Integrand size = 43, antiderivative size = 251 \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {4 a^2 (12 A+9 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^2 (7 A+6 B+5 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 a^2 (7 A+6 B+5 C) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 (12 A+9 B+8 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (9 B+4 C) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)} \] Output:
-4/15*a^2*(12*A+9*B+8*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+4/21*a^2* (7*A+6*B+5*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+2/105*a^2*(21*A+27* B+19*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)+4/21*a^2*(7*A+6*B+5*C)*sin(d*x+c)/d/ cos(d*x+c)^(3/2)+4/15*a^2*(12*A+9*B+8*C)*sin(d*x+c)/d/cos(d*x+c)^(1/2)+2/9 *C*(a+a*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(9/2)+2/63*(9*B+4*C)*(a^2+a^ 2*cos(d*x+c))*sin(d*x+c)/d/cos(d*x+c)^(7/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 12.64 (sec) , antiderivative size = 1741, normalized size of antiderivative = 6.94 \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) /Cos[c + d*x]^(3/2),x]
Output:
(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec [c + d*x] + C*Sec[c + d*x]^2)*((2*(12*A + 9*B + 8*C)*Csc[c]*Sec[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(9*d) + (Sec[c]*Sec[c + d*x]^4*(7*C* Sin[c] + 9*B*Sin[d*x] + 18*C*Sin[d*x]))/(63*d) + (2*Sec[c]*Sec[c + d*x]*(3 5*A*Sin[c] + 30*B*Sin[c] + 25*C*Sin[c] + 84*A*Sin[d*x] + 63*B*Sin[d*x] + 5 6*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]^3*(45*B*Sin[c] + 90*C*Sin[c] + 63*A*Sin[d*x] + 126*B*Sin[d*x] + 112*C*Sin[d*x]))/(315*d) + (Sec[c]*Sec [c + d*x]^2*(63*A*Sin[c] + 126*B*Sin[c] + 112*C*Sin[c] + 210*A*Sin[d*x] + 180*B*Sin[d*x] + 150*C*Sin[d*x]))/(315*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (2*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1 /2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c] ]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Si n[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2 *C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos [c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[C ot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[C ot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt [1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*C...
Time = 1.60 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {3042, 4600, 3042, 3522, 27, 3042, 3454, 27, 3042, 3447, 3042, 3500, 3042, 3227, 3042, 3116, 3042, 3119, 3120}
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.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^2 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )}{\cos (c+d x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^2 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^2 (a (9 B+4 C)+3 a (3 A+C) \cos (c+d x))}{2 \cos ^{\frac {9}{2}}(c+d x)}dx}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^2 (a (9 B+4 C)+3 a (3 A+C) \cos (c+d x))}{\cos ^{\frac {9}{2}}(c+d x)}dx}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (9 B+4 C)+3 a (3 A+C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {3 (\cos (c+d x) a+a) \left ((21 A+27 B+19 C) a^2+(21 A+9 B+11 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{7} \int \frac {(\cos (c+d x) a+a) \left ((21 A+27 B+19 C) a^2+(21 A+9 B+11 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((21 A+27 B+19 C) a^2+(21 A+9 B+11 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {3}{7} \int \frac {(21 A+9 B+11 C) \cos ^2(c+d x) a^3+(21 A+27 B+19 C) a^3+\left ((21 A+9 B+11 C) a^3+(21 A+27 B+19 C) a^3\right ) \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} \int \frac {(21 A+9 B+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^3+(21 A+27 B+19 C) a^3+\left ((21 A+9 B+11 C) a^3+(21 A+27 B+19 C) a^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {3}{7} \left (\frac {2}{5} \int \frac {15 (7 A+6 B+5 C) a^3+7 (12 A+9 B+8 C) \cos (c+d x) a^3}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^3 (21 A+27 B+19 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} \left (\frac {2}{5} \int \frac {15 (7 A+6 B+5 C) a^3+7 (12 A+9 B+8 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^3 (21 A+27 B+19 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {3}{7} \left (\frac {2}{5} \left (15 a^3 (7 A+6 B+5 C) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)}dx+7 a^3 (12 A+9 B+8 C) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 a^3 (21 A+27 B+19 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} \left (\frac {2}{5} \left (15 a^3 (7 A+6 B+5 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+7 a^3 (12 A+9 B+8 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {2 a^3 (21 A+27 B+19 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\frac {3}{7} \left (\frac {2}{5} \left (15 a^3 (7 A+6 B+5 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 a^3 (12 A+9 B+8 