Optimal. Leaf size=62 \[ -\frac {2 (-1)^{3/4} a \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}-\frac {2 a}{d f \sqrt {d \tan (e+f x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3529, 3533, 205} \[ -\frac {2 (-1)^{3/4} a \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}-\frac {2 a}{d f \sqrt {d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 3529
Rule 3533
Rubi steps
\begin {align*} \int \frac {a+i a \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx &=-\frac {2 a}{d f \sqrt {d \tan (e+f x)}}+\frac {\int \frac {i a d-a d \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{d^2}\\ &=-\frac {2 a}{d f \sqrt {d \tan (e+f x)}}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{i a d^2+a d x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{f}\\ &=-\frac {2 (-1)^{3/4} a \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}-\frac {2 a}{d f \sqrt {d \tan (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 1.16, size = 138, normalized size = 2.23 \[ -\frac {2 a e^{-i (e+f x)} \sin (e+f x) (\tan (e+f x)-i) \left (i \sqrt {i \tan (e+f x)}+\tan (e+f x) \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right )\right )}{f \sqrt {\frac {-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}} (d \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.44, size = 339, normalized size = 5.47 \[ -\frac {{\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - d^{2} f\right )} \sqrt {\frac {4 i \, a^{2}}{d^{3} f^{2}}} \log \left (\frac {{\left (-2 i \, a d e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d^{2} f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {4 i \, a^{2}}{d^{3} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a}\right ) - {\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - d^{2} f\right )} \sqrt {\frac {4 i \, a^{2}}{d^{3} f^{2}}} \log \left (\frac {{\left (-2 i \, a d e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, d^{2} f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {4 i \, a^{2}}{d^{3} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a}\right ) - {\left (-8 i \, a e^{\left (2 i \, f x + 2 i \, e\right )} - 8 i \, a\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{4 \, {\left (d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - d^{2} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.91, size = 89, normalized size = 1.44 \[ \frac {2 \, a {\left (\frac {i \, \sqrt {2} \arctan \left (-\frac {16 i \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{-8 i \, \sqrt {2} d^{\frac {3}{2}} + 8 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{\sqrt {d} f {\left (-\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} - \frac {1}{\sqrt {d \tan \left (f x + e\right )} f}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 358, normalized size = 5.77 \[ \frac {i a \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )}{4 f \,d^{2}}+\frac {i a \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 f \,d^{2}}-\frac {i a \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 f \,d^{2}}-\frac {a \sqrt {2}\, \ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )}{4 f d \left (d^{2}\right )^{\frac {1}{4}}}-\frac {a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 f d \left (d^{2}\right )^{\frac {1}{4}}}+\frac {a \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 f d \left (d^{2}\right )^{\frac {1}{4}}}-\frac {2 a}{d f \sqrt {d \tan \left (f x +e \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.45, size = 172, normalized size = 2.77 \[ \frac {a {\left (\frac {\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\left (i + 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\left (i + 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {8 \, a}{\sqrt {d \tan \left (f x + e\right )}}}{4 \, d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 50, normalized size = 0.81 \[ \frac {2\,{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{d^{3/2}\,f}-\frac {2\,a}{d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}} \]
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 \[ i a \left (\int \left (- \frac {i}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\right )\, dx + \int \frac {\tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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