Integrand size = 32, antiderivative size = 415 \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=-\frac {\sqrt [4]{-1} a^{5/2} \left (5 c^4+100 i c^3 d+690 c^2 d^2-900 i c d^3-363 d^4\right ) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{64 d^{3/2} f}-\frac {4 i \sqrt {2} a^{5/2} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a^2 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 d f}+\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 d f}+\frac {a^2 (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d f}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f} \]
-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*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))*2^(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. \(849\) vs. \(2(415)=830\).
Time = 15.66 (sec) , antiderivative size = 849, normalized size of antiderivative = 2.05 \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=-\frac {\left (\frac {1}{128}+\frac {i}{128}\right ) \cos ^3(e+f x) \left (\left (5 c^4+100 i c^3 d+690 c^2 d^2-900 i c d^3-363 d^4\right ) \left (\log \left (\frac {(2+2 i) e^{\frac {i e}{2}} \left (c+i d-i c e^{i (e+f x)}-d e^{i (e+f x)}+(1+i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )}{\sqrt {d} \left (-5 c^4-100 i c^3 d-690 c^2 d^2+900 i c d^3+363 d^4\right ) \left (i+e^{i (e+f x)}\right )}\right )-\log \left (-\frac {(2+2 i) e^{\frac {i e}{2}} \left (c+i d+i c e^{i (e+f x)}+d e^{i (e+f x)}+(1+i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )}{\sqrt {d} \left (-5 c^4-100 i c^3 d-690 c^2 d^2+900 i c d^3+363 d^4\right ) \left (-i+e^{i (e+f x)}\right )}\right )\right )+(512+512 i) (c-i d)^{5/2} d^{3/2} \log \left (2 \left (\sqrt {c-i d} \cos (e+f x)+i \sqrt {c-i d} \sin (e+f x)+\sqrt {1+\cos (2 e+2 f x)+i \sin (2 e+2 f x)} \sqrt {c+d \tan (e+f x)}\right )\right )\right ) (\cos (2 e)-i \sin (2 e)) (a+i a \tan (e+f x))^{5/2}}{d^{3/2} f (\cos (f x)+i \sin (f x))^2 \sqrt {1+\cos (2 (e+f x))+i \sin (2 (e+f x))}}+\frac {\cos ^2(e+f x) \sqrt {\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left (\left (-15 c^3+719 i c^2 d+1621 c d^2-845 i d^3\right ) \left (\frac {\cos (2 e)}{192 d}-\frac {i \sin (2 e)}{192 d}\right )+(17 i c+23 d) \sec ^2(e+f x) \left (\frac {1}{24} i d \cos (2 e)+\frac {1}{24} d \sin (2 e)\right )+\left (59 c^2-226 i c d-131 d^2\right ) \sec (e+f x) \left (-\frac {1}{96} i \cos (3 e+f x)-\frac {1}{96} \sin (3 e+f x)\right )+\sec ^3(e+f x) \left (-\frac {1}{4} i d^2 \cos (3 e+f x)-\frac {1}{4} d^2 \sin (3 e+f x)\right )\right ) (a+i a \tan (e+f x))^{5/2}}{f (\cos (f x)+i \sin (f x))^2} \]
((-1/128 - I/128)*Cos[e + f*x]^3*((5*c^4 + (100*I)*c^3*d + 690*c^2*d^2 - ( 900*I)*c*d^3 - 363*d^4)*(Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d - I*c*E^(I*(e + f*x)) - d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x) )]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/ (Sqrt[d]*(-5*c^4 - (100*I)*c^3*d - 690*c^2*d^2 + (900*I)*c*d^3 + 363*d^4)* (I + E^(I*(e + f*x))))] - Log[((-2 - 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I* (e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f* x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]) )/(Sqrt[d]*(-5*c^4 - (100*I)*c^3*d - 690*c^2*d^2 + (900*I)*c*d^3 + 363*d^4 )*(-I + E^(I*(e + f*x))))]) + (512 + 512*I)*(c - I*d)^(5/2)*d^(3/2)*Log[2* (Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[ 2*e + 2*f*x] + I*Sin[2*e + 2*f*x]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[2*e] - I*Sin[2*e])*(a + I*a*Tan[e + f*x])^(5/2))/(d^(3/2)*f*(Cos[f*x] + I*Sin[f* x])^2*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]) + (Cos[e + f*x]^2*S qrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*((-15*c^3 + (719*I)*c^ 2*d + 1621*c*d^2 - (845*I)*d^3)*(Cos[2*e]/(192*d) - ((I/192)*Sin[2*e])/d) + ((17*I)*c + 23*d)*Sec[e + f*x]^2*((I/24)*d*Cos[2*e] + (d*Sin[2*e])/24) + (59*c^2 - (226*I)*c*d - 131*d^2)*Sec[e + f*x]*((-1/96*I)*Cos[3*e + f*x] - Sin[3*e + f*x]/96) + Sec[e + f*x]^3*((-1/4*I)*d^2*Cos[3*e + f*x] - (d^2*S in[3*e + f*x])/4))*(a + I*a*Tan[e + f*x])^(5/2))/(f*(Cos[f*x] + I*Sin[f...
Time = 2.67 (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 |
\(\displaystyle \frac {a \left (\frac {\frac {a^2 \left (5 c^2+90 i c d+107 d^2\right ) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}+\frac {3 \left (\frac {\frac {2 (-1)^{3/4} a^{9/2} \left (5 i c^4-100 c^3 d+690 i c^2 d^2+900 c d^3-363 i d^4\right ) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {512 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 \left (5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right ) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}}{6 a}+\frac {a (c+17 i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 f}\right )}{8 d}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
-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 )
3.12.49.3.1 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[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 = 0.92 (sec) , antiderivative size = 1851, normalized size of antiderivative = 4.46
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1851\) |
default | \(\text {Expression too large to display}\) | \(1851\) |
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*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*(-a*(I*d-c) )^(1/2)*tan(f*x+e)^3-272*2^(1/2)*c*d^2*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e) ))^(1/2)*(I*a*d)^(1/2)*(-a*(I*d-c))^(1/2)*tan(f*x+e)^2-236*2^(1/2)*c^2*d*( a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*(-a*(I*d-c))^(1/2 )*tan(f*x+e)+768*I*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*ta n(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))*a*c*d+15*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^4+272*I*2^(1/2)*d^3*(a*(1+I*tan(f*x+e))*(c +d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*(-a*(I*d-c))^(1/2)*tan(f*x+e)^2-894*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^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*(1+I*tan(f*x+e))*(c+d*tan(f*x+e )))^(1/2))/(tan(f*x+e)+I))*a*d^2-1089*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^4-768*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^2-300*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)/(...
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 = 0.34 (sec) , antiderivative size = 1919, normalized size of antiderivative = 4.62 \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \]
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. \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Timed out} \]
Exception generated. \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
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]%%%}+%%%{%%%{%%{[3 60,0]:[1,
Timed out. \[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx=\int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \]