Integrand size = 43, antiderivative size = 259 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {(107 A-72 B+112 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {a} d}-\frac {\sqrt {2} (A-B+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d}-\frac {(21 A-56 B+16 C) \tan (c+d x)}{64 d \sqrt {a+a \cos (c+d x)}}+\frac {(43 A-8 B+48 C) \sec (c+d x) \tan (c+d x)}{96 d \sqrt {a+a \cos (c+d x)}}-\frac {(A-8 B) \sec ^2(c+d x) \tan (c+d x)}{24 d \sqrt {a+a \cos (c+d x)}}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 d \sqrt {a+a \cos (c+d x)}} \]
1/64*(107*A-72*B+112*C)*arctanh(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c))^(1/2)) /d/a^(1/2)-(A-B+C)*arctanh(1/2*sin(d*x+c)*a^(1/2)*2^(1/2)/(a+a*cos(d*x+c)) ^(1/2))*2^(1/2)/d/a^(1/2)-1/64*(21*A-56*B+16*C)*tan(d*x+c)/d/(a+a*cos(d*x+ c))^(1/2)+1/96*(43*A-8*B+48*C)*sec(d*x+c)*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1 /2)-1/24*(A-8*B)*sec(d*x+c)^2*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/4*A*se c(d*x+c)^3*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)
Time = 2.96 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.77 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=-\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (768 (A-B+C) \text {arctanh}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4(c+d x)-6 \sqrt {2} (107 A-72 B+112 C) \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4(c+d x)+(-364 A+32 B-192 C+(221 A-760 B+144 C) \cos (c+d x)-4 (43 A-8 B+48 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))-168 B \cos (3 (c+d x))+48 C \cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{384 d \sqrt {a (1+\cos (c+d x))}} \]
Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/Sqrt[a + a*Cos[c + d*x]],x]
-1/384*(Cos[(c + d*x)/2]*Sec[c + d*x]^4*(768*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[c + d*x]^4 - 6*Sqrt[2]*(107*A - 72*B + 112*C)*ArcTanh[Sqrt[2] *Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (-364*A + 32*B - 192*C + (221*A - 760* B + 144*C)*Cos[c + d*x] - 4*(43*A - 8*B + 48*C)*Cos[2*(c + d*x)] + 63*A*Co s[3*(c + d*x)] - 168*B*Cos[3*(c + d*x)] + 48*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])
Time = 1.86 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.13, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {3042, 3522, 27, 3042, 3463, 27, 3042, 3463, 27, 3042, 3463, 27, 3042, 3464, 3042, 3128, 219, 3252, 219}
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.
| \(\displaystyle \int \frac {\sec ^5(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {a \cos (c+d x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^5 \sqrt {a \sin \left (c+d x+\frac {\pi }{2}\right )+a}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {\int -\frac {(a (A-8 B)-a (7 A+8 C) \cos (c+d x)) \sec ^4(c+d x)}{2 \sqrt {\cos (c+d x) a+a}}dx}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {(a (A-8 B)-a (7 A+8 C) \cos (c+d x)) \sec ^4(c+d x)}{\sqrt {\cos (c+d x) a+a}}dx}{8 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {a (A-8 B)-a (7 A+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{8 a}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {\int -\frac {\left (a^2 (43 A-8 B+48 C)-5 a^2 (A-8 B) \cos (c+d x)\right ) \sec ^3(c+d x)}{2 \sqrt {\cos (c+d x) a+a}}dx}{3 a}+\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}}{8 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {\left (a^2 (43 A-8 B+48 C)-5 a^2 (A-8 B) \cos (c+d x)\right ) \sec ^3(c+d x)}{\sqrt {\cos (c+d x) a+a}}dx}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {a^2 (43 A-8 B+48 C)-5 a^2 (A-8 B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {\int -\frac {3 \left (a^3 (21 A-56 B+16 C)-a^3 (43 A-8 B+48 C) \cos (c+d x)\right ) \sec ^2(c+d x)}{2 \sqrt {\cos (c+d x) a+a}}dx}{2 a}+\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \int \frac {\left (a^3 (21 A-56 B+16 C)-a^3 (43 A-8 B+48 C) \cos (c+d x)\right ) \sec ^2(c+d x)}{\sqrt {\cos (c+d x) a+a}}dx}{4 a}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \int \frac {a^3 (21 A-56 B+16 C)-a^3 (43 A-8 B+48 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {\int -\frac {\left (a^4 (107 A-72 B+112 C)-a^4 (21 A-56 B+16 C) \cos (c+d x)\right ) \sec (c+d x)}{2 \sqrt {\cos (c+d x) a+a}}dx}{a}+\frac {a^3 (21 A-56 B+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {\left (a^4 (107 A-72 B+112 C)-a^4 (21 A-56 B+16 C) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {\cos (c+d x) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {a^4 (107 A-72 B+112 C)-a^4 (21 A-56 B+16 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {a^3 (107 A-72 B+112 C) \int \sqrt {\cos (c+d x) a+a} \sec (c+d x)dx-128 a^4 (A-B+C) \int \frac {1}{\sqrt {\cos (c+d x) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {a^3 (107 A-72 B+112 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-128 a^4 (A-B+C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {a^3 (107 A-72 B+112 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {256 a^4 (A-B+C) \int \frac {1}{2 a-\frac {a^2 \sin ^2(c+d x)}{\cos (c+d x) a+a}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {a^3 (107 A-72 B+112 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {128 \sqrt {2} a^{7/2} (A-B+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {-\frac {2 a^4 (107 A-72 B+112 C) \int \frac {1}{a-\frac {a^2 \sin ^2(c+d x)}{\cos (c+d x) a+a}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}-\frac {128 \sqrt {2} a^{7/2} (A-B+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a (A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 a^{7/2} (107 A-72 B+112 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}-\frac {128 \sqrt {2} a^{7/2} (A-B+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
(A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - ((a*(A - 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - ((a^2*( 43*A - 8*B + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x] ]) - (3*(-1/2*((2*a^(7/2)*(107*A - 72*B + 112*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (128*Sqrt[2]*a^(7/2)*(A - B + C)*ArcT anh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/d)/a + (a^ 3*(21*A - 56*B + 16*C)*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])))/(4*a)) /(6*a))/(8*a)
3.5.9.3.1 Defintions of rubi rules used
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]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 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*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && Eq Q[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A *b - a*B)/(b*c - a*d) Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Simp[(B*c - A*d)/(b*c - a*d) Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*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]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(2665\) vs. \(2(226)=452\).
Time = 16.62 (sec) , antiderivative size = 2666, normalized size of antiderivative = 10.29
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2666\) |
| default | \(\text {Expression too large to display}\) | \(3027\) |
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5/(a+cos(d*x+c)*a)^(1/2),x, method=_RETURNVERBOSE)
-1/24*A*cos(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-48*a*(-128*2^( 1/2)*ln(4*(a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+a)/cos(1/2*d*x+1/2*c))+1 07*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/ 2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))+107*ln(-4/(2*cos(1/2*d*x+1 /2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c) ^2)^(1/2)*a^(1/2)-2*a)))*sin(1/2*d*x+1/2*c)^8-48*(21*2^(1/2)*(a*sin(1/2*d* x+1/2*c)^2)^(1/2)*a^(1/2)+256*2^(1/2)*ln(4*(a^(1/2)*(a*sin(1/2*d*x+1/2*c)^ 2)^(1/2)+a)/cos(1/2*d*x+1/2*c))*a-214*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))* (2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/ 2)+2*a))*a-214*ln(-4/(2*cos(1/2*d*x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x +1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-2*a))*a)*sin(1/2*d* x+1/2*c)^6+(9216*2^(1/2)*ln(4*(a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+a)/c os(1/2*d*x+1/2*c))*a+824*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-77 04*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/ 2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+2*a))*a-7704*ln(-4/(2*cos(1/2*d* x+1/2*c)-2^(1/2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)-2^(1/2)*(a*sin(1/2*d*x+1/2 *c)^2)^(1/2)*a^(1/2)-2*a))*a)*sin(1/2*d*x+1/2*c)^4-4*(25*2^(1/2)*(a*sin(1/ 2*d*x+1/2*c)^2)^(1/2)*a^(1/2)+768*2^(1/2)*ln(4*(a^(1/2)*(a*sin(1/2*d*x+1/2 *c)^2)^(1/2)+a)/cos(1/2*d*x+1/2*c))*a-642*ln(4/(2*cos(1/2*d*x+1/2*c)+2^(1/ 2))*(2^(1/2)*a*cos(1/2*d*x+1/2*c)+2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2...
