Integrand size = 35, antiderivative size = 243 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {(107 A+112 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {a} d}-\frac {\sqrt {2} (A+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d}-\frac {(21 A+16 C) \tan (c+d x)}{64 d \sqrt {a+a \cos (c+d x)}}+\frac {(43 A+48 C) \sec (c+d x) \tan (c+d x)}{96 d \sqrt {a+a \cos (c+d x)}}-\frac {A \sec ^2(c+d x) \tan (c+d x)}{24 d \sqrt {a+a \cos (c+d x)}}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 d \sqrt {a+a \cos (c+d x)}} \]
1/64*(107*A+112*C)*arctanh(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c))^(1/2))/d/a^ (1/2)-(A+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+16*C)*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/ 96*(43*A+48*C)*sec(d*x+c)*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)-1/24*A*sec(d *x+c)^2*tan(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/4*A*sec(d*x+c)^3*tan(d*x+c)/ d/(a+a*cos(d*x+c))^(1/2)
Time = 1.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.72 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=-\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (768 (A+C) \text {arctanh}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4(c+d x)-6 \sqrt {2} (107 A+112 C) \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4(c+d x)+(-364 A-192 C+(221 A+144 C) \cos (c+d x)-4 (43 A+48 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))+48 C \cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{384 d \sqrt {a (1+\cos (c+d x))}} \]
-1/384*(Cos[(c + d*x)/2]*Sec[c + d*x]^4*(768*(A + C)*ArcTanh[Sin[(c + d*x) /2]]*Cos[c + d*x]^4 - 6*Sqrt[2]*(107*A + 112*C)*ArcTanh[Sqrt[2]*Sin[(c + d *x)/2]]*Cos[c + d*x]^4 + (-364*A - 192*C + (221*A + 144*C)*Cos[c + d*x] - 4*(43*A + 48*C)*Cos[2*(c + d*x)] + 63*A*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 = 1.76 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.14, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.543, Rules used = {3042, 3523, 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.
\(\displaystyle \int \frac {\sec ^5(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a \cos (c+d x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^5 \sqrt {a \sin \left (c+d x+\frac {\pi }{2}\right )+a}}dx\) |
\(\Big \downarrow \) 3523 |
\(\displaystyle \frac {\int -\frac {(a A-a (7 A+8 C) \cos (c+d x)) \sec ^4(c+d x)}{2 \sqrt {\cos (c+d x) a+a}}dx}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {(a A-a (7 A+8 C) \cos (c+d x)) \sec ^4(c+d x)}{\sqrt {\cos (c+d x) a+a}}dx}{8 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {a A-a (7 A+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{8 a}\) |
\(\Big \downarrow \) 3463 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {\int -\frac {\left (a^2 (43 A+48 C)-5 a^2 A \cos (c+d x)\right ) \sec ^3(c+d x)}{2 \sqrt {\cos (c+d x) a+a}}dx}{3 a}+\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}}{8 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {\left (a^2 (43 A+48 C)-5 a^2 A \cos (c+d x)\right ) \sec ^3(c+d x)}{\sqrt {\cos (c+d x) a+a}}dx}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {a^2 (43 A+48 C)-5 a^2 A \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3463 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {\int -\frac {3 \left (a^3 (21 A+16 C)-a^3 (43 A+48 C) \cos (c+d x)\right ) \sec ^2(c+d x)}{2 \sqrt {\cos (c+d x) a+a}}dx}{2 a}+\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \int \frac {\left (a^3 (21 A+16 C)-a^3 (43 A+48 C) \cos (c+d x)\right ) \sec ^2(c+d x)}{\sqrt {\cos (c+d x) a+a}}dx}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \int \frac {a^3 (21 A+16 C)-a^3 (43 A+48 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3463 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {\int -\frac {\left (a^4 (107 A+112 C)-a^4 (21 A+16 C) \cos (c+d x)\right ) \sec (c+d x)}{2 \sqrt {\cos (c+d x) a+a}}dx}{a}+\frac {a^3 (21 A+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {\left (a^4 (107 A+112 C)-a^4 (21 A+16 C) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {\cos (c+d x) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {a^4 (107 A+112 C)-a^4 (21 A+16 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3464 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {a^3 (107 A+112 C) \int \sqrt {\cos (c+d x) a+a} \sec (c+d x)dx-128 a^4 (A+C) \int \frac {1}{\sqrt {\cos (c+d x) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {a^3 (107 A+112 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-128 a^4 (A+C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {a^3 (107 A+112 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {256 a^4 (A+C) \int \frac {1}{2 a-\frac {a^2 \sin ^2(c+d x)}{\cos (c+d x) a+a}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {a^3 (107 A+112 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {128 \sqrt {2} a^{7/2} (A+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3252 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {-\frac {2 a^4 (107 A+112 C) \int \frac {1}{a-\frac {a^2 \sin ^2(c+d x)}{\cos (c+d x) a+a}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}-\frac {128 \sqrt {2} a^{7/2} (A+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {a^2 (43 A+48 C) \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A+16 C) \tan (c+d x)}{d \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 a^{7/2} (107 A+112 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}-\frac {128 \sqrt {2} a^{7/2} (A+C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
(A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - ((a*A*Sec [c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - ((a^2*(43*A + 4 8*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 + 112*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[ c + d*x]]])/d - (128*Sqrt[2]*a^(7/2)*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x] )/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/d)/a + (a^3*(21*A + 16*C)*Tan[c + d *x])/(d*Sqrt[a + a*Cos[c + d*x]])))/(4*a))/(6*a))/(8*a)
3.2.10.3.1 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_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + 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*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(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, 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. \(1775\) vs. \(2(210)=420\).
