Integrand size = 32, antiderivative size = 415 \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=-\frac {\sqrt [4]{-1} a^{5/2} \left (5 c^4+100 i c^3 d+690 c^2 d^2-900 i c d^3-363 d^4\right ) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{64 d^{3/2} f}-\frac {4 i \sqrt {2} a^{5/2} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a^2 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 d f}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 d f}+\frac {a^2 (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d f}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f} \] Output:
-1/64*(-1)^(1/4)*a^(5/2)*(5*c^4+100*I*c^3*d+690*c^2*d^2-900*I*c*d^3-363*d^ 4)*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))/d^(3/2)/f-4*I*2^(1/2)*a^(5/2)*(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))/f+1/ 64*a^2*(5*c^3+95*I*c^2*d+273*c*d^2-149*I*d^3)*(a+I*a*tan(f*x+e))^(1/2)*(c+ d*tan(f*x+e))^(1/2)/d/f+1/96*a^2*(5*c^2+90*I*c*d+107*d^2)*(a+I*a*tan(f*x+e ))^(1/2)*(c+d*tan(f*x+e))^(3/2)/d/f+1/24*a^2*(c+17*I*d)*(a+I*a*tan(f*x+e)) ^(1/2)*(c+d*tan(f*x+e))^(5/2)/d/f-1/4*a^2*(a+I*a*tan(f*x+e))^(1/2)*(c+d*ta n(f*x+e))^(7/2)/d/f
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1609\) vs. \(2(415)=830\).
Time = 7.15 (sec) , antiderivative size = 1609, normalized size of antiderivative = 3.88 \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2),x]
Output:
(d*(a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2))/(4*f) - (-1/6* (a*(11*c - (5*I)*d)*d*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x] ])/f + ((((3*I)/32)*(I*a*c - a*d)^3*(59*c^2 - (90*I)*c*d - 43*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))^3*Sqrt[(I*a*(c + d*Tan[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))) + (4*a^2*d^2*(a + I*a*Tan[e + f*x])^2)/(3*(I*a*c - a*d)^2*(-((a^2*c)/ (I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d))^2) - (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)])]*Sqr t[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)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d)))])))/(a*d^2*f *Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) + (a*(-1/4*(a^3*d*(5 9*c^2 - (90*I)*c*d - 43*d^2)) - (I/4)*a^3*(48*c^3 - (85*I)*c^2*d - 54*c*d^ 2 + (5*I)*d^3))*(2*a*((-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]])])/Sqrt[-(a*c) + I*a* d] - (2*(-1)^(3/4)*Sqrt[c + I*d]*Sqrt[(c/(c + I*d) + (I*d)/(c + I*d))^(...
Time = 2.70 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.05, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3042, 4039, 27, 3042, 4080, 27, 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.
| \(\displaystyle \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 4039 |
| \(\displaystyle \frac {a \int \frac {1}{2} \sqrt {i \tan (e+f x) a+a} (a (i c+15 d)+a (c+17 i d) \tan (e+f x)) (c+d \tan (e+f x))^{5/2}dx}{4 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \sqrt {i \tan (e+f x) a+a} (a (i c+15 d)+a (c+17 i d) \tan (e+f x)) (c+d \tan (e+f x))^{5/2}dx}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \sqrt {i \tan (e+f x) a+a} (a (i c+15 d)+a (c+17 i d) \tan (e+f x)) (c+d \tan (e+f x))^{5/2}dx}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 4080 |
| \(\displaystyle \frac {a \left (\frac {\int \frac {1}{2} \sqrt {i \tan (e+f x) a+a} (c+d \tan (e+f x))^{3/2} \left (\left (5 i c^2+102 d c-85 i d^2\right ) a^2+\left (5 c^2+90 i d c+107 d^2\right ) \tan (e+f x) a^2\right )dx}{3 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\int \sqrt {i \tan (e+f x) a+a} (c+d \tan (e+f x))^{3/2} \left (\left (5 i c^2+102 d c-85 i d^2\right ) a^2+\left (5 c^2+90 i d c+107 d^2\right ) \tan (e+f x) a^2\right )dx}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {\int \sqrt {i \tan (e+f x) a+a} (c+d \tan (e+f x))^{3/2} \left (\left (5 i c^2+102 d c-85 i d^2\right ) a^2+\left (5 c^2+90 i d c+107 d^2\right ) \tan (e+f x) a^2\right )dx}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 4080 |
| \(\displaystyle \frac {a \left (\frac {\frac {\int \frac {3}{2} \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (\left (5 i c^3+161 d c^2-239 i d^2 c-107 d^3\right ) a^3+\left (5 c^3+95 i d c^2+273 d^2 c-149 i d^3\right ) \tan (e+f x) a^3\right )dx}{2 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \int \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (\left (5 i c^3+161 d c^2-239 i d^2 c-107 d^3\right ) a^3+\left (5 c^3+95 i d c^2+273 d^2 c-149 i d^3\right ) \tan (e+f x) a^3\right )dx}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \int \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (\left (5 i c^3+161 d c^2-239 i d^2 c-107 d^3\right ) a^3+\left (5 c^3+95 i d c^2+273 d^2 c-149 i d^3\right ) \tan (e+f x) a^3\right )dx}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 4080 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \left (\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (5 i c^4+412 d c^3-846 i d^2 c^2-636 d^3 c+149 i d^4\right ) a^4+\left (5 c^4+100 i d c^3+690 d^2 c^2-900 i d^3 c-363 d^4\right ) \tan (e+f x) a^4\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \left (\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (5 i c^4+412 d c^3-846 i d^2 c^2-636 d^3 c+149 i d^4\right ) a^4+\left (5 c^4+100 i d c^3+690 d^2 c^2-900 i d^3 c-363 d^4\right ) \tan (e+f x) a^4\right )}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \left (\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (5 i c^4+412 d c^3-846 i d^2 c^2-636 d^3 c+149 i d^4\right ) a^4+\left (5 c^4+100 i d c^3+690 d^2 c^2-900 i d^3 c-363 d^4\right ) \tan (e+f x) a^4\right )}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 4084 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \left (\frac {512 a^4 d (c-i d)^3 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx+a^3 \left (5 i c^4-100 c^3 d+690 i c^2 d^2+900 c d^3-363 i d^4\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \left (\frac {512 a^4 d (c-i d)^3 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx+a^3 \left (5 i c^4-100 c^3 d+690 i c^2 d^2+900 c d^3-363 i d^4\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \left (\frac {a^3 \left (5 i c^4-100 c^3 d+690 i c^2 d^2+900 c d^3-363 i d^4\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-\frac {1024 i a^6 d (c-i d)^3 \int \frac {1}{a (c-i d)-\frac {2 a^2 (c+d \tan (e+f x))}{i \tan (e+f x) a+a}}d\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}}{f}}{2 a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \left (\frac {a^3 \left (5 i c^4-100 c^3 d+690 i c^2 d^2+900 c d^3-363 i d^4\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-\frac {512 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \left (\frac {\frac {a^5 \left (5 i c^4-100 c^3 d+690 i c^2 d^2+900 c d^3-363 i d^4\right ) \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f}-\frac {512 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {a \left (\frac {\frac {3 \left (\frac {\frac {2 a^5 \left (5 i c^4-100 c^3 d+690 i c^2 d^2+900 c d^3-363 i d^4\right ) \int \frac {1}{i a-\frac {d (i \tan (e+f x) a+a)}{c+d \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}}{f}-\frac {512 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (\frac {\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}+\frac {3 \left (\frac {\frac {2 (-1)^{3/4} a^{9/2} \left (5 i c^4-100 c^3 d+690 i c^2 d^2+900 c d^3-363 i d^4\right ) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {512 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
Input:
Int[(a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2),x]
Output:
-1/4*(a^2*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(7/2))/(d*f) + ( a*((a*(c + (17*I)*d)*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2) )/(3*f) + ((a^2*(5*c^2 + (90*I)*c*d + 107*d^2)*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)*((5*I)*c^4 - 100*c^3*d + (690*I)*c^2*d^2 + 900*c*d^3 - (363*I)*d^4)*ArcTanh[((-1)^(3/ 4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])] )/(Sqrt[d]*f) - ((512*I)*Sqrt[2]*a^(9/2)*(c - I*d)^(5/2)*d*ArcTanh[(Sqrt[2 ]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f* x]])])/f)/(2*a) + (a^3*(5*c^3 + (95*I)*c^2*d + 273*c*d^2 - (149*I)*d^3)*Sq rt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/f))/(4*a))/(6*a)))/(8*d )
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[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. 1850 vs. \(2 (342 ) = 684\).
Time = 0.44 (sec) , antiderivative size = 1851, normalized size of antiderivative = 4.46
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1851\) |
| default | \(\text {Expression too large to display}\) | \(1851\) |
Input:
int((a+I*a*tan(f*x+e))^(5/2)*(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOS E)
Output:
1/768/f*(a*(1+I*tan(f*x+e)))^(1/2)*(c+d*tan(f*x+e))^(1/2)*a^2*(-96*2^(1/2) *d^3*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)) )^(1/2)*tan(f*x+e)^3-272*2^(1/2)*c*d^2*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*(a *(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2-236*2^(1/2)*c^2*d*( -a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2 )*tan(f*x+e)+904*I*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*(a*(c+d*tan(f* x+e))*(1+I*tan(f*x+e)))^(1/2)*c*d^2*tan(f*x+e)-894*I*2^(1/2)*(-a*(I*d-c))^ (1/2)*(I*a*d)^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*d^3+768*I* (I*a*d)^(1/2)*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*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x +e)))*a*d^2-768*I*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*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a* d)^(1/2))*a*c*d-768*I*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*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/( I*a*d)^(1/2))*a*d^2+272*I*2^(1/2)*d^3*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*(a* (c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2-1089*I*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*(c+d*tan(f*x+e))*(1+I* tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*d^4-300*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*(c+d*tan(f*x+e))*(1+ I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*c^3*d+2700*2^(...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1919 vs. \(2 (327) = 654\).
