1/2*d*x+1/2*c)-1)/(cos(1/2*d*x+1/2*c)+1)/((2*cos(1/2*d*x+1/2*c)^2-1)/(cos (1/2*d*x+1/2*c)+1)^2)^(1/2))*(123*cot(1/2*d*x+1/2*c)-123*csc(1/2*d*x+1/2*c ))+2^(1/2)*ln(2*(((2*cos(1/2*d*x+1/2*c)^2-1)/(cos(1/2*d*x+1/2*c)+1)^2)^(1/ 2)*cos(1/2*d*x+1/2*c)+((2*cos(1/2*d*x+1/2*c)^2-1)/(cos(1/2*d*x+1/2*c)+1)^2 )^(1/2)-2*cos(1/2*d*x+1/2*c)-1)/(cos(1/2*d*x+1/2*c)+1))*(-123*cot(1/2*d*x+ 1/2*c)+123*csc(1/2*d*x+1/2*c))+arctanh(2^(1/2)*cos(1/2*d*x+1/2*c)/(cos(...
Time = 0.12 (sec) , antiderivative size = 656, normalized size of antiderivative = 2.34 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(5/2),x, algorith m="fricas")
Output:
[-1/96*(861*sqrt(2)*(A*cos(d*x + c)^2 - 2*A*cos(d*x + c) + A)*sqrt(-a)*log ((2*sqrt(2)*(cos(d*x + c)^2 + cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)) + (3*a*cos(d*x + c) + a)*sin(d*x + c))/((cos(d*x + c) - 1)*sin(d*x + c)))*sin(d*x + c) + 1218*(A*cos(d*x + c)^2 - 2*A*cos(d*x + c ) + A)*sqrt(-a)*log((2*(cos(d*x + c)^2 + cos(d*x + c))*sqrt(-a)*sqrt((a*co s(d*x + c) - a)/cos(d*x + c)) - (2*a*cos(d*x + c) + a)*sin(d*x + c))/sin(d *x + c))*sin(d*x + c) + 4*(8*A*cos(d*x + c)^6 + 30*A*cos(d*x + c)^5 + 113* A*cos(d*x + c)^4 - 294*A*cos(d*x + c)^3 - 133*A*cos(d*x + c)^2 + 252*A*cos (d*x + c))*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c)), 1/48*(861*sqrt(2)*(A*cos(d *x + c)^2 - 2*A*cos(d*x + c) + A)*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)))*sin(d*x + c) - 1218*(A*cos(d*x + c)^2 - 2*A*cos(d*x + c) + A)*sqrt(a)*arctan(sqrt((a*cos (d*x + c) - a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) - 2*(8*A*cos(d*x + c)^6 + 30*A*cos(d*x + c)^5 + 113*A*cos(d*x + c)^4 - 294*A*cos(d*x + c)^3 - 133*A*cos(d*x + c)^2 + 252*A*cos(d*x + c))*sqrt(( a*cos(d*x + c) - a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d* x + c) + a^3*d)*sin(d*x + c))]
Timed out. \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\int { \frac {{\left (A \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (-a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(5/2),x, algorith m="maxima")
Output:
integrate((A*sec(d*x + c) + A)*cos(d*x + c)^3/(-a*sec(d*x + c) + a)^(5/2), x)
Time = 0.82 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {861 \, \sqrt {2} A \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {1218 \, A \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {2 \, \sqrt {2} {\left (129 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {5}{2}} A + 560 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A a + 636 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{3} a^{2}} - \frac {3 \, \sqrt {2} {\left (33 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A + 31 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a\right )}}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{48 \, d} \] Input:
integrate(cos(d*x+c)^3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(5/2),x, algorith m="giac")
Output:
1/48*(861*sqrt(2)*A*arctan(sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/a^( 5/2) - 1218*A*arctan(1/2*sqrt(2)*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a ))/a^(5/2) - 2*sqrt(2)*(129*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(5/2)*A + 560*( a*tan(1/2*d*x + 1/2*c)^2 - a)^(3/2)*A*a + 636*sqrt(a*tan(1/2*d*x + 1/2*c)^ 2 - a)*A*a^2)/((a*tan(1/2*d*x + 1/2*c)^2 + a)^3*a^2) - 3*sqrt(2)*(33*(a*ta n(1/2*d*x + 1/2*c)^2 - a)^(3/2)*A + 31*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)* A*a)/(a^4*tan(1/2*d*x + 1/2*c)^4))/d
Timed out. \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+\frac {A}{\cos \left (c+d\,x\right )}\right )}{{\left (a-\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \] Input:
int((cos(c + d*x)^3*(A + A/cos(c + d*x)))/(a - a/cos(c + d*x))^(5/2),x)
Output:
int((cos(c + d*x)^3*(A + A/cos(c + d*x)))/(a - a/cos(c + d*x))^(5/2), x)
\[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=-\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )}{\sec \left (d x +c \right )^{3}-3 \sec \left (d x +c \right )^{2}+3 \sec \left (d x +c \right )-1}d x +\int \frac {\sqrt {-\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{3}-3 \sec \left (d x +c \right )^{2}+3 \sec \left (d x +c \right )-1}d x \right )}{a^{2}} \] Input:
int(cos(d*x+c)^3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(5/2),x)
Output:
( - sqrt(a)*(int((sqrt( - sec(c + d*x) + 1)*cos(c + d*x)**3*sec(c + d*x))/ (sec(c + d*x)**3 - 3*sec(c + d*x)**2 + 3*sec(c + d*x) - 1),x) + int((sqrt( - sec(c + d*x) + 1)*cos(c + d*x)**3)/(sec(c + d*x)**3 - 3*sec(c + d*x)**2 + 3*sec(c + d*x) - 1),x)))/a**2