Optimal. Leaf size=444 \[ \frac{d \left (-182 c^2 d^2+80 i c^3 d+15 c^4+1224 i c d^3+707 d^4\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 f (c-i d)^2 (c+i d)^5 \sqrt{c+d \tan (e+f x)}}+\frac{d \left (85 i c^2 d+15 c^3-221 c d^2+361 i d^3\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^{3/2}}+\frac{5 c^2+30 i c d-89 d^2}{20 a^2 f (-d+i c)^3 \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{4 \sqrt{2} a^{5/2} f (c-i d)^{5/2}}+\frac{-21 d+5 i c}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 1.78868, antiderivative size = 444, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{d \left (-182 c^2 d^2+80 i c^3 d+15 c^4+1224 i c d^3+707 d^4\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 f (c-i d)^2 (c+i d)^5 \sqrt{c+d \tan (e+f x)}}+\frac{d \left (85 i c^2 d+15 c^3-221 c d^2+361 i d^3\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^{3/2}}+\frac{5 c^2+30 i c d-89 d^2}{20 a^2 f (-d+i c)^3 \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{4 \sqrt{2} a^{5/2} f (c-i d)^{5/2}}+\frac{-21 d+5 i c}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3559
Rule 3596
Rule 3598
Rule 12
Rule 3544
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx &=-\frac{1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{-\frac{1}{2} a (5 i c-13 d)-4 i a d \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}} \, dx}{5 a^2 (i c-d)}\\ &=-\frac{1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{-\frac{3}{4} a^2 \left (5 c^2+20 i c d-47 d^2\right )-\frac{3}{2} a^2 (5 c+21 i d) d \tan (e+f x)}{\sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx}{15 a^4 (c+i d)^2}\\ &=-\frac{1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{\int \frac{\sqrt{a+i a \tan (e+f x)} \left (\frac{3}{8} a^3 \left (5 i c^3-35 c^2 d-135 i c d^2+361 d^3\right )+\frac{3}{2} a^3 d \left (5 i c^2-30 c d-89 i d^2\right ) \tan (e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx}{15 a^6 (i c-d)^3}\\ &=-\frac{1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac{2 \int \frac{\sqrt{a+i a \tan (e+f x)} \left (\frac{3}{16} a^4 \left (15 i c^4-90 c^3 d-260 i c^2 d^2+502 c d^3-707 i d^4\right )+\frac{3}{8} a^4 d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right ) \tan (e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{45 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} (c+d \tan (e+f x))^{3/2}}+\frac{5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}+\frac{d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 (c-i d)^2 (c+i d)^5 f \sqrt{c+d \tan (e+f x)}}-\frac{4 \int \frac{45 a^5 (i c-d)^5 \sqrt{a+i a \tan (e+f x)}}{32 \sqrt{c+d \tan (e+f x)}} \, dx}{45 a^8 (i c-d)^3 \left (c^2+d^2\right )^2}\\ &=-\frac{1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}+\frac{d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 (c-i d)^2 (c+i d)^5 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)^2}\\ &=-\frac{1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}+\frac{d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 (c-i d)^2 (c+i d)^5 f \sqrt{c+d \tan (e+f x)}}-\frac{i \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 (c-i d)^2 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)^{5/2} f}-\frac{1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac{5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}+\frac{d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt{a+i a \tan (e+f x)}}{60 a^3 (c-i d)^2 (c+i d)^5 f \sqrt{c+d \tan (e+f x)}}\\ \end{align*}
Mathematica [B] time = 9.76708, size = 928, normalized size = 2.09 \[ \frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left (\frac{\left (17 c^2+102 i d c-231 d^2\right ) \cos (2 f x) \left (\frac{1}{60} i \cos (e)-\frac{\sin (e)}{60}\right )}{(c+i d)^5}+\frac{(c+3 i d) \cos (4 f x) \left (\frac{7}{60} i \cos (e)+\frac{7 \sin (e)}{60}\right )}{(c+i d)^4}+\frac{\left (23 i \cos (e) c^5-108 d \cos (e) c^4+23 i d \sin (e) c^4-138 i d^2 \cos (e) c^3-108 d^2 \sin (e) c^3-692 d^3 \cos (e) c^2-138 i d^3 \sin (e) c^2+1623 i d^4 \cos (e) c-692 d^4 \sin (e) c+640 d^5 \cos (e)+343 i d^5 \sin (e)\right ) \left (\frac{1}{120} \cos (3 e)+\frac{1}{120} i \sin (3 e)\right )}{(c-i d)^2 (c+i d)^5 (c \cos (e)+d \sin (e))}+\frac{\cos (6 f x) \left (\frac{1}{40} i \cos (3 e)+\frac{1}{40} \sin (3 e)\right )}{(c+i d)^3}+\frac{\left (17 c^2+102 i d c-231 d^2\right ) \left (\frac{\cos (e)}{60}+\frac{1}{60} i \sin (e)\right ) \sin (2 f x)}{(c+i d)^5}+\frac{(c+3 i d) \left (\frac{7 \cos (e)}{60}-\frac{7}{60} i \sin (e)\right ) \sin (4 f x)}{(c+i d)^4}+\frac{\left (\frac{1}{40} \cos (3 e)-\frac{1}{40} i \sin (3 e)\right ) \sin (6 f x)}{(c+i d)^3}+\frac{16 \left (-\frac{1}{2} i \cos (3 e-f x) d^6+\frac{1}{2} i \cos (3 e+f x) d^6+\frac{1}{2} \sin (3 e-f x) d^6-\frac{1}{2} \sin (3 e+f x) d^6+c \cos (3 e-f x) d^5-c \cos (3 e+f x) d^5+i c \sin (3 e-f x) d^5-i c \sin (3 e+f x) d^5\right )}{3 (c-i d)^2 (c+i d)^5 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{\frac{2}{3} i d^6 \cos (3 e)-\frac{2}{3} d^6 \sin (3 e)}{(c-i d)^2 (c+i d)^5 (c \cos (e+f x)+d \sin (e+f x))^2}\right )}{f (i \tan (e+f x) a+a)^{5/2}}-\frac{i e^{3 i e} \sqrt{e^{i f x}} \log \left (2 \left (e^{i (e+f x)} \sqrt{c-i d}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )\right ) \sec ^{\frac{5}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{5/2}}{4 \sqrt{2} (c-i d)^{5/2} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} f (i \tan (e+f x) a+a)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.119, size = 10145, normalized size = 22.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.92612, size = 4513, normalized size = 10.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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