Integrand size = 32, antiderivative size = 378 \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=-\frac {\sqrt [4]{-1} a^{3/2} \left (5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right ) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{8 \sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a (c-3 i d) (5 i c+3 d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 f}+\frac {a (5 i c+7 d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}} \]
-2*I*a^(3/2)*(c-I*d)^(5/2)*arctanh(2^(1/2)*a^(1/2)*(c+d*tan(f*x+e))^(1/2)/ (c-I*d)^(1/2)/(a+I*a*tan(f*x+e))^(1/2))*2^(1/2)/f-1/8*(-1)^(1/4)*a^(3/2)*( 5*I*c^3+45*c^2*d-55*I*c*d^2-23*d^3)*arctanh((-1)^(3/4)*d^(1/2)*(a+I*a*tan( f*x+e))^(1/2)/a^(1/2)/(c+d*tan(f*x+e))^(1/2))/f/d^(1/2)+1/8*a*(c-3*I*d)*(5 *I*c+3*d)*(a+I*a*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/f+1/12*a*(5*I*c+ 7*d)*(a+I*a*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)/f+1/3*a^2*(c+I*d)*(c+ d*tan(f*x+e))^(5/2)/d/f/(a+I*a*tan(f*x+e))^(1/2)-1/3*a^2*(c+d*tan(f*x+e))^ (7/2)/d/f/(a+I*a*tan(f*x+e))^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1078\) vs. \(2(378)=756\).
Time = 7.19 (sec) , antiderivative size = 1078, normalized size of antiderivative = 2.85 \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\frac {d (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {-\frac {3 a (3 c-i d) d (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{4 f}+\frac {\frac {a \left (-\frac {3}{4} a^3 d \left (11 c^2-14 i c d-7 d^2\right )-\frac {3}{4} i a^3 \left (8 c^3-13 i c^2 d-10 c d^2+i d^3\right )\right ) \left (-\frac {2 \sqrt {2} \arctan \left (\frac {\sqrt {-a c+i a d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a c+i a d}}-\frac {2 (-1)^{3/4} \sqrt {c+i d} \sqrt {\frac {1}{\frac {c}{c+i d}+\frac {i d}{c+i d}}} \sqrt {\frac {c}{c+i d}+\frac {i d}{c+i d}} \arcsin \left (\frac {\sqrt [4]{-1} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+i d} \sqrt {\frac {c}{c+i d}+\frac {i d}{c+i d}}}\right ) \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}{\sqrt {a} \sqrt {d} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {3 (i a c-a d)^2 \left (11 c^2-14 i c d-7 d^2\right ) \sqrt {\frac {i a}{-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}}} \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )^2 \sqrt {\frac {i a (c+d \tan (e+f x))}{i a c-a d}} \sqrt {1+\frac {i a d (a+i a \tan (e+f x))}{(i a c-a d) \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )}} \left (\frac {2 i a d (a+i a \tan (e+f x))}{(i a c-a d) \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )}-\frac {2 \sqrt [4]{-1} \sqrt {a} \sqrt {d} \text {arcsinh}\left (\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {i a c-a d} \sqrt {-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}}}\right ) \sqrt {a+i a \tan (e+f x)}}{\sqrt {i a c-a d} \sqrt {-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}} \sqrt {1+\frac {i a d (a+i a \tan (e+f x))}{(i a c-a d) \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )}}}\right )}{8 d f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}}{2 a}}{3 a} \]
(d*(a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(3*f) - ((-3*a *(3*c - I*d)*d*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(4*f ) + ((a*((-3*a^3*d*(11*c^2 - (14*I)*c*d - 7*d^2))/4 - ((3*I)/4)*a^3*(8*c^3 - (13*I)*c^2*d - 10*c*d^2 + I*d^3))*((-2*Sqrt[2]*ArcTan[(Sqrt[-(a*c) + I* a*d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[2]*a*Sqrt[c + d*Tan[e + f*x]])])/Sq rt[-(a*c) + I*a*d] - (2*(-1)^(3/4)*Sqrt[c + I*d]*Sqrt[(c/(c + I*d) + (I*d) /(c + I*d))^(-1)]*Sqrt[c/(c + I*d) + (I*d)/(c + I*d)]*ArcSin[((-1)^(1/4)*S qrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + I*d]*Sqrt[c/(c + I*d) + (I*d)/(c + I*d)])]*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)])/(Sqrt[a]*Sqrt[ d]*Sqrt[c + d*Tan[e + f*x]])))/f + (3*(I*a*c - a*d)^2*(11*c^2 - (14*I)*c*d - 7*d^2)*Sqrt[(I*a)/(-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d))] *(-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d))^2*Sqrt[(I*a*(c + d*T an[e + f*x]))/(I*a*c - a*d)]*Sqrt[1 + (I*a*d*(a + I*a*Tan[e + f*x]))/((I*a *c - a*d)*(-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d)))]*(((2*I)*a *d*(a + I*a*Tan[e + f*x]))/((I*a*c - a*d)*(-((a^2*c)/(I*a*c - a*d)) - (I*a ^2*d)/(I*a*c - a*d))) - (2*(-1)^(1/4)*Sqrt[a]*Sqrt[d]*ArcSinh[((-1)^(1/4)* Sqrt[a]*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[I*a*c - a*d]*Sqrt[-((a^2 *c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d)])]*Sqrt[a + I*a*Tan[e + f*x]] )/(Sqrt[I*a*c - a*d]*Sqrt[-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a* d)]*Sqrt[1 + (I*a*d*(a + I*a*Tan[e + f*x]))/((I*a*c - a*d)*(-((a^2*c)/(...