Time = 0.24 (sec) , antiderivative size = 1919, normalized size of antiderivative = 4.62 \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((a+I*a*tan(f*x+e))^(5/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="fr icas")
Output:
1/384*(768*sqrt(2)*(d*f*e^(6*I*f*x + 6*I*e) + 3*d*f*e^(4*I*f*x + 4*I*e) + 3*d*f*e^(2*I*f*x + 2*I*e) + d*f)*sqrt(-(a^5*c^5 - 5*I*a^5*c^4*d - 10*a^5*c ^3*d^2 + 10*I*a^5*c^2*d^3 + 5*a^5*c*d^4 - I*a^5*d^5)/f^2)*log((I*sqrt(2)*f *sqrt(-(a^5*c^5 - 5*I*a^5*c^4*d - 10*a^5*c^3*d^2 + 10*I*a^5*c^2*d^3 + 5*a^ 5*c*d^4 - I*a^5*d^5)/f^2)*e^(I*f*x + I*e) + sqrt(2)*(a^2*c^2 - 2*I*a^2*c*d - a^2*d^2 + (a^2*c^2 - 2*I*a^2*c*d - a^2*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^2*c^2 - 2*I*a^2*c*d - a^2 *d^2)) - 768*sqrt(2)*(d*f*e^(6*I*f*x + 6*I*e) + 3*d*f*e^(4*I*f*x + 4*I*e) + 3*d*f*e^(2*I*f*x + 2*I*e) + d*f)*sqrt(-(a^5*c^5 - 5*I*a^5*c^4*d - 10*a^5 *c^3*d^2 + 10*I*a^5*c^2*d^3 + 5*a^5*c*d^4 - I*a^5*d^5)/f^2)*log((-I*sqrt(2 )*f*sqrt(-(a^5*c^5 - 5*I*a^5*c^4*d - 10*a^5*c^3*d^2 + 10*I*a^5*c^2*d^3 + 5 *a^5*c*d^4 - I*a^5*d^5)/f^2)*e^(I*f*x + I*e) + sqrt(2)*(a^2*c^2 - 2*I*a^2* c*d - a^2*d^2 + (a^2*c^2 - 2*I*a^2*c*d - a^2*d^2)*e^(2*I*f*x + 2*I*e))*sqr t(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqr t(a/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/(a^2*c^2 - 2*I*a^2*c*d - a^2*d^2)) - 2*sqrt(2)*((15*a^2*c^3 - 719*I*a^2*c^2*d - 1621*a^2*c*d^2 + 84 5*I*a^2*d^3)*e^(7*I*f*x + 7*I*e) + (45*a^2*c^3 - 1921*I*a^2*c^2*d - 3415*a ^2*c*d^2 + 1275*I*a^2*d^3)*e^(5*I*f*x + 5*I*e) + (45*a^2*c^3 - 1685*I*a^2* c^2*d - 2511*a^2*c*d^2 + 1135*I*a^2*d^3)*e^(3*I*f*x + 3*I*e) + 3*(5*a^2...
Timed out. \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+I*a*tan(f*x+e))**(5/2)*(c+d*tan(f*x+e))**(5/2),x)
Output:
Timed out
Exception generated. \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+I*a*tan(f*x+e))^(5/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="ma xima")
Output:
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. \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+I*a*tan(f*x+e))^(5/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="gi ac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeRecursive ass umption s
Timed out. \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((a + a*tan(e + f*x)*1i)^(5/2)*(c + d*tan(e + f*x))^(5/2),x)
Output:
int((a + a*tan(e + f*x)*1i)^(5/2)*(c + d*tan(e + f*x))^(5/2), x)
\[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\sqrt {a}\, a^{2} \left (-\left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{4}d x \right ) d^{2}-2 \left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{3}d x \right ) c d +2 \left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{3}d x \right ) d^{2} i -\left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}d x \right ) c^{2}+4 \left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}d x \right ) c d i +\left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}d x \right ) d^{2}+2 \left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )d x \right ) c^{2} i +2 \left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )d x \right ) c d +\left (\int \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}d x \right ) c^{2}\right ) \] Input:
int((a+I*a*tan(f*x+e))^(5/2)*(c+d*tan(f*x+e))^(5/2),x)
Output:
sqrt(a)*a**2*( - int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan (e + f*x)**4,x)*d**2 - 2*int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3,x)*c*d + 2*int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3,x)*d**2*i - int(sqrt(tan(e + f*x)*i + 1)*sqrt( tan(e + f*x)*d + c)*tan(e + f*x)**2,x)*c**2 + 4*int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2,x)*c*d*i + int(sqrt(tan(e + f* x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2,x)*d**2 + 2*int(sqrt(ta n(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x),x)*c**2*i + 2*int( sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x),x)*c*d + in t(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c),x)*c**2)