Optimal. Leaf size=153 \[ \frac {6 i a^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c} f}-\frac {3 i a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c f}-\frac {2 i a (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A] time = 0.16, antiderivative size = 153, 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 = {3523, 47, 50, 63, 217, 203} \[ \frac {6 i a^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{\sqrt {c} f}-\frac {3 i a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c f}-\frac {2 i a (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 203
Rule 217
Rule 3523
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c-i c \tan (e+f x)}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {2 i a (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+i a x}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {2 i a (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}}-\frac {3 i a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c f}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {2 i a (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}}-\frac {3 i a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c f}+\frac {\left (6 i a^2\right ) \operatorname {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 {2 i a (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}}-\frac {3 i a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c f}+\frac {\left (6 i a^2\right ) \operatorname {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 {6 i a^{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 )}{\sqrt {c} f}-\frac {2 i a (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}}-\frac {3 i a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c f}\\ \end {align*}
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Mathematica [A] time = 4.52, size = 155, normalized size = 1.01 \[ -\frac {2 i e^{-3 i (e+f x)} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} \left (e^{i (e+f x)} \left (3+2 e^{2 i (e+f x)}\right )-3 \left (1+e^{2 i (e+f x)}\right ) \tan ^{-1}\left (e^{i (e+f x)}\right )\right ) (a+i a \tan (e+f x))^{5/2}}{c f \sec ^{\frac {5}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 339, normalized size = 2.22 \[ \frac {3 \, \sqrt {\frac {a^{5}}{c f^{2}}} c f \log \left (\frac {2 \, {\left (4 \, {\left (a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + a^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + {\left (2 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} - 2 i \, c f\right )} \sqrt {\frac {a^{5}}{c f^{2}}}\right )}}{a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2}}\right ) - 3 \, \sqrt {\frac {a^{5}}{c f^{2}}} c f \log \left (\frac {2 \, {\left (4 \, {\left (a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + a^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + {\left (-2 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, c f\right )} \sqrt {\frac {a^{5}}{c f^{2}}}\right )}}{a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2}}\right ) + {\left (-8 i \, a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} - 12 i \, a^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{2 \, c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 299, normalized size = 1.95 \[ \frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, a^{2} \left (3 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) \left (\tan ^{2}\left (f x +e \right )\right ) a c -3 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) a c -6 i \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \tan \left (f x +e \right )-6 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) \tan \left (f x +e \right ) a c -\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \left (\tan ^{2}\left (f x +e \right )\right )+5 \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\right )}{f c \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan \left (f x +e \right )+i\right )^{2} \sqrt {c a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 541, normalized size = 3.54 \[ \frac {{\left (6 \, {\left (a^{2} \cos \left (2 \, f x + 2 \, e\right ) + i \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) + a^{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^{2} \cos \left (2 \, f x + 2 \, e\right ) + i \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) + a^{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 ) - 4 \, {\left (2 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + 2 i \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) + 3 \, a^{2}\right )} \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 ) - {\left (-3 i \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) - 3 i \, a^{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 ) - {\left (3 i \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) - 3 \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) + 3 i \, a^{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 ) - {\left (8 i \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) - 8 \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) + 12 i \, a^{2}\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 )\right )} \sqrt {a} \sqrt {c}}{{\left (-2 i \, c \cos \left (2 \, f x + 2 \, e\right ) + 2 \, c \sin \left (2 \, f x + 2 \, e\right ) - 2 i \, c\right )} f} \]
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 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]
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 \[ \int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}}}{\sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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