Time = 2.59 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.594, Rules used = {3042, 4039, 27, 3042, 4078, 3042, 4080, 27, 3042, 4080, 27, 3042, 4084, 3042, 4027, 221, 4082, 66, 221}
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.
| \(\displaystyle \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 4039 |
| \(\displaystyle \frac {a \int -\frac {(a (i c-13 d)+a (c-11 i d) \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{2 \sqrt {i \tan (e+f x) a+a}}dx}{3 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \int \frac {(a (i c-13 d)+a (c-11 i d) \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{\sqrt {i \tan (e+f x) a+a}}dx}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(a (i c-13 d)+a (c-11 i d) \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{\sqrt {i \tan (e+f x) a+a}}dx}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle -\frac {a \left (-\frac {\int \sqrt {i \tan (e+f x) a+a} (c+d \tan (e+f x))^{3/2} \left ((7 c-5 i d) d a^2+d (5 i c+7 d) \tan (e+f x) a^2\right )dx}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (-\frac {\int \sqrt {i \tan (e+f x) a+a} (c+d \tan (e+f x))^{3/2} \left ((7 c-5 i d) d a^2+d (5 i c+7 d) \tan (e+f x) a^2\right )dx}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4080 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {\int \frac {3}{2} \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (d \left (11 c^2-14 i d c-7 d^2\right ) a^3+(c-3 i d) d (5 i c+3 d) \tan (e+f x) a^3\right )dx}{2 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \int \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (d \left (11 c^2-14 i d c-7 d^2\right ) a^3+(c-3 i d) d (5 i c+3 d) \tan (e+f x) a^3\right )dx}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \int \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (d \left (11 c^2-14 i d c-7 d^2\right ) a^3+(c-3 i d) d (5 i c+3 d) \tan (e+f x) a^3\right )dx}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4080 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left ((3 c-i d) d \left (9 c^2-14 i d c-9 d^2\right ) a^4+d \left (5 i c^3+45 d c^2-55 i d^2 c-23 d^3\right ) \tan (e+f x) a^4\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left ((3 c-i d) d \left (9 c^2-14 i d c-9 d^2\right ) a^4+d \left (5 i c^3+45 d c^2-55 i d^2 c-23 d^3\right ) \tan (e+f x) a^4\right )}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left ((3 c-i d) d \left (9 c^2-14 i d c-9 d^2\right ) a^4+d \left (5 i c^3+45 d c^2-55 i d^2 c-23 d^3\right ) \tan (e+f x) a^4\right )}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4084 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {32 a^4 d (c-i d)^3 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-a^3 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {32 a^4 d (c-i d)^3 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-a^3 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {-a^3 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-\frac {64 i a^6 d (c-i d)^3 \int \frac {1}{a (c-i d)-\frac {2 a^2 (c+d \tan (e+f x))}{i \tan (e+f x) a+a}}d\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}}{f}}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {-a^3 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-\frac {32 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {-\frac {a^5 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f}-\frac {32 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {-\frac {2 a^5 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {1}{i a-\frac {d (i \tan (e+f x) a+a)}{c+d \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}}{f}-\frac {32 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a \left (-\frac {\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}+\frac {3 \left (\frac {-\frac {2 (-1)^{3/4} a^{9/2} \sqrt {d} \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {32 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}\) |
-1/3*(a^2*(c + d*Tan[e + f*x])^(7/2))/(d*f*Sqrt[a + I*a*Tan[e + f*x]]) - ( a*((-2*a*(c + I*d)*(c + d*Tan[e + f*x])^(5/2))/(f*Sqrt[a + I*a*Tan[e + f*x ]]) - ((a^2*d*((5*I)*c + 7*d)*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f* x])^(3/2))/(2*f) + (3*(((-2*(-1)^(3/4)*a^(9/2)*Sqrt[d]*(5*c^3 - (45*I)*c^2 *d - 55*c*d^2 + (23*I)*d^3)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/f - ((32*I)*Sqrt[2]*a^(9/2) *(c - I*d)^(5/2)*d*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqr t[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f)/(2*a) + (a^3*(c - (3*I)*d)*d*( (5*I)*c + 3*d)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/f))/(4 *a))/a^2))/(6*d)
3.12.50.3.1 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[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] && !GtQ[c - a*(d/b), 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*a*(b/f) Subst[Int[1/(a*c - b*d - 2* a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && N eQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[a/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[b*c*(m - 2) + a*d*(m + 2*n) + (a*c*(m - 2) + b*d*(3*m + 2*n - 4))*Tan[e + f*x], x], x], x ] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-(A*b - a*B))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(2*a*f*m)), x] + Simp[1/(2*a^2*m) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f* x])^(n - 1)*Simp[A*(a*c*m + b*d*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a *A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[B*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(f*(m + n))), x] + Simp[ 1/(a*(m + n)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1)*Sim p[a*A*c*(m + n) - B*(b*c*m + a*d*n) + (a*A*d*(m + n) - B*(b*d*m - a*c*n))*T an[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b + a*B)/b Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x], x] - Simp[B/b Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(a - b*Tan[ e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1502 vs. \(2 (305 ) = 610\).
