Optimal. Leaf size=349 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{4 \sqrt {2} a^{5/2} f (c-i d)^{3/2}}+\frac {d \left (15 c^3+65 i c^2 d-117 c d^2+317 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 f (c-i d) (c+i d)^4 \sqrt {c+d \tan (e+f x)}}+\frac {15 c^2+70 i c d-151 d^2}{60 a^2 f (-d+i c)^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-17 d+5 i c}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.24, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3559, 3596, 3598, 12, 3544, 208} \[ \frac {d \left (65 i c^2 d+15 c^3-117 c d^2+317 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 f (c-i d) (c+i d)^4 \sqrt {c+d \tan (e+f x)}}+\frac {15 c^2+70 i c d-151 d^2}{60 a^2 f (-d+i c)^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{4 \sqrt {2} a^{5/2} f (c-i d)^{3/2}}+\frac {-17 d+5 i c}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 208
Rule 3544
Rule 3559
Rule 3596
Rule 3598
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {-\frac {1}{2} a (5 i c-11 d)-3 i a d \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}} \, dx}{5 a^2 (i c-d)}\\ &=-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}+\frac {5 i c-17 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {-\frac {1}{4} a^2 \left (15 c^2+50 i c d-83 d^2\right )-a^2 (5 c+17 i d) d \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx}{15 a^4 (c+i d)^2}\\ &=-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}+\frac {5 i c-17 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {15 c^2+70 i c d-151 d^2}{60 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {\sqrt {a+i a \tan (e+f x)} \left (\frac {1}{8} a^3 \left (15 i c^3-75 c^2 d-185 i c d^2+317 d^3\right )+\frac {1}{4} a^3 d \left (15 i c^2-70 c d-151 i d^2\right ) \tan (e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{15 a^6 (i c-d)^3}\\ &=-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}+\frac {5 i c-17 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {15 c^2+70 i c d-151 d^2}{60 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {d \left (15 c^3+65 i c^2 d-117 c d^2+317 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}-\frac {2 \int \frac {15 i a^4 (c+i d)^4 \sqrt {a+i a \tan (e+f x)}}{16 \sqrt {c+d \tan (e+f x)}} \, dx}{15 a^7 (i c-d)^3 \left (c^2+d^2\right )}\\ &=-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}+\frac {5 i c-17 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {15 c^2+70 i c d-151 d^2}{60 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {d \left (15 c^3+65 i c^2 d-117 c d^2+317 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx}{8 a^3 (c-i d)}\\ &=-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}+\frac {5 i c-17 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {15 c^2+70 i c d-151 d^2}{60 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {d \left (15 c^3+65 i c^2 d-117 c d^2+317 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{a c-i a d-2 a^2 x^2} \, dx,x,\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{4 a (i c+d) f}\\ &=-\frac {i \tanh ^{-1}\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 )}{4 \sqrt {2} a^{5/2} (c-i d)^{3/2} f}-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}+\frac {5 i c-17 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {15 c^2+70 i c d-151 d^2}{60 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {d \left (15 c^3+65 i c^2 d-117 c d^2+317 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 9.24, size = 788, normalized size = 2.26 \[ \frac {\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\frac {\left (17 c^2+77 i c d-126 d^2\right ) \left (-\frac {\sin (e)}{60}+\frac {1}{60} i \cos (e)\right ) \cos (2 f x)}{(c+i d)^4}+\frac {\left (17 c^2+77 i c d-126 d^2\right ) \left (\frac {\cos (e)}{60}+\frac {1}{60} i \sin (e)\right ) \sin (2 f x)}{(c+i d)^4}+\frac {\left (\frac {1}{120} \cos (3 e)+\frac {1}{120} i \sin (3 e)\right ) \left (23 c^4 \cos (e)+23 c^3 d \sin (e)+91 i c^3 d \cos (e)+91 i c^2 d^2 \sin (e)-109 c^2 d^2 \cos (e)-109 c d^3 \sin (e)+223 i c d^3 \cos (e)+223 i d^4 \sin (e)+240 d^4 \cos (e)\right )}{(c-i d) (c+i d)^4 (-i c \cos (e)-i d \sin (e))}+\frac {2 \left (\frac {1}{2} i d^5 \sin (3 e-f x)-\frac {1}{2} i d^5 \sin (3 e+f x)+\frac {1}{2} d^5 \cos (3 e-f x)-\frac {1}{2} d^5 \cos (3 e+f x)\right )}{(c-i d) (c+i d)^4 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac {(7 c+16 i d) \left (\frac {\sin (e)}{60}+\frac {1}{60} i \cos (e)\right ) \cos (4 f x)}{(c+i d)^3}+\frac {\left (\frac {1}{40} \sin (3 e)+\frac {1}{40} i \cos (3 e)\right ) \cos (6 f x)}{(c+i d)^2}+\frac {(7 c+16 i d) \left (\frac {\cos (e)}{60}-\frac {1}{60} i \sin (e)\right ) \sin (4 f x)}{(c+i d)^3}+\frac {\left (\frac {1}{40} \cos (3 e)-\frac {1}{40} i \sin (3 e)\right ) \sin (6 f x)}{(c+i d)^2}\right ) \sqrt {\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))}}{f (a+i a \tan (e+f x))^{5/2}}-\frac {i e^{3 i e} \sqrt {e^{i f x}} \sec ^{\frac {5}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{5/2} \log \left (2 \left (\sqrt {c-i d} e^{i (e+f x)}+\sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )\right )}{4 \sqrt {2} f (c-i d)^{3/2} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt {1+e^{2 i (e+f x)}} (a+i a \tan (e+f x))^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.60, size = 1082, normalized size = 3.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.36, size = 7870, normalized size = 22.55 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________