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {2 a^3 (21 A+27 B+19 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} \left (\frac {2}{5} \left (15 a^3 (7 A+6 B+5 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 a^3 (12 A+9 B+8 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {2 a^3 (21 A+27 B+19 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {3}{7} \left (\frac {2}{5} \left (15 a^3 (7 A+6 B+5 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 a^3 (12 A+9 B+8 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 a^3 (21 A+27 B+19 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {3}{7} \left (\frac {2 a^3 (21 A+27 B+19 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2}{5} \left (15 a^3 (7 A+6 B+5 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 a^3 (12 A+9 B+8 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {2 (9 B+4 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
Input:
Int[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]
Output:
(2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((2*( 9*B + 4*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (3*((2*a^3*(21*A + 27*B + 19*C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(15*a^3*(7*A + 6*B + 5*C)*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*S in[c + d*x])/(3*d*Cos[c + d*x]^(3/2))) + 7*a^3*(12*A + 9*B + 8*C)*((-2*Ell ipticE[(c + d*x)/2, 2])/d + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]))))/5)) /7)/(9*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*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] :> Sim p[(-b^2)*(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)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m* (c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2* (n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ [m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x _)])^(m_.)*((A_.) + (B_.)*sec[(e_.) + (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.) *(x_)]^2), x_Symbol] :> Simp[d^(m + 2) Int[(b + a*Cos[e + f*x])^m*(d*Cos[ e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; Fr eeQ[{a, b, d, e, f, A, B, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(1153\) vs. \(2(230)=460\).
Time = 9.11 (sec) , antiderivative size = 1154, normalized size of antiderivative = 4.60
Input:
int((a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2),x, method=_RETURNVERBOSE)
Output:
-8*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^2*(1/4*A/si n(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin (1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(2*sin (1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2* d*x+1/2*c)^2)^(1/2))+1/4*C*(-1/144*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2* c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180*cos(1/ 2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2 *d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*c os(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c )^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin( 1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/ 2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/ 2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))- EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))))+(1/2*A+1/4*B)*(-1/6*cos(1/2*d*x+1/ 2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2 *c)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^ (1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1 /2*d*x+1/2*c),2^(1/2)))+(1/4*B+1/2*C)*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/ 2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/ 42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1...
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.21 \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (7 \, A + 6 \, B + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (7 \, A + 6 \, B + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (12 \, A + 9 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (12 \, A + 9 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (42 \, {\left (12 \, A + 9 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 30 \, {\left (7 \, A + 6 \, B + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 7 \, {\left (9 \, A + 18 \, B + 16 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 45 \, {\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right ) + 35 \, C a^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d \cos \left (d x + c\right )^{5}} \] Input:
integrate((a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(3 /2),x, algorithm="fricas")
Output:
