Optimal. Leaf size=139 \[ -\frac {8 i a^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}+\frac {2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt {c+d \tan (e+f x)}}+\frac {4 a^3 c \sqrt {c+d \tan (e+f x)}}{d^2 (i c+d) f} \]
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Rubi [A]
time = 0.22, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3634, 3673,
3618, 65, 214} \begin {gather*} \frac {4 a^3 c \sqrt {c+d \tan (e+f x)}}{d^2 f (d+i c)}+\frac {2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{d f (c-i d) \sqrt {c+d \tan (e+f x)}}-\frac {8 i a^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 3618
Rule 3634
Rule 3673
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{3/2}} \, dx &=\frac {2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt {c+d \tan (e+f x)}}-\frac {2 \int \frac {(a+i a \tan (e+f x)) \left (-a^2 (c+2 i d)+i a^2 c \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{d (i c+d)}\\ &=\frac {2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt {c+d \tan (e+f x)}}+\frac {4 a^3 c \sqrt {c+d \tan (e+f x)}}{d^2 (i c+d) f}-\frac {2 \int \frac {-2 i a^3 d+2 a^3 d \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{d (i c+d)}\\ &=\frac {2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt {c+d \tan (e+f x)}}+\frac {4 a^3 c \sqrt {c+d \tan (e+f x)}}{d^2 (i c+d) f}+\frac {\left (8 a^6 d\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {x}{2 a^3}} \left (4 a^6 d^2-2 i a^3 d x\right )} \, dx,x,2 a^3 d \tan (e+f x)\right )}{(c-i d) f}\\ &=\frac {2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt {c+d \tan (e+f x)}}+\frac {4 a^3 c \sqrt {c+d \tan (e+f x)}}{d^2 (i c+d) f}+\frac {\left (32 a^9 d\right ) \text {Subst}\left (\int \frac {1}{4 i a^6 c d+4 a^6 d^2-4 i a^6 d x^2} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(c-i d) f}\\ &=-\frac {8 i a^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}+\frac {2 (c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f \sqrt {c+d \tan (e+f x)}}+\frac {4 a^3 c \sqrt {c+d \tan (e+f x)}}{d^2 (i c+d) f}\\ \end {align*}
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Mathematica [A]
time = 4.37, size = 219, normalized size = 1.58 \begin {gather*} \frac {a^3 (\cos (e+f x)+i \sin (e+f x))^3 \left (-\frac {8 i e^{-3 i e} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2}}+\frac {2 (\cos (3 e)-i \sin (3 e)) \left (\left (-2 i c^2+c d+i d^2\right ) \cos (e+f x)+(-i c-d) d \sin (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{(c-i d) d^2 (c \cos (e+f x)+d \sin (e+f x))}\right )}{f (\cos (f x)+i \sin (f x))^3} \end {gather*}
Antiderivative was successfully verified.
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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. 1055 vs. \(2 (122 ) = 244\).
time = 0.29, size = 1056, normalized size = 7.60
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (-i \sqrt {c +d \tan \left (f x +e \right )}-\frac {i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}}{\left (c^{2}+d^{2}\right ) \sqrt {c +d \tan \left (f x +e \right )}}-\frac {4 d^{2} \left (\frac {\frac {\left (i \sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c +i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{2}-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d -2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c d \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (2 i c^{2} \sqrt {c^{2}+d^{2}}-2 i d^{2} \sqrt {c^{2}+d^{2}}+2 i c^{3}-2 i c \,d^{2}-4 c d \sqrt {c^{2}+d^{2}}-4 c^{2} d -\frac {\left (i \sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c +i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{2}-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d -2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 \left (\sqrt {c^{2}+d^{2}}+c \right ) \sqrt {c^{2}+d^{2}}}+\frac {\frac {\left (-i \sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c -i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c d \right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (2 i c^{2} \sqrt {c^{2}+d^{2}}-2 i d^{2} \sqrt {c^{2}+d^{2}}+2 i c^{3}-2 i c \,d^{2}-4 c d \sqrt {c^{2}+d^{2}}-4 c^{2} d +\frac {\left (-i \sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c -i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 \left (\sqrt {c^{2}+d^{2}}+c \right ) \sqrt {c^{2}+d^{2}}}\right )}{c^{2}+d^{2}}\right )}{f \,d^{2}}\) | \(1056\) |
default | \(\frac {2 a^{3} \left (-i \sqrt {c +d \tan \left (f x +e \right )}-\frac {i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}}{\left (c^{2}+d^{2}\right ) \sqrt {c +d \tan \left (f x +e \right )}}-\frac {4 d^{2} \left (\frac {\frac {\left (i \sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c +i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{2}-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d -2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c d \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (2 i c^{2} \sqrt {c^{2}+d^{2}}-2 i d^{2} \sqrt {c^{2}+d^{2}}+2 i c^{3}-2 i c \,d^{2}-4 c d \sqrt {c^{2}+d^{2}}-4 c^{2} d -\frac {\left (i \sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c +i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2}-i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{2}-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d -2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 \left (\sqrt {c^{2}+d^{2}}+c \right ) \sqrt {c^{2}+d^{2}}}+\frac {\frac {\left (-i \sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c -i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c d \right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (2 i c^{2} \sqrt {c^{2}+d^{2}}-2 i d^{2} \sqrt {c^{2}+d^{2}}+2 i c^{3}-2 i c \,d^{2}-4 c d \sqrt {c^{2}+d^{2}}-4 c^{2} d +\frac {\left (-i \sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c -i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 \left (\sqrt {c^{2}+d^{2}}+c \right ) \sqrt {c^{2}+d^{2}}}\right )}{c^{2}+d^{2}}\right )}{f \,d^{2}}\) | \(1056\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 620 vs. \(2 (119) = 238\).
