Integrand size = 28, antiderivative size = 71 \[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))} \, dx=-\frac {3 i \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {17}{6},\frac {1}{6},\frac {1}{2} (1-i \tan (e+f x))\right ) (1+i \tan (e+f x))^{5/6}}{10\ 2^{5/6} a f (d \sec (e+f x))^{5/3}} \] Output:
-3/20*I*hypergeom([-5/6, 17/6],[1/6],1/2-1/2*I*tan(f*x+e))*(1+I*tan(f*x+e) )^(5/6)*2^(1/6)/a/f/(d*sec(f*x+e))^(5/3)
Time = 1.79 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.68 \[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))} \, dx=-\frac {3 \sec ^2(e+f x) \left (-26+6 \cos (2 (e+f x))+\frac {128 e^{2 i (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-e^{2 i (e+f x)}\right )}{\left (1+e^{2 i (e+f x)}\right )^{2/3}}+16 i \sin (2 (e+f x))\right )}{220 a f (d \sec (e+f x))^{5/3} (-i+\tan (e+f x))} \] Input:
Integrate[1/((d*Sec[e + f*x])^(5/3)*(a + I*a*Tan[e + f*x])),x]
Output:
(-3*Sec[e + f*x]^2*(-26 + 6*Cos[2*(e + f*x)] + (128*E^((2*I)*(e + f*x))*Hy pergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(e + f*x))])/(1 + E^((2*I)*(e + f *x)))^(2/3) + (16*I)*Sin[2*(e + f*x)]))/(220*a*f*(d*Sec[e + f*x])^(5/3)*(- I + Tan[e + f*x]))
Time = 0.47 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3986, 3042, 4006, 80, 27, 79}
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.
\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x)) (d \sec (e+f x))^{5/3}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x)) (d \sec (e+f x))^{5/3}}dx\) |
\(\Big \downarrow \) 3986 |
\(\displaystyle \frac {(a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6} \int \frac {1}{(a-i a \tan (e+f x))^{5/6} (i \tan (e+f x) a+a)^{11/6}}dx}{(d \sec (e+f x))^{5/3}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6} \int \frac {1}{(a-i a \tan (e+f x))^{5/6} (i \tan (e+f x) a+a)^{11/6}}dx}{(d \sec (e+f x))^{5/3}}\) |
\(\Big \downarrow \) 4006 |
\(\displaystyle \frac {a^2 (a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6} \int \frac {1}{(a-i a \tan (e+f x))^{11/6} (i \tan (e+f x) a+a)^{17/6}}d\tan (e+f x)}{f (d \sec (e+f x))^{5/3}}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {(1+i \tan (e+f x))^{5/6} (a-i a \tan (e+f x))^{5/6} \int \frac {4\ 2^{5/6}}{(i \tan (e+f x)+1)^{17/6} (a-i a \tan (e+f x))^{11/6}}d\tan (e+f x)}{4\ 2^{5/6} f (d \sec (e+f x))^{5/3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(1+i \tan (e+f x))^{5/6} (a-i a \tan (e+f x))^{5/6} \int \frac {1}{(i \tan (e+f x)+1)^{17/6} (a-i a \tan (e+f x))^{11/6}}d\tan (e+f x)}{f (d \sec (e+f x))^{5/3}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {3 i (1+i \tan (e+f x))^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {17}{6},\frac {1}{6},\frac {1}{2} (1-i \tan (e+f x))\right )}{10\ 2^{5/6} a f (d \sec (e+f x))^{5/3}}\) |
Input:
Int[1/((d*Sec[e + f*x])^(5/3)*(a + I*a*Tan[e + f*x])),x]
Output:
(((-3*I)/10)*Hypergeometric2F1[-5/6, 17/6, 1/6, (1 - I*Tan[e + f*x])/2]*(1 + I*Tan[e + f*x])^(5/6))/(2^(5/6)*a*f*(d*Sec[e + f*x])^(5/3))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_.), x_Symbol] :> Simp[(d*Sec[e + f*x])^m/((a + b*Tan[e + f*x])^(m/ 2)*(a - b*Tan[e + f*x])^(m/2)) Int[(a + b*Tan[e + f*x])^(m/2 + n)*(a - b* Tan[e + f*x])^(m/2), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*( c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
\[\int \frac {1}{\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}} \left (a +i a \tan \left (f x +e \right )\right )}d x\]
Input:
int(1/(d*sec(f*x+e))^(5/3)/(a+I*a*tan(f*x+e)),x)
Output:
int(1/(d*sec(f*x+e))^(5/3)/(a+I*a*tan(f*x+e)),x)
\[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}} \,d x } \] Input:
integrate(1/(d*sec(f*x+e))^(5/3)/(a+I*a*tan(f*x+e)),x, algorithm="fricas")
Output:
1/440*(440*a*d^2*f*e^(4*I*f*x + 4*I*e)*integral(-16/55*I*2^(1/3)*(d/(e^(2* I*f*x + 2*I*e) + 1))^(1/3)*e^(-2/3*I*f*x - 2/3*I*e)/(a*d^2*f), x) - 3*2^(1 /3)*(d/(e^(2*I*f*x + 2*I*e) + 1))^(1/3)*(11*I*e^(6*I*f*x + 6*I*e) - 15*I*e ^(4*I*f*x + 4*I*e) - 31*I*e^(2*I*f*x + 2*I*e) - 5*I)*e^(1/3*I*f*x + 1/3*I* e))*e^(-4*I*f*x - 4*I*e)/(a*d^2*f)
\[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))} \, dx=- \frac {i \int \frac {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}} \tan {\left (e + f x \right )} - i \left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}\, dx}{a} \] Input:
integrate(1/(d*sec(f*x+e))**(5/3)/(a+I*a*tan(f*x+e)),x)
Output:
-I*Integral(1/((d*sec(e + f*x))**(5/3)*tan(e + f*x) - I*(d*sec(e + f*x))** (5/3)), x)/a
Exception generated. \[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(d*sec(f*x+e))^(5/3)/(a+I*a*tan(f*x+e)),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}} \,d x } \] Input:
integrate(1/(d*sec(f*x+e))^(5/3)/(a+I*a*tan(f*x+e)),x, algorithm="giac")
Output:
integrate(1/((d*sec(f*x + e))^(5/3)*(I*a*tan(f*x + e) + a)), x)
Timed out. \[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))} \, dx=\int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}\,\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )} \,d x \] Input:
int(1/((d/cos(e + f*x))^(5/3)*(a + a*tan(e + f*x)*1i)),x)
Output:
int(1/((d/cos(e + f*x))^(5/3)*(a + a*tan(e + f*x)*1i)), x)
\[ \int \frac {1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))} \, dx=\frac {\int \frac {1}{\sec \left (f x +e \right )^{\frac {5}{3}} \tan \left (f x +e \right ) i +\sec \left (f x +e \right )^{\frac {5}{3}}}d x}{d^{\frac {5}{3}} a} \] Input:
int(1/(d*sec(f*x+e))^(5/3)/(a+I*a*tan(f*x+e)),x)
Output:
int(1/(sec(e + f*x)**(2/3)*sec(e + f*x)*tan(e + f*x)*i + sec(e + f*x)**(2/ 3)*sec(e + f*x)),x)/(d**(2/3)*a*d)