Integrand size = 26, antiderivative size = 90 \[ \int (e \cos (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx=-\frac {2 i a (e \cos (c+d x))^{3/2}}{3 d}+\frac {2 a (e \cos (c+d x))^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (e \cos (c+d x))^{3/2} \tan (c+d x)}{3 d} \] Output:
-2/3*I*a*(e*cos(d*x+c))^(3/2)/d+2/3*a*(e*cos(d*x+c))^(3/2)*InverseJacobiAM (1/2*d*x+1/2*c,2^(1/2))/d/cos(d*x+c)^(3/2)+2/3*a*(e*cos(d*x+c))^(3/2)*tan( d*x+c)/d
Time = 0.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.11 \[ \int (e \cos (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx=\frac {2 a e \sqrt {\cos (c+d x)} \sqrt {e \cos (c+d x)} \left (\operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (i \cos (c)+\sin (c))+\sqrt {\cos (c+d x)} (\cos (d x)+i \sin (d x))\right ) (\cos (d x)-i \sin (d x)) (-i+\tan (c+d x))}{3 d} \] Input:
Integrate[(e*Cos[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x]),x]
Output:
(2*a*e*Sqrt[Cos[c + d*x]]*Sqrt[e*Cos[c + d*x]]*(EllipticF[(c + d*x)/2, 2]* (I*Cos[c] + Sin[c]) + Sqrt[Cos[c + d*x]]*(Cos[d*x] + I*Sin[d*x]))*(Cos[d*x ] - I*Sin[d*x])*(-I + Tan[c + d*x]))/(3*d)
Time = 0.67 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.36, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3998, 3042, 3967, 3042, 4256, 3042, 4258, 3042, 3120}
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.
| \(\displaystyle \int (a+i a \tan (c+d x)) (e \cos (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (c+d x)) (e \cos (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3998 |
| \(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \int \frac {i \tan (c+d x) a+a}{(e \sec (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \int \frac {i \tan (c+d x) a+a}{(e \sec (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (a \int \frac {1}{(e \sec (c+d x))^{3/2}}dx-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (a \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (a \left (\frac {\int \sqrt {e \sec (c+d x)}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (a \left (\frac {\int \sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (a \left (\frac {\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (a \left (\frac {\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (a \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{3 d e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}\right )\) |
Input:
Int[(e*Cos[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x]),x]
Output:
(e*Cos[c + d*x])^(3/2)*(e*Sec[c + d*x])^(3/2)*((((-2*I)/3)*a)/(d*(e*Sec[c + d*x])^(3/2)) + a*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e *Sec[c + d*x]])/(3*d*e^2) + (2*Sin[c + d*x])/(3*d*e*Sqrt[e*Sec[c + d*x]])) )
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* n]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (76 ) = 152\).
Time = 4.54 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.87
| method | result | size |
| default | \(-\frac {2 a \,e^{2} \left (4 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-4 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\) | \(168\) |
| parts | \(-\frac {2 a \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, e^{2} \left (4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-e \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, d}-\frac {2 i a \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}\) | \(209\) |
| risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \sqrt {2}\, e a \sqrt {e \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) {\mathrm e}^{-i \left (d x +c \right )}}}{3 d}+\frac {2 \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right ) e a \sqrt {e \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) {\mathrm e}^{-i \left (d x +c \right )}}\, \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{3 d \sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(224\) |
Input:
int((e*cos(d*x+c))^(3/2)*(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-2/3/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*(4*I*sin (1/2*d*x+1/2*c)^5+4*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-4*I*sin(1/2*d* x+1/2*c)^3-2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+(sin(1/2*d*x+1/2*c)^2 )^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1 /2))+I*sin(1/2*d*x+1/2*c))/d
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64 \[ \int (e \cos (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx=-\frac {2 \, {\left (i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} a e e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 2 i \, \sqrt {\frac {1}{2}} a e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{3 \, d} \] Input:
integrate((e*cos(d*x+c))^(3/2)*(a+I*a*tan(d*x+c)),x, algorithm="fricas")
Output:
-2/3*(I*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*a*e*e^(1/2*I*d*x + 1/2*I *c) + 2*I*sqrt(1/2)*a*e^(3/2)*weierstrassPInverse(-4, 0, e^(I*d*x + I*c))) /d
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(3/2)*(a+I*a*tan(d*x+c)),x)
Output:
Timed out
\[ \int (e \cos (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \,d x } \] Input:
integrate((e*cos(d*x+c))^(3/2)*(a+I*a*tan(d*x+c)),x, algorithm="maxima")
Output:
integrate((e*cos(d*x + c))^(3/2)*(I*a*tan(d*x + c) + a), x)
\[ \int (e \cos (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \,d x } \] Input:
integrate((e*cos(d*x+c))^(3/2)*(a+I*a*tan(d*x+c)),x, algorithm="giac")
Output:
integrate((e*cos(d*x + c))^(3/2)*(I*a*tan(d*x + c) + a), x)
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right ) \,d x \] Input:
int((e*cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i),x)
Output:
int((e*cos(c + d*x))^(3/2)*(a + a*tan(c + d*x)*1i), x)
\[ \int (e \cos (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx=\sqrt {e}\, a e \left (\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \tan \left (d x +c \right )d x \right ) i +\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) \] Input:
int((e*cos(d*x+c))^(3/2)*(a+I*a*tan(d*x+c)),x)
Output:
sqrt(e)*a*e*(int(sqrt(cos(c + d*x))*cos(c + d*x)*tan(c + d*x),x)*i + int(s qrt(cos(c + d*x))*cos(c + d*x),x))