Integrand size = 34, antiderivative size = 236 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\frac {85 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{3/2} d}-\frac {15 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {2} a^{3/2} d}-\frac {A \cos ^2(c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {35 A \sin (c+d x)}{8 a d \sqrt {a-a \sec (c+d x)}}+\frac {25 A \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a-a \sec (c+d x)}}+\frac {4 A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt {a-a \sec (c+d x)}} \]
85/8*A*arctan(a^(1/2)*tan(d*x+c)/(a-a*sec(d*x+c))^(1/2))/a^(3/2)/d-A*cos(d *x+c)^2*sin(d*x+c)/d/(a-a*sec(d*x+c))^(3/2)-15/2*A*arctan(1/2*a^(1/2)*tan( d*x+c)*2^(1/2)/(a-a*sec(d*x+c))^(1/2))/a^(3/2)/d*2^(1/2)+35/8*A*sin(d*x+c) /a/d/(a-a*sec(d*x+c))^(1/2)+25/12*A*cos(d*x+c)*sin(d*x+c)/a/d/(a-a*sec(d*x +c))^(1/2)+4/3*A*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 = 2.37 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\frac {A \sec (c+d x) \left (-16 \cos ^4(c+d x)+112 (-1+\cos (c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4,\frac {3}{2},1+\sec (c+d x)\right )+\frac {5 (-1+\cos (c+d x)) \left (27 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right )-24 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right )+\cos (c+d x) \left (21+2 \cos (c+d x)+8 \cos ^2(c+d x)\right ) \sqrt {1+\sec (c+d x)}\right )}{\sqrt {1+\sec (c+d x)}}\right ) \tan (c+d x)}{16 d (a-a \sec (c+d x))^{3/2}} \]
(A*Sec[c + d*x]*(-16*Cos[c + d*x]^4 + 112*(-1 + Cos[c + d*x])*Hypergeometr ic2F1[1/2, 4, 3/2, 1 + Sec[c + d*x]] + (5*(-1 + Cos[c + d*x])*(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])/(16*d*(a - a*Sec[c + d*x]) ^(3/2))
Time = 1.64 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.11, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 4508, 3042, 4510, 27, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) (A \sec (c+d x)+A)}{(a-a \sec (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A \csc \left (c+d x+\frac {\pi }{2}\right )+A}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (8 a A+7 a \sec (c+d x) A)}{\sqrt {a-a \sec (c+d x)}}dx}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {8 a A+7 a \csc \left (c+d x+\frac {\pi }{2}\right ) A}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}-\frac {\int -\frac {5 \cos ^2(c+d x) \left (5 A a^2+4 A \sec (c+d x) a^2\right )}{\sqrt {a-a \sec (c+d x)}}dx}{3 a}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \int \frac {\cos ^2(c+d x) \left (5 A a^2+4 A \sec (c+d x) a^2\right )}{\sqrt {a-a \sec (c+d x)}}dx}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \int \frac {5 A a^2+4 A \csc \left (c+d x+\frac {\pi }{2}\right ) a^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {5 \left (\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}-\frac {\int -\frac {3 \cos (c+d x) \left (7 A a^3+5 A \sec (c+d x) a^3\right )}{2 \sqrt {a-a \sec (c+d x)}}dx}{2 a}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \int \frac {\cos (c+d x) \left (7 A a^3+5 A \sec (c+d x) a^3\right )}{\sqrt {a-a \sec (c+d x)}}dx}{4 a}+\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \int \frac {7 A a^3+5 A \csc \left (c+d x+\frac {\pi }{2}\right ) a^3}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {7 a^3 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}-\frac {\int -\frac {17 A a^4+7 A \sec (c+d x) a^4}{2 \sqrt {a-a \sec (c+d x)}}dx}{a}\right )}{4 a}+\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {\int \frac {17 A a^4+7 A \sec (c+d x) a^4}{\sqrt {a-a \sec (c+d x)}}dx}{2 a}+\frac {7 a^3 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{4 a}+\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {\int \frac {17 A a^4+7 A \csc \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {7 a^3 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{4 a}+\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4408 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {24 a^4 A \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}}dx+17 a^3 A \int \sqrt {a-a \sec (c+d x)}dx}{2 a}+\frac {7 a^3 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{4 a}+\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {24 a^4 A \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+17 a^3 A \int \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}+\frac {7 a^3 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{4 a}+\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {24 a^4 A \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {34 a^4 A \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{a-a \sec (c+d x)}+a}d\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}}{d}}{2 a}+\frac {7 a^3 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{4 a}+\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {24 a^4 A \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {34 a^{7/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}}{2 a}+\frac {7 a^3 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{4 a}+\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {\frac {34 a^{7/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}-\frac {48 a^4 A \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{a-a \sec (c+d x)}+2 a}d\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}}{d}}{2 a}+\frac {7 a^3 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{4 a}+\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {5 \left (\frac {5 a^2 A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}+\frac {3 \left (\frac {\frac {34 a^{7/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}-\frac {24 \sqrt {2} a^{7/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{d}}{2 a}+\frac {7 a^3 A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}\right )}{4 a}\right )}{3 a}+\frac {8 a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a-a \sec (c+d x)}}}{2 a^2}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}\) |
-((A*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a - a*Sec[c + d*x])^(3/2))) + ((8*a* A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a - a*Sec[c + d*x]]) + (5*((5*a^2 *A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a - a*Sec[c + d*x]]) + (3*(((34*a^ (7/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/d - (24*S qrt[2]*a^(7/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/d)/(2*a) + (7*a^3*A*Sin[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]]))) /(4*a)))/(3*a))/(2*a^2)