Time = 11.98 (sec) , antiderivative size = 1776, normalized size of antiderivative = 7.31
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1776\) |
default | \(\text {Expression too large to display}\) | \(2069\) |
-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)/ cos(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)-7 704*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...
Time = 0.34 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.28 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {3 \, {\left ({\left (107 \, A + 112 \, C\right )} \cos \left (d x + c\right )^{5} + {\left (107 \, A + 112 \, C\right )} \cos \left (d x + c\right )^{4}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) - 4 \, {\left (3 \, {\left (21 \, A + 16 \, C\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (43 \, A + 48 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, A \cos \left (d x + c\right ) - 48 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + \frac {384 \, \sqrt {2} {\left ({\left (A + C\right )} a \cos \left (d x + c\right )^{5} + {\left (A + C\right )} a \cos \left (d x + c\right )^{4}\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} + \frac {2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt {a}}}{768 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}} \]
1/768*(3*((107*A + 112*C)*cos(d*x + c)^5 + (107*A + 112*C)*cos(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 + co s(d*x + c)^2)) - 4*(3*(21*A + 16*C)*cos(d*x + c)^3 - 2*(43*A + 48*C)*cos(d *x + c)^2 + 8*A*cos(d*x + c) - 48*A)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c) + 384*sqrt(2)*((A + C)*a*cos(d*x + c)^5 + (A + C)*a*cos(d*x + c)^4)*log(- (cos(d*x + c)^2 + 2*sqrt(2)*sqrt(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. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 657419 vs. \(2 (210) = 420\).
Time = 286.61 (sec) , antiderivative size = 657419, normalized size of antiderivative = 2705.43 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\text {Too large to display} \]
-1/768*((1280*(334*sqrt(2)*sin(5/2*d*x + 5/2*c) - 365*sqrt(2)*sin(3/2*d*x + 3/2*c))*cos(3*d*x + 3*c)^3 - 1280*(334*sqrt(2)*cos(5/2*d*x + 5/2*c) - 36 5*sqrt(2)*cos(3/2*d*x + 3/2*c))*sin(3*d*x + 3*c)^3 - 34800*sqrt(2)*cos(2*d *x + 2*c)^2*sin(3/2*d*x + 3/2*c) - 34800*sqrt(2)*sin(2*d*x + 2*c)^2*sin(3/ 2*d*x + 3/2*c) + 348*(4*sqrt(2)*cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c) - 4* sqrt(2)*cos(3*d*x + 3*c)*sin(3/2*d*x + 3/2*c) - sqrt(2)*sin(3/2*d*x + 3/2* c))*cos(12*d*x + 12*c)^2 + 5568*(4*sqrt(2)*cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c) - 4*sqrt(2)*cos(3*d*x + 3*c)*sin(3/2*d*x + 3/2*c) - sqrt(2)*sin(3/2 *d*x + 3/2*c))*cos(11*d*x + 11*c)^2 + 34800*(4*sqrt(2)*cos(3/2*d*x + 3/2*c )*sin(3*d*x + 3*c) - 4*sqrt(2)*cos(3*d*x + 3*c)*sin(3/2*d*x + 3/2*c) - sqr t(2)*sin(3/2*d*x + 3/2*c))*cos(10*d*x + 10*c)^2 + 139200*(4*sqrt(2)*cos(3/ 2*d*x + 3/2*c)*sin(3*d*x + 3*c) - 4*sqrt(2)*cos(3*d*x + 3*c)*sin(3/2*d*x + 3/2*c) - sqrt(2)*sin(3/2*d*x + 3/2*c))*cos(9*d*x + 9*c)^2 + 334428*(4*sqr t(2)*cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c) - 4*sqrt(2)*cos(3*d*x + 3*c)*si n(3/2*d*x + 3/2*c) - sqrt(2)*sin(3/2*d*x + 3/2*c))*cos(8*d*x + 8*c)^2 + 55 6800*(4*sqrt(2)*cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c) - 4*sqrt(2)*cos(3*d* x + 3*c)*sin(3/2*d*x + 3/2*c) - sqrt(2)*sin(3/2*d*x + 3/2*c))*cos(7*d*x + 7*c)^2 + 673728*(4*sqrt(2)*cos(3/2*d*x + 3/2*c)*sin(3*d*x + 3*c) - 4*sqrt( 2)*cos(3*d*x + 3*c)*sin(3/2*d*x + 3/2*c) - sqrt(2)*sin(3/2*d*x + 3/2*c))*c os(6*d*x + 6*c)^2 + 556800*(4*sqrt(2)*cos(3/2*d*x + 3/2*c)*sin(3*d*x + ...
Time = 0.43 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.42 \[ \int \frac {\left (A+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} + 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} + 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 + 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 + 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} + 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} - 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} - 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 ) + 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} \]
-1/384*(192*sqrt(2)*(A*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) + C*sqrt(a))*log(-s in(1/2*d*x + 1/2*c) + 1)/(a*sgn(cos(1/2*d*x + 1/2*c))) - 3*(107*A + 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 + 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 + 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 - 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 - 96*C*sqrt(a)*sin(1/2*d*x + 1/2*c)^3 + 63*A*sqrt(a)*si n(1/2*d*x + 1/2*c) + 48*C*sqrt(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. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^5\,\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]