Optimal. Leaf size=255 \[ -\frac {2 \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}}+\frac {\text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} f \sqrt {b \tan ^3(e+f x)}}-\frac {\text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} f \sqrt {b \tan ^3(e+f x)}}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3739, 3555,
3557, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\tan ^{\frac {3}{2}}(e+f x) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2} f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rule 3739
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \tan ^3(e+f x)}} \, dx &=\frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan ^{\frac {3}{2}}(e+f x)} \, dx}{\sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \int \sqrt {\tan (e+f x)} \, dx}{\sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}}-\frac {\left (2 \tan ^{\frac {3}{2}}(e+f x)\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} f \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} f \sqrt {b \tan ^3(e+f x)}}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 43, normalized size = 0.17 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\tan ^2(e+f x)\right ) \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 211, normalized size = 0.83
method | result | size |
derivativedivides | \(-\frac {\tan \left (f x +e \right ) \left (\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}{b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+2 \sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+8 \left (b^{2}\right )^{\frac {1}{4}}\right )}{4 f \sqrt {b \left (\tan ^{3}\left (f x +e \right )\right )}\, \left (b^{2}\right )^{\frac {1}{4}}}\) | \(211\) |
default | \(-\frac {\tan \left (f x +e \right ) \left (\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}{b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+2 \sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+8 \left (b^{2}\right )^{\frac {1}{4}}\right )}{4 f \sqrt {b \left (\tan ^{3}\left (f x +e \right )\right )}\, \left (b^{2}\right )^{\frac {1}{4}}}\) | \(211\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 133, normalized size = 0.52 \begin {gather*} -\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right )}{\sqrt {b}} + \frac {8}{\sqrt {b} \sqrt {\tan \left (f x + e\right )}}}{4 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \frac {1}{\sqrt {b \tan ^{3}{\left (e + f x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 263, normalized size = 1.03 \begin {gather*} -\frac {1}{4} \, b^{2} {\left (\frac {2 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{4} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {2 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{4} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} - \frac {\sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (f x + e\right ) + \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{4} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {\sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (f x + e\right ) - \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{4} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {8}{\sqrt {b \tan \left (f x + e\right )} b^{2} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )}\right )} \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 \frac {1}{\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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