Integrand size = 26, antiderivative size = 60 \[ \int \sqrt {e \cos (c+d x)} (a+i a \tan (c+d x)) \, dx=-\frac {2 i a \sqrt {e \cos (c+d x)}}{d}+\frac {2 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)}} \]
-2*I*a*(e*cos(d*x+c))^(1/2)/d+2*a*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x +1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/d/cos(d *x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.68 (sec) , antiderivative size = 192, normalized size of antiderivative = 3.20 \[ \int \sqrt {e \cos (c+d x)} (a+i a \tan (c+d x)) \, dx=\frac {a \cos (c) \sqrt {e \cos (c+d x)} \sin (c) (\cos (d x)-i \sin (d x)) \left (\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) (-i \csc (c)-\sec (c)) \sec (c) \sin (d x+\arctan (\tan (c)))+\sqrt {\sin ^2(d x+\arctan (\tan (c)))} \left (2 \cos (d x+\arctan (\tan (c))) \csc (c) (i \csc (c)+\sec (c))+\sec (c) \left (-2 i \cos (c+d x) \csc ^2(c) \sqrt {\sec ^2(c)}+(i \csc (c)+\sec (c)) \sin (d x+\arctan (\tan (c)))\right )\right )\right ) (-i+\tan (c+d x))}{d \sqrt {\sec ^2(c)} \sqrt {\sin ^2(d x+\arctan (\tan (c)))}} \]
(a*Cos[c]*Sqrt[e*Cos[c + d*x]]*Sin[c]*(Cos[d*x] - I*Sin[d*x])*(Hypergeomet ricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*((-I)*Csc[c] - Se c[c])*Sec[c]*Sin[d*x + ArcTan[Tan[c]]] + Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2] *(2*Cos[d*x + ArcTan[Tan[c]]]*Csc[c]*(I*Csc[c] + Sec[c]) + Sec[c]*((-2*I)* Cos[c + d*x]*Csc[c]^2*Sqrt[Sec[c]^2] + (I*Csc[c] + Sec[c])*Sin[d*x + ArcTa n[Tan[c]]])))*(-I + Tan[c + d*x]))/(d*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan [Tan[c]]]^2])
Time = 0.49 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.42, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 3998, 3042, 3967, 3042, 4258, 3042, 3119}
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.
\(\displaystyle \int (a+i a \tan (c+d x)) \sqrt {e \cos (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+i a \tan (c+d x)) \sqrt {e \cos (c+d x)}dx\) |
\(\Big \downarrow \) 3998 |
\(\displaystyle \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {i \tan (c+d x) a+a}{\sqrt {e \sec (c+d x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {i \tan (c+d x) a+a}{\sqrt {e \sec (c+d x)}}dx\) |
\(\Big \downarrow \) 3967 |
\(\displaystyle \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)} \left (a \int \frac {1}{\sqrt {e \sec (c+d x)}}dx-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)} \left (a \int \frac {1}{\sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}\right )\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)} \left (\frac {a \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)} \left (\frac {a \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}\right )\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)} \left (\frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i a}{d \sqrt {e \sec (c+d x)}}\right )\) |
Sqrt[e*Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]*(((-2*I)*a)/(d*Sqrt[e*Sec[c + d* x]]) + (2*a*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]))
3.7.58.3.1 Defintions of rubi rules used
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[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_.), x_Symbol] :> Simp[(d*Cos[e + f*x])^m*(d*Sec[e + f*x])^m Int[( a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m , n}, x] && !IntegerQ[m]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Time = 4.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.80
method | result | size |
default | \(\frac {2 a e \left (2 i \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}-i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(108\) |
parts | \(\frac {2 a \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {2 i a \sqrt {e \cos \left (d x +c \right )}}{d}\) | \(161\) |
risch | \(-\frac {2 i \sqrt {2}\, a \sqrt {e \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) {\mathrm e}^{-i \left (d x +c \right )}}}{d}-\frac {i \left (-\frac {2 \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}{e \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \left (-2 i E\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i F\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {2}\, a \sqrt {e \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) {\mathrm e}^{-i \left (d x +c \right )}}\, \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(301\) |
2/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a*e*(2*I*sin(1/2* d*x+1/2*c)^3+EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2 -1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)-I*sin(1/2*d*x+1/2*c))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.45 \[ \int \sqrt {e \cos (c+d x)} (a+i a \tan (c+d x)) \, dx=\frac {2 i \, \sqrt {2} a \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}{d} \]
\[ \int \sqrt {e \cos (c+d x)} (a+i a \tan (c+d x)) \, dx=i a \left (\int \left (- i \sqrt {e \cos {\left (c + d x \right )}}\right )\, dx + \int \sqrt {e \cos {\left (c + d x \right )}} \tan {\left (c + d x \right )}\, dx\right ) \]
\[ \int \sqrt {e \cos (c+d x)} (a+i a \tan (c+d x)) \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \,d x } \]
\[ \int \sqrt {e \cos (c+d x)} (a+i a \tan (c+d x)) \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \,d x } \]
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+i a \tan (c+d x)) \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right ) \,d x \]