Optimal. Leaf size=250 \[ -\frac {\sqrt [4]{-1} a^{3/2} (i c+3 d) \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a^2 (c+i d) \sqrt {c+d \tan (e+f x)}}{d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}} \]
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Rubi [A]
time = 0.63, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {3637, 3676,
3682, 3625, 214, 3680, 65, 223, 212} \begin {gather*} -\frac {\sqrt [4]{-1} a^{3/2} (3 d+i c) \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}+\frac {a^2 (c+i d) \sqrt {c+d \tan (e+f x)}}{d f \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 214
Rule 223
Rule 3625
Rule 3637
Rule 3676
Rule 3680
Rule 3682
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \, dx &=-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}+\frac {a \int \frac {\left (-\frac {1}{2} a (i c-5 d)-\frac {1}{2} a (c-3 i d) \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx}{d}\\ &=\frac {a^2 (c+i d) \sqrt {c+d \tan (e+f x)}}{d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}-\frac {\int \frac {\sqrt {a+i a \tan (e+f x)} \left (-\frac {1}{2} a^2 (3 c-i d) d-\frac {1}{2} a^2 d (i c+3 d) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{a d}\\ &=\frac {a^2 (c+i d) \sqrt {c+d \tan (e+f x)}}{d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}+(2 a (c-i d)) \int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx-\frac {1}{2} (c-3 i d) \int \frac {(a-i a \tan (e+f x)) \sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {a^2 (c+i d) \sqrt {c+d \tan (e+f x)}}{d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}-\frac {\left (a^2 (c-3 i d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}-\frac {\left (4 a^3 (i c+d)\right ) \text {Subst}\left (\int \frac {1}{a c-i a d-2 a^2 x^2} \, dx,x,\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{f}\\ &=-\frac {2 i \sqrt {2} a^{3/2} \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a^2 (c+i d) \sqrt {c+d \tan (e+f x)}}{d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}+\frac {(a (i c+3 d)) \text {Subst}\left (\int \frac {1}{\sqrt {c+i d-\frac {i d x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{f}\\ &=-\frac {2 i \sqrt {2} a^{3/2} \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a^2 (c+i d) \sqrt {c+d \tan (e+f x)}}{d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}+\frac {(a (i c+3 d)) \text {Subst}\left (\int \frac {1}{1+\frac {i d x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}\\ &=-\frac {\sqrt [4]{-1} a^{3/2} (i c+3 d) \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a^2 (c+i d) \sqrt {c+d \tan (e+f x)}}{d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt {a+i a \tan (e+f x)}}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(559\) vs. \(2(250)=500\).
time = 6.29, size = 559, normalized size = 2.24 \begin {gather*} \frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \cos (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x))^{3/2} \left (\frac {\cos (e+f x) \left ((-i c-3 d) \log \left (\frac {(2+2 i) e^{\frac {i e}{2}} \left (-i d+d e^{i (e+f x)}+i c \left (i+e^{i (e+f x)}\right )-(1+i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )}{\sqrt {d} (i c+3 d) \left (i+e^{i (e+f x)}\right )}\right )+(i c+3 d) \log \left (\frac {(2+2 i) e^{\frac {i e}{2}} \left (c+i d+i c e^{i (e+f x)}+d e^{i (e+f x)}+(1+i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )}{\sqrt {d} (i c+3 d) \left (-i+e^{i (e+f x)}\right )}\right )-(4+4 i) \sqrt {c-i d} \sqrt {d} \log \left (2 \left (\sqrt {c-i d} \cos (e+f x)+i \sqrt {c-i d} \sin (e+f x)+\sqrt {1+\cos (2 (e+f x))+i \sin (2 (e+f x))} \sqrt {c+d \tan (e+f x)}\right )\right )\right ) (\cos (e)-i \sin (e))}{\sqrt {d} \sqrt {1+\cos (2 (e+f x))+i \sin (2 (e+f x))}}+(1+i) \cos (e) \sqrt {c+d \tan (e+f x)}+(1-i) \sin (e) \sqrt {c+d \tan (e+f x)}\right )}{f} \end {gather*}
Warning: Unable to verify antiderivative.
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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. 865 vs. \(2 (201 ) = 402\).
time = 0.52, size = 866, normalized size = 3.46 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 773 vs. \(2 (200) = 400\).
time = 0.79, size = 773, normalized size = 3.09 \begin {gather*} \frac {2 i \, \sqrt {2} a \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}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (i \, f x + i \, e\right )} - 2 \, \sqrt {2} f \sqrt {-\frac {a^{3} c - i \, a^{3} d}{f^{2}}} \log \left (-\frac {{\left (i \, \sqrt {2} f \sqrt {-\frac {a^{3} c - i \, a^{3} d}{f^{2}}} e^{\left (i \, f x + i \, e\right )} - \sqrt {2} {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\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}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a}\right ) + 2 \, \sqrt {2} f \sqrt {-\frac {a^{3} c - i \, a^{3} d}{f^{2}}} \log \left (-\frac {{\left (-i \, \sqrt {2} f \sqrt {-\frac {a^{3} c - i \, a^{3} d}{f^{2}}} e^{\left (i \, f x + i \, e\right )} - \sqrt {2} {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\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}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a}\right ) - f \sqrt {\frac {-i \, a^{3} c^{2} - 6 \, a^{3} c d + 9 i \, a^{3} d^{2}}{d f^{2}}} \log \left (\frac {{\left (2 i \, d f \sqrt {\frac {-i \, a^{3} c^{2} - 6 \, a^{3} c d + 9 i \, a^{3} d^{2}}{d f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} {\left (i \, a c + 3 \, a d + {\left (i \, a c + 3 \, a d\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}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{i \, a c + 3 \, a d}\right ) + f \sqrt {\frac {-i \, a^{3} c^{2} - 6 \, a^{3} c d + 9 i \, a^{3} d^{2}}{d f^{2}}} \log \left (\frac {{\left (-2 i \, d f \sqrt {\frac {-i \, a^{3} c^{2} - 6 \, a^{3} c d + 9 i \, a^{3} d^{2}}{d f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} {\left (i \, a c + 3 \, a d + {\left (i \, a c + 3 \, a d\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}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{i \, a c + 3 \, a d}\right )}{2 \, f} \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*} \int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \sqrt {c + d \tan {\left (e + f x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.00 \begin {gather*} \int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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