Optimal. Leaf size=349 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{4 \sqrt {2} a^{5/2} (c-i d)^{3/2} f}-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}+\frac {5 i c-17 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {15 c^2+70 i c d-151 d^2}{60 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {d \left (15 c^3+65 i c^2 d-117 c d^2+317 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}} \]
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Rubi [A]
time = 0.87, 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 = {3640, 3677,
3679, 12, 3625, 214} \begin {gather*} -\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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 214
Rule 3625
Rule 3640
Rule 3677
Rule 3679
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 {\text {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*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(788\) vs. \(2(349)=698\).
time = 9.35, size = 788, normalized size = 2.26 \begin {gather*} -\frac {i e^{3 i e} \sqrt {e^{i 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 ) \sec ^{\frac {5}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{5/2}}{4 \sqrt {2} (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)}} f (a+i a \tan (e+f x))^{5/2}}+\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+77 i c d-126 d^2\right ) \cos (2 f x) \left (\frac {1}{60} i \cos (e)-\frac {\sin (e)}{60}\right )}{(c+i d)^4}+\frac {(7 c+16 i d) \cos (4 f x) \left (\frac {1}{60} i \cos (e)+\frac {\sin (e)}{60}\right )}{(c+i d)^3}+\frac {\left (23 c^4 \cos (e)+91 i c^3 d \cos (e)-109 c^2 d^2 \cos (e)+223 i c d^3 \cos (e)+240 d^4 \cos (e)+23 c^3 d \sin (e)+91 i c^2 d^2 \sin (e)-109 c d^3 \sin (e)+223 i d^4 \sin (e)\right ) \left (\frac {1}{120} \cos (3 e)+\frac {1}{120} i \sin (3 e)\right )}{(c-i d) (c+i d)^4 (-i c \cos (e)-i 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)^2}+\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 {(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}+\frac {2 \left (\frac {1}{2} d^5 \cos (3 e-f x)-\frac {1}{2} d^5 \cos (3 e+f x)+\frac {1}{2} i d^5 \sin (3 e-f x)-\frac {1}{2} i d^5 \sin (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))}\right )}{f (a+i a \tan (e+f x))^{5/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 7869 vs. \(2 (290 ) = 580\).
time = 0.64, size = 7870, normalized size = 22.55
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(7870\) |
default | \(\text {Expression too large to display}\) | \(7870\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1097 vs. \(2 (283) = 566\).
time = 2.22, size = 1097, normalized size = 3.14 \begin {gather*} -\frac {\sqrt {2} {\left (-3 i \, c^{4} + 6 \, c^{3} d + 6 \, c d^{3} + 3 i \, d^{4} + {\left (-23 i \, c^{4} + 68 \, c^{3} d + 18 i \, c^{2} d^{2} + 332 \, c d^{3} - 463 i \, d^{4}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-57 i \, c^{4} + 200 \, c^{3} d + 178 i \, c^{2} d^{2} + 464 \, c d^{3} - 269 i \, d^{4}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - 4 \, {\left (12 i \, c^{4} - 43 \, c^{3} d - 43 i \, c^{2} d^{2} - 43 \, c d^{3} - 55 i \, d^{4}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-17 i \, c^{4} + 46 \, c^{3} d + 12 i \, c^{2} d^{2} + 46 \, c d^{3} + 29 i \, d^{4}\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}} + 30 \, {\left ({\left (a^{3} c^{6} + 2 i \, a^{3} c^{5} d + a^{3} c^{4} d^{2} + 4 i \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} + 2 i \, a^{3} c d^{5} - a^{3} d^{6}\right )} f e^{\left (7 i \, f x + 7 i \, e\right )} + {\left (a^{3} c^{6} + 4 i \, a^{3} c^{5} d - 5 \, a^{3} c^{4} d^{2} - 5 \, a^{3} c^{2} d^{4} - 4 i \, a^{3} c d^{5} + a^{3} d^{6}\right )} f e^{\left (5 i \, f x + 5 i \, e\right )}\right )} \sqrt {-\frac {i}{8 \, {\left (i \, a^{5} c^{3} + 3 \, a^{5} c^{2} d - 3 i \, a^{5} c d^{2} - a^{5} d^{3}\right )} f^{2}}} \log \left (-4 \, {\left (i \, a^{3} c^{2} + 2 \, a^{3} c d - i \, a^{3} d^{2}\right )} f \sqrt {-\frac {i}{8 \, {\left (i \, a^{5} c^{3} + 3 \, a^{5} c^{2} d - 3 i \, a^{5} c d^{2} - a^{5} d^{3}\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 ) - 30 \, {\left ({\left (a^{3} c^{6} + 2 i \, a^{3} c^{5} d + a^{3} c^{4} d^{2} + 4 i \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} + 2 i \, a^{3} c d^{5} - a^{3} d^{6}\right )} f e^{\left (7 i \, f x + 7 i \, e\right )} + {\left (a^{3} c^{6} + 4 i \, a^{3} c^{5} d - 5 \, a^{3} c^{4} d^{2} - 5 \, a^{3} c^{2} d^{4} - 4 i \, a^{3} c d^{5} + a^{3} d^{6}\right )} f e^{\left (5 i \, f x + 5 i \, e\right )}\right )} \sqrt {-\frac {i}{8 \, {\left (i \, a^{5} c^{3} + 3 \, a^{5} c^{2} d - 3 i \, a^{5} c d^{2} - a^{5} d^{3}\right )} f^{2}}} \log \left (-4 \, {\left (-i \, a^{3} c^{2} - 2 \, a^{3} c d + i \, a^{3} d^{2}\right )} f \sqrt {-\frac {i}{8 \, {\left (i \, a^{5} c^{3} + 3 \, a^{5} c^{2} d - 3 i \, a^{5} c d^{2} - a^{5} d^{3}\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 )}{120 \, {\left ({\left (a^{3} c^{6} + 2 i \, a^{3} c^{5} d + a^{3} c^{4} d^{2} + 4 i \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} + 2 i \, a^{3} c d^{5} - a^{3} d^{6}\right )} f e^{\left (7 i \, f x + 7 i \, e\right )} + {\left (a^{3} c^{6} + 4 i \, a^{3} c^{5} d - 5 \, a^{3} c^{4} d^{2} - 5 \, a^{3} c^{2} d^{4} - 4 i \, a^{3} c d^{5} + a^{3} d^{6}\right )} f e^{\left (5 i \, f x + 5 i \, e\right )}\right )}} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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