Optimal. Leaf size=167 \[ -\frac {15 i}{32 a^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {5 i}{16 a^2 f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}+\frac {i}{4 a^2 f (1+i \tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)}}+\frac {15 i \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{32 \sqrt {2} a^2 \sqrt {c} f} \]
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Rubi [A] time = 0.20, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3522, 3487, 51, 63, 206} \[ -\frac {15 i}{32 a^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {5 i}{16 a^2 f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}+\frac {i}{4 a^2 f (1+i \tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)}}+\frac {15 i \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{32 \sqrt {2} a^2 \sqrt {c} f} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)}} \, dx &=\frac {\int \cos ^4(e+f x) (c-i c \tan (e+f x))^{3/2} \, dx}{a^2 c^2}\\ &=\frac {\left (i c^3\right ) \operatorname {Subst}\left (\int \frac {1}{(c-x)^3 (c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{a^2 f}\\ &=\frac {i}{4 a^2 f (1+i \tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)}}+\frac {\left (5 i c^2\right ) \operatorname {Subst}\left (\int \frac {1}{(c-x)^2 (c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{8 a^2 f}\\ &=\frac {i}{4 a^2 f (1+i \tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)}}+\frac {5 i}{16 a^2 f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}+\frac {(15 i c) \operatorname {Subst}\left (\int \frac {1}{(c-x) (c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{32 a^2 f}\\ &=-\frac {15 i}{32 a^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {i}{4 a^2 f (1+i \tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)}}+\frac {5 i}{16 a^2 f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}+\frac {(15 i) \operatorname {Subst}\left (\int \frac {1}{(c-x) \sqrt {c+x}} \, dx,x,-i c \tan (e+f x)\right )}{64 a^2 f}\\ &=-\frac {15 i}{32 a^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {i}{4 a^2 f (1+i \tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)}}+\frac {5 i}{16 a^2 f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}+\frac {(15 i) \operatorname {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{32 a^2 f}\\ &=\frac {15 i \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{32 \sqrt {2} a^2 \sqrt {c} f}-\frac {15 i}{32 a^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {i}{4 a^2 f (1+i \tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)}}+\frac {5 i}{16 a^2 f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 2.47, size = 141, normalized size = 0.84 \[ -\frac {i e^{-4 i (e+f x)} \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} \left (-11 e^{2 i (e+f x)}-e^{4 i (e+f x)}+8 e^{6 i (e+f x)}-15 e^{4 i (e+f x)} \sqrt {1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\sqrt {1+e^{2 i (e+f x)}}\right )-2\right )}{32 \sqrt {2} a^2 c f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 295, normalized size = 1.77 \[ \frac {{\left (15 i \, \sqrt {\frac {1}{2}} a^{2} c f \sqrt {\frac {1}{a^{4} c f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (240 i \, a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + 240 i \, a^{2} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {1}{a^{4} c f^{2}}} + 240 i\right )} e^{\left (-i \, f x - i \, e\right )}}{256 \, a^{2} f}\right ) - 15 i \, \sqrt {\frac {1}{2}} a^{2} c f \sqrt {\frac {1}{a^{4} c f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-240 i \, a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 240 i \, a^{2} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {1}{a^{4} c f^{2}}} + 240 i\right )} e^{\left (-i \, f x - i \, e\right )}}{256 \, a^{2} f}\right ) + \sqrt {2} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-8 i \, e^{\left (6 i \, f x + 6 i \, e\right )} + i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 11 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i\right )}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{64 \, a^{2} c f} \]
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 \[ \int \frac {1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 121, normalized size = 0.72 \[ -\frac {2 i c^{3} \left (\frac {\frac {\frac {7 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{8}-\frac {9 \sqrt {c -i c \tan \left (f x +e \right )}\, c}{4}}{\left (-c -i c \tan \left (f x +e \right )\right )^{2}}-\frac {15 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 \sqrt {c}}}{8 c^{3}}+\frac {1}{8 c^{3} \sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 169, normalized size = 1.01 \[ -\frac {i \, {\left (\frac {4 \, {\left (15 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} c - 50 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} c^{2} + 32 \, c^{3}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a^{2} - 4 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} a^{2} c + 4 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} a^{2} c^{2}} + \frac {15 \, \sqrt {2} \sqrt {c} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a^{2}}\right )}}{128 \, c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.00, size = 156, normalized size = 0.93 \[ -\frac {\frac {{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,15{}\mathrm {i}}{32\,a^2\,f}+\frac {c^2\,1{}\mathrm {i}}{a^2\,f}-\frac {c\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,25{}\mathrm {i}}{16\,a^2\,f}}{-4\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}+{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}+4\,c^2\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,15{}\mathrm {i}}{64\,a^2\,\sqrt {-c}\,f} \]
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 \[ - \frac {\int \frac {1}{\sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - 2 i \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} - \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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