Optimal. Leaf size=159 \[ -\frac {i a^{3/2} c^{5/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}+\frac {a c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {i c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f} \]
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Rubi [A]
time = 0.11, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3604, 51, 38,
65, 223, 209} \begin {gather*} -\frac {i a^{3/2} c^{5/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}+\frac {a c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {i c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 51
Rule 65
Rule 209
Rule 223
Rule 3604
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2} \, dx &=\frac {(a c) \text {Subst}\left (\int \sqrt {a+i a x} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f}+\frac {\left (a c^2\right ) \text {Subst}\left (\int \sqrt {a+i a x} \sqrt {c-i c x} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {i c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f}+\frac {\left (a^2 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac {a c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {i c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {\left (i a c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{f}\\ &=\frac {a c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {i c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {\left (i a c^3\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}\right )}{f}\\ &=-\frac {i a^{3/2} c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}+\frac {a c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}-\frac {i c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f}\\ \end {align*}
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Mathematica [A]
time = 3.12, size = 169, normalized size = 1.06 \begin {gather*} -\frac {i e^{-2 i (e+f x)} \left (\frac {c}{1+e^{2 i (e+f x)}}\right )^{5/2} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \left (e^{i (e+f x)} \left (-3+8 e^{2 i (e+f x)}+3 e^{4 i (e+f x)}\right )+3 \left (1+e^{2 i (e+f x)}\right )^3 \text {ArcTan}\left (e^{i (e+f x)}\right )\right ) (a+i a \tan (e+f x))^{3/2}}{3 f \sec ^{\frac {3}{2}}(e+f x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 186, normalized size = 1.17
method | result | size |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{2} a \left (-2 i \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+3 a c \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right )-2 i \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}+3 \tan \left (f x +e \right ) \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{6 f \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}\) | \(186\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{2} a \left (-2 i \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+3 a c \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right )-2 i \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}+3 \tan \left (f x +e \right ) \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{6 f \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}\) | \(186\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 949 vs. \(2 (126) = 252\).
time = 0.65, size = 949, normalized size = 5.97 \begin {gather*} -\frac {{\left (12 \, a c^{2} \cos \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 32 \, a c^{2} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 12 \, a c^{2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 12 i \, a c^{2} \sin \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 32 i \, a c^{2} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 12 i \, a c^{2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 6 \, {\left (a c^{2} \cos \left (6 \, f x + 6 \, e\right ) + 3 \, a c^{2} \cos \left (4 \, f x + 4 \, e\right ) + 3 \, a c^{2} \cos \left (2 \, f x + 2 \, e\right ) + i \, a c^{2} \sin \left (6 \, f x + 6 \, e\right ) + 3 i \, a c^{2} \sin \left (4 \, f x + 4 \, e\right ) + 3 i \, a c^{2} \sin \left (2 \, f x + 2 \, e\right ) + a c^{2}\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 6 \, {\left (a c^{2} \cos \left (6 \, f x + 6 \, e\right ) + 3 \, a c^{2} \cos \left (4 \, f x + 4 \, e\right ) + 3 \, a c^{2} \cos \left (2 \, f x + 2 \, e\right ) + i \, a c^{2} \sin \left (6 \, f x + 6 \, e\right ) + 3 i \, a c^{2} \sin \left (4 \, f x + 4 \, e\right ) + 3 i \, a c^{2} \sin \left (2 \, f x + 2 \, e\right ) + a c^{2}\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), -\sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 3 \, {\left (i \, a c^{2} \cos \left (6 \, f x + 6 \, e\right ) + 3 i \, a c^{2} \cos \left (4 \, f x + 4 \, e\right ) + 3 i \, a c^{2} \cos \left (2 \, f x + 2 \, e\right ) - a c^{2} \sin \left (6 \, f x + 6 \, e\right ) - 3 \, a c^{2} \sin \left (4 \, f x + 4 \, e\right ) - 3 \, a c^{2} \sin \left (2 \, f x + 2 \, e\right ) + i \, a c^{2}\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 3 \, {\left (-i \, a c^{2} \cos \left (6 \, f x + 6 \, e\right ) - 3 i \, a c^{2} \cos \left (4 \, f x + 4 \, e\right ) - 3 i \, a c^{2} \cos \left (2 \, f x + 2 \, e\right ) + a c^{2} \sin \left (6 \, f x + 6 \, e\right ) + 3 \, a c^{2} \sin \left (4 \, f x + 4 \, e\right ) + 3 \, a c^{2} \sin \left (2 \, f x + 2 \, e\right ) - i \, a c^{2}\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right )\right )} \sqrt {a} \sqrt {c}}{-12 \, f {\left (i \, \cos \left (6 \, f x + 6 \, e\right ) + 3 i \, \cos \left (4 \, f x + 4 \, e\right ) + 3 i \, \cos \left (2 \, f x + 2 \, e\right ) - \sin \left (6 \, f x + 6 \, e\right ) - 3 \, \sin \left (4 \, f x + 4 \, e\right ) - 3 \, \sin \left (2 \, f x + 2 \, e\right ) + i\right )}} \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. 455 vs. \(2 (126) = 252\).
time = 1.14, size = 455, normalized size = 2.86 \begin {gather*} \frac {3 \, \sqrt {\frac {a^{3} c^{5}}{f^{2}}} {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {4 \, {\left (2 \, {\left (a c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + a c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - \sqrt {\frac {a^{3} c^{5}}{f^{2}}} {\left (i \, f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, f\right )}\right )}}{a c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + a c^{2}}\right ) - 3 \, \sqrt {\frac {a^{3} c^{5}}{f^{2}}} {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {4 \, {\left (2 \, {\left (a c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + a c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - \sqrt {\frac {a^{3} c^{5}}{f^{2}}} {\left (-i \, f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, f\right )}\right )}}{a c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + a c^{2}}\right ) - 4 \, {\left (3 i \, a c^{2} e^{\left (5 i \, f x + 5 i \, e\right )} + 8 i \, a c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} - 3 i \, a c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{12 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.01 \begin {gather*} \int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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