Optimal. Leaf size=364 \[ \frac {2 \tan (e+f x)}{b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}} \]
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Rubi [A] time = 0.15, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3658, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \tan (e+f x)}{b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3474
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{5/2}} \, dx &=\frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan ^{\frac {15}{2}}(e+f x)} \, dx}{b^2 \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan ^{\frac {11}{2}}(e+f x)} \, dx}{b^2 \sqrt {b \tan ^3(e+f x)}}\\ &=\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan ^{\frac {7}{2}}(e+f x)} \, dx}{b^2 \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan ^{\frac {3}{2}}(e+f x)} \, dx}{b^2 \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \tan (e+f x)}{b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \int \sqrt {\tan (e+f x)} \, dx}{b^2 \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \tan (e+f x)}{b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{b^2 f \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \tan (e+f x)}{b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\left (2 \tan ^{\frac {3}{2}}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{b^2 f \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \tan (e+f x)}{b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{b^2 f \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \tan (e+f x)}{b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \operatorname {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} b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \operatorname {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} b^2 f \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \tan (e+f x)}{b^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} b^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} b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\tan ^{\frac {3}{2}}(e+f x) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\tan ^{\frac {3}{2}}(e+f x) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}\\ &=-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \tan (e+f x)}{b^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} b^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} b^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} b^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} b^2 f \sqrt {b \tan ^3(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 45, normalized size = 0.12 \[ -\frac {2 \tan (e+f x) \, _2F_1\left (-\frac {13}{4},1;-\frac {9}{4};-\tan ^2(e+f x)\right )}{13 f \left (b \tan ^3(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \tan \left (f x + e\right )^{3}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 272, normalized size = 0.75 \[ \frac {\tan \left (f x +e \right ) \left (585 \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {13}{2}} \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}{b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+4680 \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{6}\left (f x +e \right )\right ) b^{6}-936 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{4}\left (f x +e \right )\right )+520 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{2}\left (f x +e \right )\right )-360 b^{6} \left (b^{2}\right )^{\frac {1}{4}}\right )}{2340 f \,b^{6} \left (b \left (\tan ^{3}\left (f x +e \right )\right )\right )^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.92, size = 172, normalized size = 0.47 \[ \frac {\frac {585 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right )\right )}}{b^{\frac {5}{2}}} + \frac {8 \, {\left (\frac {585 \, \sqrt {b}}{\sqrt {\tan \left (f x + e\right )}} - \frac {117 \, \sqrt {b}}{\tan \left (f x + e\right )^{\frac {5}{2}}} + \frac {65 \, \sqrt {b}}{\tan \left (f x + e\right )^{\frac {9}{2}}} - \frac {45 \, \sqrt {b}}{\tan \left (f x + e\right )^{\frac {13}{2}}}\right )}}{b^{3}}}{2340 \, f} \]
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 \[ \int \frac {1}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^3\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \tan ^{3}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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