Integrand size = 28, antiderivative size = 154 \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {10 (e \cos (c+d x))^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{33 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \cos (c+d x) (e \cos (c+d x))^{3/2} \sin (c+d x)}{11 a^2 d}+\frac {10 (e \cos (c+d x))^{3/2} \tan (c+d x)}{33 a^2 d}+\frac {4 i \cos ^2(c+d x) (e \cos (c+d x))^{3/2}}{11 d \left (a^2+i a^2 \tan (c+d x)\right )} \] Output:
10/33*(e*cos(d*x+c))^(3/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/a^2/d/co s(d*x+c)^(3/2)+2/11*cos(d*x+c)*(e*cos(d*x+c))^(3/2)*sin(d*x+c)/a^2/d+10/33 *(e*cos(d*x+c))^(3/2)*tan(d*x+c)/a^2/d+4/11*I*cos(d*x+c)^2*(e*cos(d*x+c))^ (3/2)/d/(a^2+I*a^2*tan(d*x+c))
Time = 1.41 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.85 \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {(e \cos (c+d x))^{3/2} \left (-20 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt {\cos (c+d x)} (-28 i \cos (c+d x)+4 i \cos (3 (c+d x))+13 \sin (c+d x)-7 \sin (3 (c+d x)))\right )}{66 a^2 d \cos ^{\frac {7}{2}}(c+d x) (-i+\tan (c+d x))^2} \] Input:
Integrate[(e*Cos[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^2,x]
Output:
((e*Cos[c + d*x])^(3/2)*(-20*EllipticF[(c + d*x)/2, 2]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + Sqrt[Cos[c + d*x]]*((-28*I)*Cos[c + d*x] + (4*I)*Co s[3*(c + d*x)] + 13*Sin[c + d*x] - 7*Sin[3*(c + d*x)])))/(66*a^2*d*Cos[c + d*x]^(7/2)*(-I + Tan[c + d*x])^2)
Time = 0.86 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.21, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3998, 3042, 3981, 3042, 4256, 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.
\(\displaystyle \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2}dx\) |
\(\Big \downarrow \) 3998 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \int \frac {1}{(e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)^2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \int \frac {1}{(e \sec (c+d x))^{3/2} (i \tan (c+d x) a+a)^2}dx\) |
\(\Big \downarrow \) 3981 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (\frac {7 e^2 \int \frac {1}{(e \sec (c+d x))^{7/2}}dx}{11 a^2}+\frac {4 i e^2}{11 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (\frac {7 e^2 \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{11 a^2}+\frac {4 i e^2}{11 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (\frac {7 e^2 \left (\frac {5 \int \frac {1}{(e \sec (c+d x))^{3/2}}dx}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a^2}+\frac {4 i e^2}{11 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (\frac {7 e^2 \left (\frac {5 \int \frac {1}{\left (e \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a^2}+\frac {4 i e^2}{11 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (\frac {7 e^2 \left (\frac {5 \left (\frac {\int \sqrt {e \sec (c+d x)}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a^2}+\frac {4 i e^2}{11 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (\frac {7 e^2 \left (\frac {5 \left (\frac {\int \sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a^2}+\frac {4 i e^2}{11 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (\frac {7 e^2 \left (\frac {5 \left (\frac {\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a^2}+\frac {4 i e^2}{11 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (\frac {7 e^2 \left (\frac {5 \left (\frac {\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a^2}+\frac {4 i e^2}{11 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \left (\frac {7 e^2 \left (\frac {5 \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{3 d e^2}+\frac {2 \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\right )}{7 e^2}+\frac {2 \sin (c+d x)}{7 d e (e \sec (c+d x))^{5/2}}\right )}{11 a^2}+\frac {4 i e^2}{11 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{7/2}}\right )\) |
Input:
Int[(e*Cos[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^2,x]
Output:
(e*Cos[c + d*x])^(3/2)*(e*Sec[c + d*x])^(3/2)*((7*e^2*((2*Sin[c + d*x])/(7 *d*e*(e*Sec[c + d*x])^(5/2)) + (5*((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]])))/(7*e^2)))/(11*a^2) + (((4*I)/11)*e^2)/(d*(e*Sec[c + d*x]) ^(7/2)*(a^2 + I*a^2*Tan[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 _)])^(n_), x_Symbol] :> Simp[2*d^2*(d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + f*x])^(n + 1)/(b*f*(m + 2*n))), x] - Simp[d^2*((m - 2)/(b^2*(m + 2*n))) Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[ {a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, -1] && ((ILtQ[n/2, 0] && IGtQ[m - 1/2, 0]) || EqQ[n, -2] || IGtQ[m + n, 0] || (IntegersQ[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]
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. 314 vs. \(2 (135 ) = 270\).
Time = 10.14 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.05
method | result | size |
default | \(-\frac {2 e^{2} \left (-384 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+384 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1152 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-960 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-1440 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+1008 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+960 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-552 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-360 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+176 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+72 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-28 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-6 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{33 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\) | \(315\) |
Input:
int((e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-2/33/a^2/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^2*(-384 *I*sin(1/2*d*x+1/2*c)^13+384*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+1152 *I*sin(1/2*d*x+1/2*c)^11-960*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-1440 *I*sin(1/2*d*x+1/2*c)^9+1008*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+960*I *sin(1/2*d*x+1/2*c)^7-552*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-360*I*si n(1/2*d*x+1/2*c)^5+176*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+72*I*sin(1/ 2*d*x+1/2*c)^3-28*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+5*(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))-6*I*sin(1/2*d*x+1/2*c))/d
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.75 \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {{\left (-80 i \, \sqrt {\frac {1}{2}} e^{\frac {3}{2}} e^{\left (5 i \, d x + 5 i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \sqrt {\frac {1}{2}} {\left (-11 i \, e e^{\left (6 i \, d x + 6 i \, c\right )} + 41 i \, e e^{\left (4 i \, d x + 4 i \, c\right )} + 15 i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, e\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{132 \, a^{2} d} \] Input:
integrate((e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
1/132*(-80*I*sqrt(1/2)*e^(3/2)*e^(5*I*d*x + 5*I*c)*weierstrassPInverse(-4, 0, e^(I*d*x + I*c)) + sqrt(1/2)*(-11*I*e*e^(6*I*d*x + 6*I*c) + 41*I*e*e^( 4*I*d*x + 4*I*c) + 15*I*e*e^(2*I*d*x + 2*I*c) + 3*I*e)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*e^(-1/2*I*d*x - 1/2*I*c))*e^(-5*I*d*x - 5*I*c)/(a^2*d)
Timed out. \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(3/2)/(a+I*a*tan(d*x+c))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
integrate((e*cos(d*x + c))^(3/2)/(I*a*tan(d*x + c) + a)^2, x)
Timed out. \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \] Input:
int((e*cos(c + d*x))^(3/2)/(a + a*tan(c + d*x)*1i)^2,x)
Output:
int((e*cos(c + d*x))^(3/2)/(a + a*tan(c + d*x)*1i)^2, x)
\[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )}{\tan \left (d x +c \right )^{2}-2 \tan \left (d x +c \right ) i -1}d x \right ) e}{a^{2}} \] Input:
int((e*cos(d*x+c))^(3/2)/(a+I*a*tan(d*x+c))^2,x)
Output:
( - sqrt(e)*int((sqrt(cos(c + d*x))*cos(c + d*x))/(tan(c + d*x)**2 - 2*tan (c + d*x)*i - 1),x)*e)/a**2