Optimal. Leaf size=157 \[ \frac {\sqrt [4]{-1} d^{3/2} \text {ArcTan}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 f}-\frac {d \sqrt {d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {d \sqrt {d \tan (e+f x)}}{6 a f (a+i a \tan (e+f x))^2}+\frac {d \sqrt {d \tan (e+f x)}}{8 f \left (a^3+i a^3 \tan (e+f x)\right )} \]
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Rubi [A]
time = 0.24, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3639, 3677, 12,
16, 3630, 3614, 211} \begin {gather*} \frac {\sqrt [4]{-1} d^{3/2} \text {ArcTan}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 f}+\frac {d \sqrt {d \tan (e+f x)}}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {d \sqrt {d \tan (e+f x)}}{6 a f (a+i a \tan (e+f x))^2}-\frac {d \sqrt {d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 16
Rule 211
Rule 3614
Rule 3630
Rule 3639
Rule 3677
Rubi steps
\begin {align*} \int \frac {(d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx &=-\frac {d \sqrt {d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}-\frac {\int \frac {-\frac {a d^2}{2}+\frac {7}{2} i a d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2} \, dx}{6 a^2}\\ &=-\frac {d \sqrt {d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {d \sqrt {d \tan (e+f x)}}{6 a f (a+i a \tan (e+f x))^2}-\frac {\int \frac {6 i a^2 d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))} \, dx}{24 a^4 d}\\ &=-\frac {d \sqrt {d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {d \sqrt {d \tan (e+f x)}}{6 a f (a+i a \tan (e+f x))^2}-\frac {\left (i d^2\right ) \int \frac {\tan (e+f x)}{\sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))} \, dx}{4 a^2}\\ &=-\frac {d \sqrt {d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {d \sqrt {d \tan (e+f x)}}{6 a f (a+i a \tan (e+f x))^2}-\frac {(i d) \int \frac {\sqrt {d \tan (e+f x)}}{a+i a \tan (e+f x)} \, dx}{4 a^2}\\ &=-\frac {d \sqrt {d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {d \sqrt {d \tan (e+f x)}}{6 a f (a+i a \tan (e+f x))^2}+\frac {d \sqrt {d \tan (e+f x)}}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {i \int \frac {\frac {1}{2} i a d^2-\frac {1}{2} a d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{8 a^4}\\ &=-\frac {d \sqrt {d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {d \sqrt {d \tan (e+f x)}}{6 a f (a+i a \tan (e+f x))^2}+\frac {d \sqrt {d \tan (e+f x)}}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {\left (i d^4\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} i a d^3+\frac {1}{2} a d^2 x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{16 a^2 f}\\ &=\frac {\sqrt [4]{-1} d^{3/2} \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 f}-\frac {d \sqrt {d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {d \sqrt {d \tan (e+f x)}}{6 a f (a+i a \tan (e+f x))^2}+\frac {d \sqrt {d \tan (e+f x)}}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 2.38, size = 158, normalized size = 1.01 \begin {gather*} \frac {d^2 (i \cos (3 (e+f x))+\sin (3 (e+f x))) \left (5 \cos (e+f x)-5 \cos (3 (e+f x))+3 i \sin (e+f x)-3 i \sin (3 (e+f x))+6 \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right ) (\cos (3 (e+f x))+i \sin (3 (e+f x))) \sqrt {i \tan (e+f x)}\right )}{48 a^3 f \sqrt {d \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 103, normalized size = 0.66
method | result | size |
derivativedivides | \(\frac {2 d^{4} \left (\frac {-\left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}+\frac {10 i d \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+d^{2} \sqrt {d \tan \left (f x +e \right )}}{16 d^{2} \left (i d \tan \left (f x +e \right )+d \right )^{3}}-\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{16 d^{2} \sqrt {i d}}\right )}{f \,a^{3}}\) | \(103\) |
default | \(\frac {2 d^{4} \left (\frac {-\left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}+\frac {10 i d \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+d^{2} \sqrt {d \tan \left (f x +e \right )}}{16 d^{2} \left (i d \tan \left (f x +e \right )+d \right )^{3}}-\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{16 d^{2} \sqrt {i d}}\right )}{f \,a^{3}}\) | \(103\) |
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: RuntimeError} \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. 359 vs. \(2 (134) = 268\).
time = 0.38, size = 359, normalized size = 2.29 \begin {gather*} \frac {{\left (12 \, a^{3} f \sqrt {-\frac {i \, d^{3}}{64 \, a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-\frac {2 \, {\left (i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {i \, d^{3}}{64 \, a^{6} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{d}\right ) - 12 \, a^{3} f \sqrt {-\frac {i \, d^{3}}{64 \, a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-\frac {2 \, {\left (i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 8 \, {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {i \, d^{3}}{64 \, a^{6} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{d}\right ) + {\left (4 \, d e^{\left (6 i \, f x + 6 i \, e\right )} + 4 \, d e^{\left (4 i \, f x + 4 i \, e\right )} - d e^{\left (2 i \, f x + 2 i \, e\right )} - d\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{48 \, a^{3} 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*} \frac {i \int \frac {\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\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}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 160, normalized size = 1.02 \begin {gather*} -\frac {1}{24} \, d {\left (\frac {3 i \, \sqrt {2} \sqrt {d} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a^{3} f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {3 i \, \sqrt {d \tan \left (f x + e\right )} d^{3} \tan \left (f x + e\right )^{2} + 10 \, \sqrt {d \tan \left (f x + e\right )} d^{3} \tan \left (f x + e\right ) - 3 i \, \sqrt {d \tan \left (f x + e\right )} d^{3}}{{\left (d \tan \left (f x + e\right ) - i \, d\right )}^{3} a^{3} f}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.22, size = 155, normalized size = 0.99 \begin {gather*} \frac {\frac {5\,d^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{12\,a^3\,f}-\frac {d^4\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}{8\,a^3\,f}+\frac {d^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}\,1{}\mathrm {i}}{8\,a^3\,f}}{-d^3\,{\mathrm {tan}\left (e+f\,x\right )}^3+d^3\,{\mathrm {tan}\left (e+f\,x\right )}^2\,3{}\mathrm {i}+3\,d^3\,\mathrm {tan}\left (e+f\,x\right )-d^3\,1{}\mathrm {i}}-\frac {\sqrt {\frac {1}{256}{}\mathrm {i}}\,{\left (-d\right )}^{3/2}\,\mathrm {atan}\left (\frac {16\,\sqrt {\frac {1}{256}{}\mathrm {i}}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {-d}}\right )\,2{}\mathrm {i}}{a^3\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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