Time = 0.46 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.29 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {3 \, {\left ({\left (107 \, A - 72 \, B + 112 \, C\right )} \cos \left (d x + c\right )^{5} + {\left (107 \, A - 72 \, B + 112 \, C\right )} \cos \left (d x + c\right )^{4}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) - 4 \, {\left (3 \, {\left (21 \, A - 56 \, B + 16 \, C\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (43 \, A - 8 \, B + 48 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (A - 8 \, B\right )} \cos \left (d x + c\right ) - 48 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + \frac {384 \, \sqrt {2} {\left ({\left (A - B + C\right )} a \cos \left (d x + c\right )^{5} + {\left (A - B + C\right )} a \cos \left (d x + c\right )^{4}\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} + \frac {2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt {a}}}{768 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5/(a+a*cos(d*x+c))^(1 /2),x, algorithm="fricas")
1/768*(3*((107*A - 72*B + 112*C)*cos(d*x + c)^5 + (107*A - 72*B + 112*C)*c os(d*x + c)^4)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*sqrt (a*cos(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - 2)*sin(d*x + c) + 8*a)/(cos(d *x + c)^3 + cos(d*x + c)^2)) - 4*(3*(21*A - 56*B + 16*C)*cos(d*x + c)^3 - 2*(43*A - 8*B + 48*C)*cos(d*x + c)^2 + 8*(A - 8*B)*cos(d*x + c) - 48*A)*sq rt(a*cos(d*x + c) + a)*sin(d*x + c) + 384*sqrt(2)*((A - B + C)*a*cos(d*x + c)^5 + (A - B + C)*a*cos(d*x + c)^4)*log(-(cos(d*x + c)^2 + 2*sqrt(2)*sqr t(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(a) - 2*cos(d*x + c) - 3)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))/sqrt(a))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4)
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\text {Timed out} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5/(a+a*cos(d*x+c))^(1 /2),x, algorithm="maxima")
Time = 0.51 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.66 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=-\frac {\frac {192 \, \sqrt {2} {\left (A \sqrt {a} - B \sqrt {a} + C \sqrt {a}\right )} \log \left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {192 \, \sqrt {2} {\left (A \sqrt {a} - B \sqrt {a} + C \sqrt {a}\right )} \log \left (-\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3 \, {\left (107 \, A - 72 \, B + 112 \, C\right )} \log \left ({\left | \frac {1}{2} \, \sqrt {2} + \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {3 \, {\left (107 \, A - 72 \, B + 112 \, C\right )} \log \left ({\left | -\frac {1}{2} \, \sqrt {2} + \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (504 \, A \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1344 \, B \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 384 \, C \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 412 \, A \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1952 \, B \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 192 \, C \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 50 \, A \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1072 \, B \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, C \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 63 \, A \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 216 \, B \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, C \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{384 \, d} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5/(a+a*cos(d*x+c))^(1 /2),x, algorithm="giac")
-1/384*(192*sqrt(2)*(A*sqrt(a) - B*sqrt(a) + C*sqrt(a))*log(sin(1/2*d*x + 1/2*c) + 1)/(a*sgn(cos(1/2*d*x + 1/2*c))) - 192*sqrt(2)*(A*sqrt(a) - B*sqr t(a) + C*sqrt(a))*log(-sin(1/2*d*x + 1/2*c) + 1)/(a*sgn(cos(1/2*d*x + 1/2* c))) - 3*(107*A - 72*B + 112*C)*log(abs(1/2*sqrt(2) + sin(1/2*d*x + 1/2*c) ))/(sqrt(a)*sgn(cos(1/2*d*x + 1/2*c))) + 3*(107*A - 72*B + 112*C)*log(abs( -1/2*sqrt(2) + sin(1/2*d*x + 1/2*c)))/(sqrt(a)*sgn(cos(1/2*d*x + 1/2*c))) - 2*sqrt(2)*(504*A*sqrt(a)*sin(1/2*d*x + 1/2*c)^7 - 1344*B*sqrt(a)*sin(1/2 *d*x + 1/2*c)^7 + 384*C*sqrt(a)*sin(1/2*d*x + 1/2*c)^7 - 412*A*sqrt(a)*sin (1/2*d*x + 1/2*c)^5 + 1952*B*sqrt(a)*sin(1/2*d*x + 1/2*c)^5 - 192*C*sqrt(a )*sin(1/2*d*x + 1/2*c)^5 + 50*A*sqrt(a)*sin(1/2*d*x + 1/2*c)^3 - 1072*B*sq rt(a)*sin(1/2*d*x + 1/2*c)^3 - 96*C*sqrt(a)*sin(1/2*d*x + 1/2*c)^3 + 63*A* sqrt(a)*sin(1/2*d*x + 1/2*c) + 216*B*sqrt(a)*sin(1/2*d*x + 1/2*c) + 48*C*s qrt(a)*sin(1/2*d*x + 1/2*c))/((2*sin(1/2*d*x + 1/2*c)^2 - 1)^4*a*sgn(cos(1 /2*d*x + 1/2*c))))/d
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^5\,\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]