-2/315*(15*I*sqrt(2)*(7*A + 6*B + 5*C)*a^2*cos(d*x + c)^5*weierstrassPInve rse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(7*A + 6*B + 5*C) *a^2*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*I*sqrt(2)*(12*A + 9*B + 8*C)*a^2*cos(d*x + c)^5*weierstrassZeta(- 4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*I*sq rt(2)*(12*A + 9*B + 8*C)*a^2*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierst rassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (42*(12*A + 9*B + 8* C)*a^2*cos(d*x + c)^4 + 30*(7*A + 6*B + 5*C)*a^2*cos(d*x + c)^3 + 7*(9*A + 18*B + 16*C)*a^2*cos(d*x + c)^2 + 45*(B + 2*C)*a^2*cos(d*x + c) + 35*C*a^ 2)*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^5)
\[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=a^{2} \left (\int \frac {A}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {2 A \sec {\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {A \sec ^{2}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {B \sec {\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {2 B \sec ^{2}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {B \sec ^{3}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {C \sec ^{2}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {2 C \sec ^{3}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {C \sec ^{4}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx\right ) \] Input:
integrate((a+a*sec(d*x+c))**2*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/cos(d*x+c)* *(3/2),x)
Output:
a**2*(Integral(A/cos(c + d*x)**(3/2), x) + Integral(2*A*sec(c + d*x)/cos(c + d*x)**(3/2), x) + Integral(A*sec(c + d*x)**2/cos(c + d*x)**(3/2), x) + Integral(B*sec(c + d*x)/cos(c + d*x)**(3/2), x) + Integral(2*B*sec(c + d*x )**2/cos(c + d*x)**(3/2), x) + Integral(B*sec(c + d*x)**3/cos(c + d*x)**(3 /2), x) + Integral(C*sec(c + d*x)**2/cos(c + d*x)**(3/2), x) + Integral(2* C*sec(c + d*x)**3/cos(c + d*x)**(3/2), x) + Integral(C*sec(c + d*x)**4/cos (c + d*x)**(3/2), x))
Timed out. \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(3 /2),x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(3 /2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^2/c os(d*x + c)^(3/2), x)
Time = 16.10 (sec) , antiderivative size = 688, normalized size of antiderivative = 2.74 \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
int(((a + a/cos(c + d*x))^2*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/cos(c + d*x)^(3/2),x)
Output:
(8*((B*a^2*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (2*C*a^2*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)))*h ypergeom([-3/4, 1/2], 5/4, cos(c + d*x)^2))/(21*d) - (8*hypergeom([-1/4, 1 /2], 7/4, cos(c + d*x)^2)*((9*A*a^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (18*B*a^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (16*C*a^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (5*C*a^2*sin(c + d*x))/(cos(c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^(1/2))))/(135*d) + (2*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((81*A*a^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^ (1/2)) + (9*A*a^2*sin(c + d*x))/(cos(c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^( 1/2)) + (72*B*a^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^( 1/2)) + (18*B*a^2*sin(c + d*x))/(cos(c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^( 1/2)) + (64*C*a^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^( 1/2)) + (21*C*a^2*sin(c + d*x))/(cos(c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^( 1/2)) + (5*C*a^2*sin(c + d*x))/(cos(c + d*x)^(9/2)*(1 - cos(c + d*x)^2)^(1 /2))))/(45*d) + (2*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)*((14*A*a^2* sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (11*B*a^2* sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (3*B*a^2*s in(c + d*x))/(cos(c + d*x)^(7/2)*(1 - cos(c + d*x)^2)^(1/2)) + (8*C*a^2*si n(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (6*C*a^2*...
\[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=a^{2} \left (\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{2}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{2}}d x \right ) b +2 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{2}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{2}}d x \right ) a +2 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{2}}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{2}}d x \right ) c +2 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )}{\cos \left (d x +c \right )^{2}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )}{\cos \left (d x +c \right )^{2}}d x \right ) b \right ) \] Input:
int((a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2),x)
Output:
a**2*(int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*a + int((sqrt(cos(c + d*x) )*sec(c + d*x)**4)/cos(c + d*x)**2,x)*c + int((sqrt(cos(c + d*x))*sec(c + d*x)**3)/cos(c + d*x)**2,x)*b + 2*int((sqrt(cos(c + d*x))*sec(c + d*x)**3) /cos(c + d*x)**2,x)*c + int((sqrt(cos(c + d*x))*sec(c + d*x)**2)/cos(c + d *x)**2,x)*a + 2*int((sqrt(cos(c + d*x))*sec(c + d*x)**2)/cos(c + d*x)**2,x )*b + int((sqrt(cos(c + d*x))*sec(c + d*x)**2)/cos(c + d*x)**2,x)*c + 2*in t((sqrt(cos(c + d*x))*sec(c + d*x))/cos(c + d*x)**2,x)*a + int((sqrt(cos(c + d*x))*sec(c + d*x))/cos(c + d*x)**2,x)*b)