Integrand size = 30, antiderivative size = 86 \[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 i 2^{2/3} a \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {7}{6},\frac {13}{6},\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{7/3} \sqrt [3]{1+i \tan (c+d x)}}{7 d (a+i a \tan (c+d x))^{3/2}} \]
3/7*I*2^(2/3)*a*hypergeom([1/3, 7/6],[13/6],1/2-1/2*I*tan(d*x+c))*(e*sec(d *x+c))^(7/3)*(1+I*tan(d*x+c))^(1/3)/d/(a+I*a*tan(d*x+c))^(3/2)
Time = 1.69 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.37 \[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {3 i \sqrt [3]{2} e e^{i (c+d x)} \left (\frac {e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{4/3} \left (4+\left (1+e^{2 i (c+d x)}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{6},\frac {5}{3},-e^{2 i (c+d x)}\right )\right )}{5 d \sqrt {a+i a \tan (c+d x)}} \]
(((-3*I)/5)*2^(1/3)*e*E^(I*(c + d*x))*((e*E^(I*(c + d*x)))/(1 + E^((2*I)*( c + d*x))))^(4/3)*(4 + (1 + E^((2*I)*(c + d*x)))^(5/6)*Hypergeometric2F1[2 /3, 5/6, 5/3, -E^((2*I)*(c + d*x))]))/(d*Sqrt[a + I*a*Tan[c + d*x]])
Time = 0.48 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, 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.
\(\displaystyle \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}}dx\) |
\(\Big \downarrow \) 3986 |
\(\displaystyle \frac {(e \sec (c+d x))^{7/3} \int (a-i a \tan (c+d x))^{7/6} (i \tan (c+d x) a+a)^{2/3}dx}{(a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{7/6}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(e \sec (c+d x))^{7/3} \int (a-i a \tan (c+d x))^{7/6} (i \tan (c+d x) a+a)^{2/3}dx}{(a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{7/6}}\) |
\(\Big \downarrow \) 4006 |
\(\displaystyle \frac {a^2 (e \sec (c+d x))^{7/3} \int \frac {\sqrt [6]{a-i a \tan (c+d x)}}{\sqrt [3]{i \tan (c+d x) a+a}}d\tan (c+d x)}{d (a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{7/6}}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {a^2 \sqrt [3]{1+i \tan (c+d x)} (e \sec (c+d x))^{7/3} \int \frac {\sqrt [3]{2} \sqrt [6]{a-i a \tan (c+d x)}}{\sqrt [3]{i \tan (c+d x)+1}}d\tan (c+d x)}{\sqrt [3]{2} d (a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^2 \sqrt [3]{1+i \tan (c+d x)} (e \sec (c+d x))^{7/3} \int \frac {\sqrt [6]{a-i a \tan (c+d x)}}{\sqrt [3]{i \tan (c+d x)+1}}d\tan (c+d x)}{d (a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {3 i 2^{2/3} a \sqrt [3]{1+i \tan (c+d x)} (e \sec (c+d x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {7}{6},\frac {13}{6},\frac {1}{2} (1-i \tan (c+d x))\right )}{7 d (a+i a \tan (c+d x))^{3/2}}\) |
(((3*I)/7)*2^(2/3)*a*Hypergeometric2F1[1/3, 7/6, 13/6, (1 - I*Tan[c + d*x] )/2]*(e*Sec[c + d*x])^(7/3)*(1 + I*Tan[c + d*x])^(1/3))/(d*(a + I*a*Tan[c + d*x])^(3/2))
3.5.37.3.1 Defintions of rubi rules used
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]
\[\int \frac {\left (e \sec \left (d x +c \right )\right )^{\frac {7}{3}}}{\sqrt {a +i a \tan \left (d x +c \right )}}d x\]
\[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {7}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
1/5*(-6*I*2^(5/6)*e^2*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(e/(e^(2*I*d*x + 2 *I*c) + 1))^(1/3)*e^(4/3*I*d*x + 4/3*I*c) + 5*a*d*integral(-2/5*I*2^(5/6)* e^2*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(e/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)* e^(1/3*I*d*x + 1/3*I*c)/(a*d), x))/(a*d)
Timed out. \[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {7}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
\[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {7}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{7/3}}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]