Optimal. Leaf size=262 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{4 \sqrt {2} a^{5/2} f \sqrt {c-i d}}+\frac {\left (15 c^2+50 i c d-67 d^2\right ) \sqrt {c+d \tan (e+f x)}}{60 a^2 f (-d+i c)^3 \sqrt {a+i a \tan (e+f x)}}+\frac {(-13 d+5 i c) \sqrt {c+d \tan (e+f x)}}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2}}-\frac {\sqrt {c+d \tan (e+f x)}}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.82, antiderivative size = 262, normalized size of antiderivative = 1.00, 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 12
Rule 208
Rule 3544
Rule 3559
Rule 3596
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*}
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Mathematica [A] time = 5.31, 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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fricas [B] time = 0.56, size = 568, normalized size = 2.17 \[ \frac {{\left ({\left (-30 i \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 90 i \, a^{3} c d^{2} - 30 \, a^{3} d^{3}\right )} f \sqrt {\frac {i}{8 \, {\left (-i \, a^{5} c - a^{5} d\right )} f^{2}}} e^{\left (5 i \, f x + 5 i \, e\right )} \log \left (-4 \, {\left (i \, a^{3} c + a^{3} d\right )} f \sqrt {\frac {i}{8 \, {\left (-i \, a^{5} c - a^{5} d\right )} f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) + {\left (30 i \, a^{3} c^{3} - 90 \, a^{3} c^{2} d - 90 i \, a^{3} c d^{2} + 30 \, a^{3} d^{3}\right )} f \sqrt {\frac {i}{8 \, {\left (-i \, a^{5} c - a^{5} d\right )} f^{2}}} e^{\left (5 i \, f x + 5 i \, e\right )} \log \left (-4 \, {\left (-i \, a^{3} c - a^{3} d\right )} f \sqrt {\frac {i}{8 \, {\left (-i \, a^{5} c - a^{5} d\right )} f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) - \sqrt {2} {\left (3 \, c^{2} + 6 i \, c d - 3 \, d^{2} + {\left (23 \, c^{2} + 74 i \, c d - 83 \, d^{2}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (34 \, c^{2} + 104 i \, c d - 102 \, d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (14 \, c^{2} + 36 i \, c d - 22 \, d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-5 i \, f x - 5 i \, e\right )}}{{\left (120 i \, a^{3} c^{3} - 360 \, a^{3} c^{2} d - 360 i \, a^{3} c d^{2} + 120 \, a^{3} d^{3}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [B] time = 0.43, size = 5218, normalized size = 19.92 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}} \sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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