Integrand size = 23, antiderivative size = 138 \[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\frac {\sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{3 d}-\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{\sqrt {7} d}-\frac {5 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{3 \sqrt {7} d}-\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d} \] Output:
1/3*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))*7^(1/2)/d-3/7*InverseJacobi AM(1/2*d*x+1/2*Pi+1/2*c,2/7*14^(1/2))*7^(1/2)/d-5/21*EllipticPi(cos(1/2*d* x+1/2*c),2,2/7*14^(1/2))*7^(1/2)/d-1/3*(3-4*cos(d*x+c))^(1/2)*tan(d*x+c)/d +1/2*(3-4*cos(d*x+c))^(1/2)*sec(d*x+c)*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 2.36 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.72 \[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\frac {-\frac {12 \sqrt {-3+4 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),8\right )}{\sqrt {3-4 \cos (c+d x)}}+\frac {6 \sqrt {-3+4 \cos (c+d x)} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),8\right )}{\sqrt {3-4 \cos (c+d x)}}+\frac {2 i \left (21 E\left (i \text {arcsinh}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )-12 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {3-4 \cos (c+d x)}\right ),-\frac {1}{7}\right )-8 \operatorname {EllipticPi}\left (-\frac {1}{3},i \text {arcsinh}\left (\sqrt {3-4 \cos (c+d x)}\right ),-\frac {1}{7}\right )\right ) \sin (c+d x)}{3 \sqrt {7} \sqrt {\sin ^2(c+d x)}}-\sqrt {3-4 \cos (c+d x)} (-3+2 \cos (c+d x)) \sec (c+d x) \tan (c+d x)}{6 d} \] Input:
Integrate[Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]^3,x]
Output:
((-12*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8])/Sqrt[3 - 4*Cos[ c + d*x]] + (6*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticPi[2, (c + d*x)/2, 8])/Sq rt[3 - 4*Cos[c + d*x]] + (((2*I)/3)*(21*EllipticE[I*ArcSinh[Sqrt[3 - 4*Cos [c + d*x]]], -1/7] - 12*EllipticF[I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/ 7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7])*Sin[c + d*x])/(Sqrt[7]*Sqrt[Sin[c + d*x]^2]) - Sqrt[3 - 4*Cos[c + d*x]]*(-3 + 2* Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d)
Time = 1.01 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 3275, 25, 3042, 3534, 25, 3042, 3538, 27, 3042, 3133, 3481, 3042, 3141, 3285}
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.
| \(\displaystyle \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3275 |
| \(\displaystyle \frac {1}{2} \int -\frac {\left (2 \cos ^2(c+d x)-3 \cos (c+d x)+2\right ) \sec ^2(c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}-\frac {1}{2} \int \frac {\left (2 \cos ^2(c+d x)-3 \cos (c+d x)+2\right ) \sec ^2(c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}-\frac {1}{2} \int \frac {2 \sin \left (c+d x+\frac {\pi }{2}\right )^2-3 \sin \left (c+d x+\frac {\pi }{2}\right )+2}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{3} \int -\frac {\left (-4 \cos ^2(c+d x)-6 \cos (c+d x)+5\right ) \sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {\left (-4 \cos ^2(c+d x)-6 \cos (c+d x)+5\right ) \sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {-4 \sin \left (c+d x+\frac {\pi }{2}\right )^2-6 \sin \left (c+d x+\frac {\pi }{2}\right )+5}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \sqrt {3-4 \cos (c+d x)}dx+\frac {1}{4} \int \frac {4 (5-9 \cos (c+d x)) \sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx\right )-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \sqrt {3-4 \cos (c+d x)}dx+\int \frac {(5-9 \cos (c+d x)) \sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx\right )-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\int \frac {5-9 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 3133 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \frac {5-9 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{d}\right )-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-9 \int \frac {1}{\sqrt {3-4 \cos (c+d x)}}dx+5 \int \frac {\sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx+\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{d}\right )-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-9 \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+5 \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{d}\right )-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 3141 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (5 \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {18 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{d}\right )-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}\) |
