Optimal. Leaf size=193 \[ -\frac {i c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{32 \sqrt {2} a^3 f}+\frac {i c^4 \sqrt {c-i c \tan (e+f x)}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac {i c^3 \sqrt {c-i c \tan (e+f x)}}{24 a^3 f (c+i c \tan (e+f x))^2}-\frac {i c^2 \sqrt {c-i c \tan (e+f x)}}{32 a^3 f (c+i c \tan (e+f x))} \]
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Rubi [A] time = 0.21, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3522, 3487, 47, 51, 63, 206} \[ \frac {i c^4 \sqrt {c-i c \tan (e+f x)}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac {i c^3 \sqrt {c-i c \tan (e+f x)}}{24 a^3 f (c+i c \tan (e+f x))^2}-\frac {i c^2 \sqrt {c-i c \tan (e+f x)}}{32 a^3 f (c+i c \tan (e+f x))}-\frac {i c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{32 \sqrt {2} a^3 f} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 206
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx &=\frac {\int \cos ^6(e+f x) (c-i c \tan (e+f x))^{9/2} \, dx}{a^3 c^3}\\ &=\frac {\left (i c^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+x}}{(c-x)^4} \, dx,x,-i c \tan (e+f x)\right )}{a^3 f}\\ &=\frac {i c^4 \sqrt {c-i c \tan (e+f x)}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac {\left (i c^4\right ) \operatorname {Subst}\left (\int \frac {1}{(c-x)^3 \sqrt {c+x}} \, dx,x,-i c \tan (e+f x)\right )}{6 a^3 f}\\ &=\frac {i c^4 \sqrt {c-i c \tan (e+f x)}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac {i c^3 \sqrt {c-i c \tan (e+f x)}}{24 a^3 f (c+i c \tan (e+f x))^2}-\frac {\left (i c^3\right ) \operatorname {Subst}\left (\int \frac {1}{(c-x)^2 \sqrt {c+x}} \, dx,x,-i c \tan (e+f x)\right )}{16 a^3 f}\\ &=\frac {i c^4 \sqrt {c-i c \tan (e+f x)}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac {i c^3 \sqrt {c-i c \tan (e+f x)}}{24 a^3 f (c+i c \tan (e+f x))^2}-\frac {i c^2 \sqrt {c-i c \tan (e+f x)}}{32 a^3 f (c+i c \tan (e+f x))}-\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int \frac {1}{(c-x) \sqrt {c+x}} \, dx,x,-i c \tan (e+f x)\right )}{64 a^3 f}\\ &=\frac {i c^4 \sqrt {c-i c \tan (e+f x)}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac {i c^3 \sqrt {c-i c \tan (e+f x)}}{24 a^3 f (c+i c \tan (e+f x))^2}-\frac {i c^2 \sqrt {c-i c \tan (e+f x)}}{32 a^3 f (c+i c \tan (e+f x))}-\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{32 a^3 f}\\ &=-\frac {i c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{32 \sqrt {2} a^3 f}+\frac {i c^4 \sqrt {c-i c \tan (e+f x)}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac {i c^3 \sqrt {c-i c \tan (e+f x)}}{24 a^3 f (c+i c \tan (e+f x))^2}-\frac {i c^2 \sqrt {c-i c \tan (e+f x)}}{32 a^3 f (c+i c \tan (e+f x))}\\ \end {align*}
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Mathematica [A] time = 2.95, size = 152, normalized size = 0.79 \[ \frac {c (\cos (3 (e+f x))-i \sin (3 (e+f x))) \left (\sqrt {c-i c \tan (e+f x)} \left (39 i \cos (e+f x)+11 i \cos (3 (e+f x))+20 \sin (e+f x) \cos ^2(e+f x)\right )+3 \sqrt {2} \sqrt {c} (\sin (3 (e+f x))-i \cos (3 (e+f x))) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )\right )}{192 a^3 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 304, normalized size = 1.58 \[ -\frac {{\left (3 \, \sqrt {\frac {1}{2}} a^{3} f \sqrt {-\frac {c^{3}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {c^{3}}{a^{6} f^{2}}} + i \, c^{2}\right )} e^{\left (-i \, f x - i \, e\right )}}{16 \, a^{3} f}\right ) - 3 \, \sqrt {\frac {1}{2}} a^{3} f \sqrt {-\frac {c^{3}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {c^{3}}{a^{6} f^{2}}} - i \, c^{2}\right )} e^{\left (-i \, f x - i \, e\right )}}{16 \, a^{3} f}\right ) - \sqrt {2} {\left (3 i \, c e^{\left (6 i \, f x + 6 i \, e\right )} + 17 i \, c e^{\left (4 i \, f x + 4 i \, e\right )} + 22 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, c\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{192 \, a^{3} f} \]
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 \[ \int \frac {{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 117, normalized size = 0.61 \[ \frac {2 i c^{4} \left (\frac {\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{64 c^{2}}-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{12 c}-\frac {\sqrt {c -i c \tan \left (f x +e \right )}}{16}}{\left (-c -i c \tan \left (f x +e \right )\right )^{3}}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{128 c^{\frac {5}{2}}}\right )}{f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 192, normalized size = 0.99 \[ \frac {i \, {\left (\frac {3 \, \sqrt {2} c^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a^{3}} + \frac {4 \, {\left (3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c^{3} - 16 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{4} - 12 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} c^{5}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{3} a^{3} - 6 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} a^{3} c + 12 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{3} c^{2} - 8 \, a^{3} c^{3}}\right )}}{384 \, c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.94, size = 181, normalized size = 0.94 \[ \frac {\frac {c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{8\,a^3\,f}+\frac {c^3\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,1{}\mathrm {i}}{6\,a^3\,f}-\frac {c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,1{}\mathrm {i}}{32\,a^3\,f}}{6\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-12\,c^2\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )-{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3+8\,c^3}+\frac {\sqrt {2}\,{\left (-c\right )}^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,1{}\mathrm {i}}{64\,a^3\,f} \]
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 \[ \frac {i \left (\int \frac {c \sqrt {- i c \tan {\left (e + f x \right )} + c}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx + \int \left (- \frac {i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\right )\, dx\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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