time = 1.24, size = 620, normalized size = 4.46 \begin {gather*} \frac {\sqrt {\frac {64 i \, a^{6}}{{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} f^{2}}} {\left ({\left (c^{2} d^{2} - 2 i \, c d^{3} - d^{4}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{2} d^{2} + d^{4}\right )} f\right )} \log \left (\frac {{\left (8 \, a^{3} c + \sqrt {\frac {64 i \, a^{6}}{{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} f^{2}}} {\left ({\left (i \, c^{2} + 2 \, c d - i \, d^{2}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, c^{2} + 2 \, c d - i \, d^{2}\right )} f\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}} + 8 \, {\left (a^{3} c - i \, a^{3} d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a^{3}}\right ) - \sqrt {\frac {64 i \, a^{6}}{{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} f^{2}}} {\left ({\left (c^{2} d^{2} - 2 i \, c d^{3} - d^{4}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{2} d^{2} + d^{4}\right )} f\right )} \log \left (\frac {{\left (8 \, a^{3} c + \sqrt {\frac {64 i \, a^{6}}{{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} f^{2}}} {\left ({\left (-i \, c^{2} - 2 \, c d + i \, d^{2}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, c^{2} - 2 \, c d + i \, d^{2}\right )} f\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}} + 8 \, {\left (a^{3} c - i \, a^{3} d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a^{3}}\right ) + 16 \, {\left (-i \, a^{3} c^{2} + a^{3} c d + {\left (-i \, a^{3} c^{2} + i \, a^{3} d^{2}\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}}}{4 \, {\left ({\left (c^{2} d^{2} - 2 i \, c d^{3} - d^{4}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{2} d^{2} + d^{4}\right )} f\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*} - i a^{3} \left (\int \frac {i}{c \sqrt {c + d \tan {\left (e + f x \right )}} + d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}}\, dx + \int \left (- \frac {3 \tan {\left (e + f x \right )}}{c \sqrt {c + d \tan {\left (e + f x \right )}} + d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}}\right )\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{c \sqrt {c + d \tan {\left (e + f x \right )}} + d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}}\, dx + \int \left (- \frac {3 i \tan ^{2}{\left (e + f x \right )}}{c \sqrt {c + d \tan {\left (e + f x \right )}} + d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 247 vs. \(2 (119) = 238\).
time = 0.78, size = 247, normalized size = 1.78 \begin {gather*} \frac {16 \, a^{3} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (-i \, c f - d f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {2 i \, \sqrt {d \tan \left (f x + e\right ) + c} a^{3}}{d^{2} f} + \frac {2 \, {\left (-i \, a^{3} c^{2} + 2 \, a^{3} c d + i \, a^{3} d^{2}\right )}}{{\left (c d^{2} f - i \, d^{3} f\right )} \sqrt {d \tan \left (f x + e\right ) + c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.59, size = 182, normalized size = 1.31 \begin {gather*} -\frac {a^3\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,2{}\mathrm {i}}{d^2\,f}+\frac {a^3\,\mathrm {atan}\left (\frac {\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,\left (2\,c^4\,f^2+4\,c^2\,d^2\,f^2+2\,d^4\,f^2\right )}{2\,f\,{\left (-c+d\,1{}\mathrm {i}\right )}^{3/2}\,\left (f\,c^3+1{}\mathrm {i}\,f\,c^2\,d+f\,c\,d^2+1{}\mathrm {i}\,f\,d^3\right )}\right )\,8{}\mathrm {i}}{f\,{\left (-c+d\,1{}\mathrm {i}\right )}^{3/2}}-\frac {\left (a^3\,c^2+a^3\,c\,d\,2{}\mathrm {i}-a^3\,d^2\right )\,2{}\mathrm {i}}{d^2\,f\,\left (c-d\,1{}\mathrm {i}\right )\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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