Optimal. Leaf size=117 \[ \frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac {16 a^3}{3 d^2 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{3 d f (d \tan (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3646, 3709,
3613, 214} \begin {gather*} \frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {d} \tan (e+f x)+\sqrt {d}}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac {16 a^3}{3 d^2 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3 \tan (e+f x)+a^3\right )}{3 d f (d \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 3613
Rule 3646
Rule 3709
Rubi steps
\begin {align*} \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{5/2}} \, dx &=-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{3 d f (d \tan (e+f x))^{3/2}}+\frac {2 \int \frac {4 a^3 d^2+3 a^3 d^2 \tan (e+f x)+a^3 d^2 \tan ^2(e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{3 d^3}\\ &=-\frac {16 a^3}{3 d^2 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{3 d f (d \tan (e+f x))^{3/2}}+\frac {2 \int \frac {3 a^3 d^3-3 a^3 d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{3 d^5}\\ &=-\frac {16 a^3}{3 d^2 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{3 d f (d \tan (e+f x))^{3/2}}-\frac {\left (12 a^6 d\right ) \text {Subst}\left (\int \frac {1}{-18 a^6 d^6+d x^2} \, dx,x,\frac {3 a^3 d^3+3 a^3 d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f}\\ &=\frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac {16 a^3}{3 d^2 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{3 d f (d \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 4.17, size = 272, normalized size = 2.32 \begin {gather*} \frac {a^3 (1+\tan (e+f x))^3 \left (-8 \cos ^2(e+f x) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\tan ^2(e+f x)\right ) \sin (e+f x)-72 \cos (e+f x) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\tan ^2(e+f x)\right ) \sin ^2(e+f x)+8 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(e+f x)\right ) \sin ^3(e+f x) \tan (e+f x)-9 \sqrt {2} \cos ^3(e+f x) \left (2 \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-2 \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )+\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )\right ) \tan ^{\frac {5}{2}}(e+f x)\right )}{12 f (\cos (e+f x)+\sin (e+f x))^3 (d \tan (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(302\) vs.
\(2(98)=196\).
time = 0.21, size = 303, normalized size = 2.59
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (-\frac {d}{3 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {3}{\sqrt {d \tan \left (f x +e \right )}}+\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f \,d^{2}}\) | \(303\) |
default | \(\frac {2 a^{3} \left (-\frac {d}{3 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {3}{\sqrt {d \tan \left (f x +e \right )}}+\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f \,d^{2}}\) | \(303\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 129, normalized size = 1.10 \begin {gather*} \frac {\frac {3 \, a^{3} {\left (\frac {\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 {\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 )}}{d} - \frac {2 \, {\left (9 \, a^{3} d \tan \left (f x + e\right ) + a^{3} d\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} d}}{3 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.03, size = 244, normalized size = 2.09 \begin {gather*} \left [\frac {3 \, \sqrt {2} a^{3} \sqrt {d} \log \left (\frac {\tan \left (f x + e\right )^{2} + \frac {2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) + 1\right )}}{\sqrt {d}} + 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{2} - 2 \, {\left (9 \, a^{3} \tan \left (f x + e\right ) + a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}}{3 \, d^{3} f \tan \left (f x + e\right )^{2}}, -\frac {2 \, {\left (3 \, \sqrt {2} a^{3} d \sqrt {-\frac {1}{d}} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-\frac {1}{d}} {\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{2} + {\left (9 \, a^{3} \tan \left (f x + e\right ) + a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}\right )}}{3 \, d^{3} f \tan \left (f x + e\right )^{2}}\right ] \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*} a^{3} \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {3 \tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {3 \tan ^{2}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs.
\(2 (103) = 206\).
time = 0.87, size = 299, normalized size = 2.56 \begin {gather*} \frac {\sqrt {2} {\left (a^{3} d \sqrt {{\left | d \right |}} + a^{3} {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{2 \, d^{4} f} - \frac {\sqrt {2} {\left (a^{3} d \sqrt {{\left | d \right |}} + a^{3} {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{2 \, d^{4} f} + \frac {{\left (\sqrt {2} a^{3} d \sqrt {{\left | d \right |}} - \sqrt {2} a^{3} {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d^{4} f} + \frac {{\left (\sqrt {2} a^{3} d \sqrt {{\left | d \right |}} - \sqrt {2} a^{3} {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d^{4} f} - \frac {2 \, {\left (9 \, a^{3} d \tan \left (f x + e\right ) + a^{3} d\right )}}{3 \, \sqrt {d \tan \left (f x + e\right )} d^{3} f \tan \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.32, size = 102, normalized size = 0.87 \begin {gather*} \frac {2\,\sqrt {2}\,a^3\,\mathrm {atanh}\left (\frac {32\,\sqrt {2}\,a^6\,d^{5/2}\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{32\,a^6\,d^3\,f+32\,a^6\,d^3\,f\,\mathrm {tan}\left (e+f\,x\right )}\right )}{d^{5/2}\,f}-\frac {\frac {2\,a^3\,d}{3}+6\,a^3\,d\,\mathrm {tan}\left (e+f\,x\right )}{d^2\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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