3.2.74.3.1 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]))*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]
Time = 29.94 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.39
| method | result | size |
| default | \(-\frac {A \sqrt {2}\, \left (8 \cos \left (d x +c \right )^{4} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+26 \cos \left (d x +c \right )^{3} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+73 \cos \left (d x +c \right )^{2} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+255 \sqrt {2}\, \cos \left (d x +c \right ) \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )-50 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {2}\, \cos \left (d x +c \right )+360 \cos \left (d x +c \right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-255 \sqrt {2}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )-105 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-360 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )\right ) \csc \left (d x +c \right )}{48 a d \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-a \left (\sec \left (d x +c \right )-1\right )}}\) | \(328\) |
-1/48*A/a/d*2^(1/2)*(8*cos(d*x+c)^4*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^( 1/2)+26*cos(d*x+c)^3*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+73*cos(d*x +c)^2*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+255*2^(1/2)*cos(d*x+c)*ar ctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-50*(-cos(d*x+c)/(cos(d*x+c)+1))^( 1/2)*2^(1/2)*cos(d*x+c)+360*cos(d*x+c)*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(co s(d*x+c)+1))^(1/2))-255*2^(1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)) -105*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-360*arctan(1/2*2^(1/2)/(-c os(d*x+c)/(cos(d*x+c)+1))^(1/2)))/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)/(-a*( sec(d*x+c)-1))^(1/2)*csc(d*x+c)
Time = 0.29 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.42 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\left [-\frac {180 \, \sqrt {2} {\left (A \cos \left (d x + c\right ) - A\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} + {\left (3 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 255 \, {\left (A \cos \left (d x + c\right ) - A\right )} \sqrt {-a} \log \left (\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} - {\left (2 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 2 \, {\left (8 \, A \cos \left (d x + c\right )^{5} + 26 \, A \cos \left (d x + c\right )^{4} + 73 \, A \cos \left (d x + c\right )^{3} - 50 \, A \cos \left (d x + c\right )^{2} - 105 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{48 \, {\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}, \frac {180 \, \sqrt {2} {\left (A \cos \left (d x + c\right ) - A\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 255 \, {\left (A \cos \left (d x + c\right ) - A\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - {\left (8 \, A \cos \left (d x + c\right )^{5} + 26 \, A \cos \left (d x + c\right )^{4} + 73 \, A \cos \left (d x + c\right )^{3} - 50 \, A \cos \left (d x + c\right )^{2} - 105 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{24 \, {\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}\right ] \]
[-1/48*(180*sqrt(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)))*si n(d*x + c) + 255*(A*cos(d*x + c) - A)*sqrt(-a)*log((2*(cos(d*x + c)^2 + co s(d*x + c))*sqrt(-a)*sqrt((a*cos(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) + 2*(8*A*cos(d*x + c) ^5 + 26*A*cos(d*x + c)^4 + 73*A*cos(d*x + c)^3 - 50*A*cos(d*x + c)^2 - 105 *A*cos(d*x + c))*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)))/((a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c)), 1/24*(180*sqrt(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)/(sqr t(a)*sin(d*x + c)))*sin(d*x + c) - 255*(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) - (8*A*cos(d*x + c)^5 + 26*A*cos(d*x + c)^4 + 73*A*cos(d* x + c)^3 - 50*A*cos(d*x + c)^2 - 105*A*cos(d*x + c))*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)))/((a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c))]
\[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=A \left (\int \frac {\cos ^{3}{\left (c + d x \right )}}{- a \sqrt {- a \sec {\left (c + d x \right )} + a} \sec {\left (c + d x \right )} + a \sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx + \int \frac {\cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{- a \sqrt {- a \sec {\left (c + d x \right )} + a} \sec {\left (c + d x \right )} + a \sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx\right ) \]
A*(Integral(cos(c + d*x)**3/(-a*sqrt(-a*sec(c + d*x) + a)*sec(c + d*x) + a *sqrt(-a*sec(c + d*x) + a)), x) + Integral(cos(c + d*x)**3*sec(c + d*x)/(- a*sqrt(-a*sec(c + d*x) + a)*sec(c + d*x) + a*sqrt(-a*sec(c + d*x) + a)), x ))
\[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\int { \frac {{\left (A \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (-a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Time = 1.07 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {180 \, \sqrt {2} A \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {255 \, A \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {12 \, \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {\sqrt {2} {\left (63 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {5}{2}} A + 272 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A a + 324 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{3} a}}{24 \, d} \]
1/24*(180*sqrt(2)*A*arctan(sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/a^( 3/2) - 255*A*arctan(1/2*sqrt(2)*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a) )/a^(3/2) - 12*sqrt(2)*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)*A/(a^2*tan(1/2*d *x + 1/2*c)^2) - sqrt(2)*(63*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(5/2)*A + 272* (a*tan(1/2*d*x + 1/2*c)^2 - a)^(3/2)*A*a + 324*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))/d
Timed out. \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+\frac {A}{\cos \left (c+d\,x\right )}\right )}{{\left (a-\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]