Integrand size = 28, antiderivative size = 92 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=-\frac {2 \cos ^{\frac {5}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d (e \cos (c+d x))^{5/2}}+\frac {4 i \cos ^2(c+d x)}{3 d (e \cos (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )} \] Output:
-2/3*cos(d*x+c)^(5/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/a^2/d/(e*cos( d*x+c))^(5/2)+4/3*I*cos(d*x+c)^2/d/(e*cos(d*x+c))^(5/2)/(a^2+I*a^2*tan(d*x +c))
Time = 1.01 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\frac {2 \sqrt {\cos (c+d x)} (\cos (d x)+i \sin (d x))^2 \left (\operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (2 c)+i \sin (2 c))+2 \sqrt {\cos (c+d x)} (-i \cos (c-d x)+\sin (c-d x))\right )}{3 a^2 d (e \cos (c+d x))^{5/2} (-i+\tan (c+d x))^2} \] Input:
Integrate[1/((e*Cos[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^2),x]
Output:
(2*Sqrt[Cos[c + d*x]]*(Cos[d*x] + I*Sin[d*x])^2*(EllipticF[(c + d*x)/2, 2] *(Cos[2*c] + I*Sin[2*c]) + 2*Sqrt[Cos[c + d*x]]*((-I)*Cos[c - d*x] + Sin[c - d*x])))/(3*a^2*d*(e*Cos[c + d*x])^(5/2)*(-I + Tan[c + d*x])^2)
Time = 0.57 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.25, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3998, 3042, 3981, 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.
\(\displaystyle \int \frac {1}{(a+i a \tan (c+d x))^2 (e \cos (c+d x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+i a \tan (c+d x))^2 (e \cos (c+d x))^{5/2}}dx\) |
\(\Big \downarrow \) 3998 |
\(\displaystyle \frac {\int \frac {(e \sec (c+d x))^{5/2}}{(i \tan (c+d x) a+a)^2}dx}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(e \sec (c+d x))^{5/2}}{(i \tan (c+d x) a+a)^2}dx}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3981 |
\(\displaystyle \frac {-\frac {e^2 \int \sqrt {e \sec (c+d x)}dx}{3 a^2}+\frac {4 i e^2 \sqrt {e \sec (c+d x)}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {e^2 \int \sqrt {e \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 a^2}+\frac {4 i e^2 \sqrt {e \sec (c+d x)}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {-\frac {e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 a^2}+\frac {4 i e^2 \sqrt {e \sec (c+d x)}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a^2}+\frac {4 i e^2 \sqrt {e \sec (c+d x)}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {-\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{3 a^2 d}+\frac {4 i e^2 \sqrt {e \sec (c+d x)}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\) |
Input:
Int[1/((e*Cos[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^2),x]
Output:
((-2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]] )/(3*a^2*d) + (((4*I)/3)*e^2*Sqrt[e*Sec[c + d*x]])/(d*(a^2 + I*a^2*Tan[c + d*x])))/((e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2))
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[(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. 170 vs. \(2 (81 ) = 162\).
Time = 3.46 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.86
method | result | size |
default | \(-\frac {2 \left (-8 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+8 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-2 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 e^{2} a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\) | \(171\) |
Input:
int(1/(e*cos(d*x+c))^(5/2)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-2/3/e^2/a^2/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(-8*I* sin(1/2*d*x+1/2*c)^5+8*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+8*I*sin(1/2 *d*x+1/2*c)^3-4*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))-2*I*sin(1/2*d*x+1/2*c))/d
Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=-\frac {4 \, {\left (-i \, \sqrt {\frac {1}{2}} \sqrt {e} e^{\left (i \, d x + i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) - i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{3 \, a^{2} d e^{3}} \] Input:
integrate(1/(e*cos(d*x+c))^(5/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas ")
Output:
-4/3*(-I*sqrt(1/2)*sqrt(e)*e^(I*d*x + I*c)*weierstrassPInverse(-4, 0, e^(I *d*x + I*c)) - I*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*e^(-1/2*I*d*x - 1/2*I*c))*e^(-I*d*x - I*c)/(a^2*d*e^3)
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))**(5/2)/(a+I*a*tan(d*x+c))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(e*cos(d*x+c))^(5/2)/(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.
\[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(1/(e*cos(d*x+c))^(5/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
integrate(1/((e*cos(d*x + c))^(5/2)*(I*a*tan(d*x + c) + a)^2), x)
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \] Input:
int(1/((e*cos(c + d*x))^(5/2)*(a + a*tan(c + d*x)*1i)^2),x)
Output:
int(1/((e*cos(c + d*x))^(5/2)*(a + a*tan(c + d*x)*1i)^2), x)
\[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=-\frac {\int \frac {1}{\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \tan \left (d x +c \right )^{2}-2 \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \tan \left (d x +c \right ) i -\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}d x}{\sqrt {e}\, a^{2} e^{2}} \] Input:
int(1/(e*cos(d*x+c))^(5/2)/(a+I*a*tan(d*x+c))^2,x)
Output:
( - int(1/(sqrt(cos(c + d*x))*cos(c + d*x)**2*tan(c + d*x)**2 - 2*sqrt(cos (c + d*x))*cos(c + d*x)**2*tan(c + d*x)*i - sqrt(cos(c + d*x))*cos(c + d*x )**2),x))/(sqrt(e)*a**2*e**2)