Optimal. Leaf size=262 \[ \frac{\left (15 c^2+50 i c d-67 d^2\right ) \sqrt{c+d \tan (e+f x)}}{60 a^2 f (-d+i c)^3 \sqrt{a+i a \tan (e+f x)}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{4 \sqrt{2} a^{5/2} f \sqrt{c-i d}}+\frac{(-13 d+5 i c) \sqrt{c+d \tan (e+f x)}}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2}}-\frac{\sqrt{c+d \tan (e+f x)}}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.81782, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {3559, 3596, 12, 3544, 208} \[ \frac{\left (15 c^2+50 i c d-67 d^2\right ) \sqrt{c+d \tan (e+f x)}}{60 a^2 f (-d+i c)^3 \sqrt{a+i a \tan (e+f x)}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right )}{4 \sqrt{2} a^{5/2} f \sqrt{c-i d}}+\frac{(-13 d+5 i c) \sqrt{c+d \tan (e+f x)}}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2}}-\frac{\sqrt{c+d \tan (e+f x)}}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3559
Rule 3596
Rule 12
Rule 3544
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}} \, dx &=-\frac{\sqrt{c+d \tan (e+f x)}}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2}}-\frac{\int \frac{-\frac{1}{2} a (5 i c-9 d)-2 i a d \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}} \, dx}{5 a^2 (i c-d)}\\ &=-\frac{\sqrt{c+d \tan (e+f x)}}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2}}+\frac{(5 i c-13 d) \sqrt{c+d \tan (e+f x)}}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2}}-\frac{\int \frac{-\frac{1}{4} a^2 \left (15 c^2+40 i c d-41 d^2\right )-\frac{1}{2} a^2 (5 c+13 i d) d \tan (e+f x)}{\sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{15 a^4 (c+i d)^2}\\ &=-\frac{\sqrt{c+d \tan (e+f x)}}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2}}+\frac{(5 i c-13 d) \sqrt{c+d \tan (e+f x)}}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{\left (15 c^2+50 i c d-67 d^2\right ) \sqrt{c+d \tan (e+f x)}}{60 a^2 (i c-d)^3 f \sqrt{a+i a \tan (e+f x)}}-\frac{\int -\frac{15 a^3 (i c-d)^3 \sqrt{a+i a \tan (e+f x)}}{8 \sqrt{c+d \tan (e+f x)}} \, dx}{15 a^6 (i c-d)^3}\\ &=-\frac{\sqrt{c+d \tan (e+f x)}}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2}}+\frac{(5 i c-13 d) \sqrt{c+d \tan (e+f x)}}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{\left (15 c^2+50 i c d-67 d^2\right ) \sqrt{c+d \tan (e+f x)}}{60 a^2 (i c-d)^3 f \sqrt{a+i a \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}\\ &=-\frac{\sqrt{c+d \tan (e+f x)}}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2}}+\frac{(5 i c-13 d) \sqrt{c+d \tan (e+f x)}}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{\left (15 c^2+50 i c d-67 d^2\right ) \sqrt{c+d \tan (e+f x)}}{60 a^2 (i c-d)^3 f \sqrt{a+i a \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 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} \sqrt{c-i d} f}-\frac{\sqrt{c+d \tan (e+f x)}}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2}}+\frac{(5 i c-13 d) \sqrt{c+d \tan (e+f x)}}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{\left (15 c^2+50 i c d-67 d^2\right ) \sqrt{c+d \tan (e+f x)}}{60 a^2 (i c-d)^3 f \sqrt{a+i a \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 5.13662, size = 309, normalized size = 1.18 \[ \frac{\sec ^{\frac{5}{2}}(e+f x) \left (\frac{2 i \sqrt{c+d \tan (e+f x)} \left (4 i \left (5 c^2+17 i c d-20 d^2\right ) \sin (2 (e+f x))+\left (26 c^2+80 i c d-86 d^2\right ) \cos (2 (e+f x))+11 c^2+30 i c d-19 d^2\right )}{15 (c+i d)^3 \sqrt{\sec (e+f x)}}-\frac{i \sqrt{2} e^{2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \log \left (2 \left (\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )\right )}{\sqrt{c-i d}}\right )}{8 f (a+i a \tan (e+f x))^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.117, size = 5218, normalized size = 19.9 \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 = 1.96159, size = 1590, normalized size = 6.07 \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(-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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