Time = 0.86 (sec) , antiderivative size = 1503, normalized size of antiderivative = 3.98
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1503\) |
| default | \(\text {Expression too large to display}\) | \(1503\) |
1/96/f*(a*(1+I*tan(f*x+e)))^(1/2)*(c+d*tan(f*x+e))^(1/2)*a*(48*I*ln((3*a*c +I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*( 1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*(I*a*d)^(1/2)*a*c -54*I*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*2^(1/2)*(- a*(I*d-c))^(1/2)*d^2+66*I*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a *d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*c^2+48*I*ln((3*a*c+I*a*tan(f*x+e)*c-I *a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(1+I*tan(f*x+e))*(c+ d*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*(I*a*d)^(1/2)*a*d-15*ln(1/2*(2*I*a*d *tan(f*x+e)+I*a*c+2*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1 /2)+a*d)/(I*a*d)^(1/2))*2^(1/2)*(-a*(I*d-c))^(1/2)*a*c^3+165*2^(1/2)*(-a*( I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1+I*tan(f*x+e))*(c+d* tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*c*d^2-48*I*ln(1/2*( 2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I* a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*2^(1/2)*(-a*(I*d-c))^(1/2)*a*c+135*I*ln(1/2 *(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*( I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*2^(1/2)*(-a*(I*d-c))^(1/2)*a*c^2*d+28*2^( 1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)) )^(1/2)*d^2*tan(f*x+e)-69*I*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1+I*tan (f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*2^(1/2) *(-a*(I*d-c))^(1/2)*a*d^3-48*I*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1493 vs. \(2 (290) = 580\).
Time = 0.30 (sec) , antiderivative size = 1493, normalized size of antiderivative = 3.95 \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \]
1/48*(48*sqrt(2)*(f*e^(4*I*f*x + 4*I*e) + 2*f*e^(2*I*f*x + 2*I*e) + f)*sqr t(-(a^3*c^5 - 5*I*a^3*c^4*d - 10*a^3*c^3*d^2 + 10*I*a^3*c^2*d^3 + 5*a^3*c* d^4 - I*a^3*d^5)/f^2)*log((I*sqrt(2)*f*sqrt(-(a^3*c^5 - 5*I*a^3*c^4*d - 10 *a^3*c^3*d^2 + 10*I*a^3*c^2*d^3 + 5*a^3*c*d^4 - I*a^3*d^5)/f^2)*e^(I*f*x + I*e) + sqrt(2)*(a*c^2 - 2*I*a*c*d - a*d^2 + (a*c^2 - 2*I*a*c*d - a*d^2)*e ^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I *f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/(a *c^2 - 2*I*a*c*d - a*d^2)) - 48*sqrt(2)*(f*e^(4*I*f*x + 4*I*e) + 2*f*e^(2* I*f*x + 2*I*e) + f)*sqrt(-(a^3*c^5 - 5*I*a^3*c^4*d - 10*a^3*c^3*d^2 + 10*I *a^3*c^2*d^3 + 5*a^3*c*d^4 - I*a^3*d^5)/f^2)*log((-I*sqrt(2)*f*sqrt(-(a^3* c^5 - 5*I*a^3*c^4*d - 10*a^3*c^3*d^2 + 10*I*a^3*c^2*d^3 + 5*a^3*c*d^4 - I* a^3*d^5)/f^2)*e^(I*f*x + I*e) + sqrt(2)*(a*c^2 - 2*I*a*c*d - a*d^2 + (a*c^ 2 - 2*I*a*c*d - a*d^2)*e^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2 *I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/(a*c^2 - 2*I*a*c*d - a*d^2)) + 2*sqrt(2)*((33*I*a*c^ 2 + 94*a*c*d - 49*I*a*d^2)*e^(5*I*f*x + 5*I*e) - 2*(-33*I*a*c^2 - 68*a*c*d + 19*I*a*d^2)*e^(3*I*f*x + 3*I*e) - 3*(-11*I*a*c^2 - 14*a*c*d + 7*I*a*d^2 )*e^(I*f*x + I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I* f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)) + 3*(f*e^(4*I*f*x + 4 *I*e) + 2*f*e^(2*I*f*x + 2*I*e) + f)*sqrt((-25*I*a^3*c^6 - 450*a^3*c^5*...
\[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \]
Exception generated. \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(3*d-c>0)', see `assume?` for mor e details)
Exception generated. \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-18,[0,10,0,0,0]%%%}+%%%{120,[0,9,0,0,1]%%%}+%%%{%%%{%%{[- 360,0]:[1
Timed out. \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \]