Integrand size = 35, antiderivative size = 155 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=-\frac {2 i a^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{3/2} f}-\frac {2 i a (a+i a \tan (e+f x))^{3/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 i a^2 \sqrt {a+i a \tan (e+f x)}}{c f \sqrt {c-i c \tan (e+f x)}} \]
-2*I*a^(5/2)*arctan(c^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c-I*c*tan(f* x+e))^(1/2))/c^(3/2)/f+2*I*a^2*(a+I*a*tan(f*x+e))^(1/2)/c/f/(c-I*c*tan(f*x +e))^(1/2)-2/3*I*a*(a+I*a*tan(f*x+e))^(3/2)/f/(c-I*c*tan(f*x+e))^(3/2)
Time = 2.95 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.28 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {2 a^{5/2} \sec ^2(e+f x) \left (\sqrt {a} (-3+\cos (2 (e+f x))+i \sin (2 (e+f x))) \sqrt {1-i \tan (e+f x)}+3 \arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) (\cos (2 (e+f x))-i \sin (2 (e+f x))) \sqrt {a+i a \tan (e+f x)}\right )}{3 c f \sqrt {1-i \tan (e+f x)} (i+\tan (e+f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
(2*a^(5/2)*Sec[e + f*x]^2*(Sqrt[a]*(-3 + Cos[2*(e + f*x)] + I*Sin[2*(e + f *x)])*Sqrt[1 - I*Tan[e + f*x]] + 3*ArcSin[Sqrt[a + I*a*Tan[e + f*x]]/(Sqrt [2]*Sqrt[a])]*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)])*Sqrt[a + I*a*Tan[e + f*x]]))/(3*c*f*Sqrt[1 - I*Tan[e + f*x]]*(I + Tan[e + f*x])*Sqrt[a + I*a*T an[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])
Time = 0.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3042, 4006, 57, 57, 45, 218}
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.
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4006 |
| \(\displaystyle \frac {a c \int \frac {(i \tan (e+f x) a+a)^{3/2}}{(c-i c \tan (e+f x))^{5/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {a c \left (-\frac {a \int \frac {\sqrt {i \tan (e+f x) a+a}}{(c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{c}-\frac {2 i (a+i a \tan (e+f x))^{3/2}}{3 c (c-i c \tan (e+f x))^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {a c \left (-\frac {a \left (-\frac {a \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c-i c \tan (e+f x)}}d\tan (e+f x)}{c}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{c \sqrt {c-i c \tan (e+f x)}}\right )}{c}-\frac {2 i (a+i a \tan (e+f x))^{3/2}}{3 c (c-i c \tan (e+f x))^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 45 |
| \(\displaystyle \frac {a c \left (-\frac {a \left (-\frac {2 a \int \frac {1}{i a+\frac {i c (i \tan (e+f x) a+a)}{c-i c \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c-i c \tan (e+f x)}}}{c}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{c \sqrt {c-i c \tan (e+f x)}}\right )}{c}-\frac {2 i (a+i a \tan (e+f x))^{3/2}}{3 c (c-i c \tan (e+f x))^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a c \left (-\frac {a \left (\frac {2 i \sqrt {a} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{c \sqrt {c-i c \tan (e+f x)}}\right )}{c}-\frac {2 i (a+i a \tan (e+f x))^{3/2}}{3 c (c-i c \tan (e+f x))^{3/2}}\right )}{f}\) |
(a*c*((((-2*I)/3)*(a + I*a*Tan[e + f*x])^(3/2))/(c*(c - I*c*Tan[e + f*x])^ (3/2)) - (a*(((2*I)*Sqrt[a]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(S qrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/c^(3/2) - ((2*I)*Sqrt[a + I*a*Tan[e + f*x]])/(c*Sqrt[c - I*c*Tan[e + f*x]])))/c))/f
3.11.26.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (124 ) = 248\).