| \(\Big \downarrow \) 3285 |
| \(\displaystyle \frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}+\frac {1}{2} \left (\frac {1}{3} \left (-\frac {18 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{d}-\frac {10 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{\sqrt {7} d}\right )-\frac {2 \sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\right )\) |
Input:
Int[Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]^3,x]
Output:
(Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (((2*Sqrt[7]* EllipticE[(c + Pi + d*x)/2, 8/7])/d - (18*EllipticF[(c + Pi + d*x)/2, 8/7] )/(Sqrt[7]*d) - (10*EllipticPi[2, (c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d))/3 - (2*Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a - b]/d)*EllipticE[(1/2)*(c + Pi/2 + d*x), -2*(b/(a - b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a - b]))*EllipticF[(1/2)*(c + Pi/2 + d*x), -2*(b/(a - b))], x] /; FreeQ [{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^(m + 1)*((c + d*Sin[e + f*x])^n/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ (n - 1)*Simp[a*c*(m + 1) + b*d*n + (a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(m + n + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && IntegersQ[2*m, 2*n]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a - b)*Sqrt[c - d]))*EllipticPi[ -2*(b/(a - b)), (1/2)*(e + Pi/2 + f*x), -2*(d/(c - d))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2 , 0] && GtQ[c - d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(125)=250\).
Time = 6.43 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.96
| method | result | size |
| default | \(-\frac {\sqrt {-\left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}+\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{3 \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}-\frac {3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{7 \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}+\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{3 \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}-\frac {5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticPi}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, \frac {2 \sqrt {14}}{7}\right )}{21 \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+7}\, d}\) | \(408\) |
Input:
int((3-4*cos(d*x+c))^(1/2)*sec(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-(-(8*cos(1/2*d*x+1/2*c)^2-7)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-cos(1/2*d*x+1/ 2*c)*(8*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/ 2*c)^2-1)^2+2/3*cos(1/2*d*x+1/2*c)*(8*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2 *c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2-1)-3/7*(sin(1/2*d*x+1/2*c)^2)^(1/2)*( 56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)/(8*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),2/7*14^(1/2))+1/3*(sin(1/2*d*x+1/2 *c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)/(8*sin(1/2*d*x+1/2*c)^4-sin (1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))-5/21*( sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)/(8*sin(1/2*d *x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,2/ 7*14^(1/2)))/sin(1/2*d*x+1/2*c)/(-8*cos(1/2*d*x+1/2*c)^2+7)^(1/2)/d
\[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate((3-4*cos(d*x+c))^(1/2)*sec(d*x+c)^3,x, algorithm="fricas")
Output:
integral(sqrt(-4*cos(d*x + c) + 3)*sec(d*x + c)^3, x)
\[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int \sqrt {3 - 4 \cos {\left (c + d x \right )}} \sec ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate((3-4*cos(d*x+c))**(1/2)*sec(d*x+c)**3,x)
Output:
Integral(sqrt(3 - 4*cos(c + d*x))*sec(c + d*x)**3, x)
\[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate((3-4*cos(d*x+c))^(1/2)*sec(d*x+c)^3,x, algorithm="maxima")
Output:
integrate(sqrt(-4*cos(d*x + c) + 3)*sec(d*x + c)^3, x)
\[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate((3-4*cos(d*x+c))^(1/2)*sec(d*x+c)^3,x, algorithm="giac")
Output:
integrate(sqrt(-4*cos(d*x + c) + 3)*sec(d*x + c)^3, x)
Timed out. \[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int \frac {\sqrt {3-4\,\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^3} \,d x \] Input:
int((3 - 4*cos(c + d*x))^(1/2)/cos(c + d*x)^3,x)
Output:
int((3 - 4*cos(c + d*x))^(1/2)/cos(c + d*x)^3, x)
\[ \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\int \sqrt {-4 \cos \left (d x +c \right )+3}\, \sec \left (d x +c \right )^{3}d x \] Input:
int((3-4*cos(d*x+c))^(1/2)*sec(d*x+c)^3,x)
Output:
int(sqrt( - 4*cos(c + d*x) + 3)*sec(c + d*x)**3,x)