Time = 0.81 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.25
| method | result | size |
| derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (9 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{2}\left (f x +e \right )\right )+3 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{3}\left (f x +e \right )\right )-3 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c -12 i \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )-9 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-8 \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{2}\left (f x +e \right )\right )+4 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{3 f \,c^{2} \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan \left (f x +e \right )+i\right )^{3} \sqrt {a c}}\) | \(348\) |
| default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (9 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{2}\left (f x +e \right )\right )+3 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{3}\left (f x +e \right )\right )-3 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c -12 i \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )-9 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-8 \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{2}\left (f x +e \right )\right )+4 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{3 f \,c^{2} \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan \left (f x +e \right )+i\right )^{3} \sqrt {a c}}\) | \(348\) |
1/3/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a^2/c^2*(9*I* ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))* a*c*tan(f*x+e)^2+3*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^( 1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^3-3*I*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x +e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*a*c-12*I*(a*c)^(1/2)*(a*c*(1+tan(f *x+e)^2))^(1/2)*tan(f*x+e)-9*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/ 2)*(a*c)^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)-8*(a*c)^(1/2)*(a*c*(1+tan(f*x+ e)^2))^(1/2)*tan(f*x+e)^2+4*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c *(1+tan(f*x+e)^2))^(1/2)/(tan(f*x+e)+I)^3/(a*c)^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (115) = 230\).
Time = 0.27 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.37 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {3 \, c^{2} f \sqrt {\frac {a^{5}}{c^{3} f^{2}}} \log \left (\frac {4 \, {\left (2 \, {\left (a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + a^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (i \, c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, c^{2} f\right )} \sqrt {\frac {a^{5}}{c^{3} f^{2}}}\right )}}{a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2}}\right ) - 3 \, c^{2} f \sqrt {\frac {a^{5}}{c^{3} f^{2}}} \log \left (\frac {4 \, {\left (2 \, {\left (a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + a^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (-i \, c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c^{2} f\right )} \sqrt {\frac {a^{5}}{c^{3} f^{2}}}\right )}}{a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2}}\right ) - 4 \, {\left (i \, a^{2} e^{\left (5 i \, f x + 5 i \, e\right )} - 2 i \, a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} - 3 i \, a^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{6 \, c^{2} f} \]
1/6*(3*c^2*f*sqrt(a^5/(c^3*f^2))*log(4*(2*(a^2*e^(3*I*f*x + 3*I*e) + a^2*e ^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I *e) + 1)) - (I*c^2*f*e^(2*I*f*x + 2*I*e) - I*c^2*f)*sqrt(a^5/(c^3*f^2)))/( a^2*e^(2*I*f*x + 2*I*e) + a^2)) - 3*c^2*f*sqrt(a^5/(c^3*f^2))*log(4*(2*(a^ 2*e^(3*I*f*x + 3*I*e) + a^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) - (-I*c^2*f*e^(2*I*f*x + 2*I*e) + I *c^2*f)*sqrt(a^5/(c^3*f^2)))/(a^2*e^(2*I*f*x + 2*I*e) + a^2)) - 4*(I*a^2*e ^(5*I*f*x + 5*I*e) - 2*I*a^2*e^(3*I*f*x + 3*I*e) - 3*I*a^2*e^(I*f*x + I*e) )*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)))/(c^ 2*f)
\[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}}}{\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (115) = 230\).
Time = 0.59 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.49 \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {{\left (-6 i \, a^{2} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) - 6 i \, a^{2} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), -\sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) - 4 i \, a^{2} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 12 i \, a^{2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 3 \, a^{2} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) - 3 \, a^{2} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 4 \, a^{2} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 12 \, a^{2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a}}{6 \, c^{\frac {3}{2}} f} \]
1/6*(-6*I*a^2*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) , sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) - 6*I*a^2*arct an2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), -sin(1/2*arctan2 (sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) - 4*I*a^2*cos(3/2*arctan2(sin(2 *f*x + 2*e), cos(2*f*x + 2*e))) + 12*I*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e ), cos(2*f*x + 2*e))) + 3*a^2*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2* f*x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) - 3*a^2*log(co s(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(sin (2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 4*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f* x + 2*e))) - 12*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))* sqrt(a)/(c^(3/2)*f)
\[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]