Optimal. Leaf size=141 \[ \frac {2 \sqrt {2} a^3 \text {ArcTan}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {4 a^3}{d^3 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3646, 3709,
3610, 3613, 211} \begin {gather*} \frac {2 \sqrt {2} a^3 \text {ArcTan}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {4 a^3}{d^3 f \sqrt {d \tan (e+f x)}}-\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {2 \left (a^3 \tan (e+f x)+a^3\right )}{5 d f (d \tan (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 3610
Rule 3613
Rule 3646
Rule 3709
Rubi steps
\begin {align*} \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx &=-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}}+\frac {2 \int \frac {6 a^3 d^2+5 a^3 d^2 \tan (e+f x)+a^3 d^2 \tan ^2(e+f x)}{(d \tan (e+f x))^{5/2}} \, dx}{5 d^3}\\ &=-\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}}+\frac {2 \int \frac {5 a^3 d^3-5 a^3 d^3 \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{5 d^5}\\ &=-\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {4 a^3}{d^3 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}}+\frac {2 \int \frac {-5 a^3 d^4-5 a^3 d^4 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{5 d^7}\\ &=-\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {4 a^3}{d^3 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}}-\frac {\left (20 a^6 d\right ) \text {Subst}\left (\int \frac {1}{50 a^6 d^8+d x^2} \, dx,x,\frac {-5 a^3 d^4+5 a^3 d^4 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f}\\ &=\frac {2 \sqrt {2} a^3 \tan ^{-1}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {4 a^3}{d^3 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 2.26, size = 271, normalized size = 1.92 \begin {gather*} -\frac {a^3 (1+\cot (e+f x))^3 \left (8 \cos ^2(e+f x) \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\tan ^2(e+f x)\right ) \sin (e+f x)+5 \left (8 \cos (e+f x) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\tan ^2(e+f x)\right ) \sin ^2(e+f x)+24 \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\tan ^2(e+f x)\right ) \sin ^3(e+f x)+\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 {7}{2}}(e+f x)\right )\right )}{20 d^3 f (\cos (e+f x)+\sin (e+f x))^3 \sqrt {d \tan (e+f x)}} \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. \(322\) vs.
\(2(120)=240\).
time = 0.21, size = 323, normalized size = 2.29
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (-\frac {d}{5 \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {1}{\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{d \sqrt {d \tan \left (f x +e \right )}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d}\right )}{f \,d^{2}}\) | \(323\) |
default | \(\frac {2 a^{3} \left (-\frac {d}{5 \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {1}{\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{d \sqrt {d \tan \left (f x +e \right )}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d}\right )}{f \,d^{2}}\) | \(323\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 147, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (\frac {5 \, a^{3} {\left (\frac {\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 {\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}}\right )}}{d^{2}} + \frac {10 \, a^{3} d^{2} \tan \left (f x + e\right )^{2} + 5 \, a^{3} d^{2} \tan \left (f x + e\right ) + a^{3} d^{2}}{\left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} d^{2}}\right )}}{5 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 275, normalized size = 1.95 \begin {gather*} \left [\frac {5 \, \sqrt {2} a^{3} d \sqrt {-\frac {1}{d}} \log \left (-\frac {2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-\frac {1}{d}} {\left (\tan \left (f x + e\right ) - 1\right )} - \tan \left (f x + e\right )^{2} + 4 \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{3} - 2 \, {\left (10 \, a^{3} \tan \left (f x + e\right )^{2} + 5 \, a^{3} \tan \left (f x + e\right ) + a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}}{5 \, d^{4} f \tan \left (f x + e\right )^{3}}, -\frac {2 \, {\left (5 \, \sqrt {2} a^{3} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {d} \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{3} + {\left (10 \, a^{3} \tan \left (f x + e\right )^{2} + 5 \, a^{3} \tan \left (f x + e\right ) + a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}\right )}}{5 \, d^{4} f \tan \left (f x + e\right )^{3}}\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 {7}{2}}}\, dx + \int \frac {3 \tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {3 \tan ^{2}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{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. 322 vs.
\(2 (127) = 254\).
time = 0.85, size = 322, normalized size = 2.28 \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^{5} 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^{5} 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^{5} 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^{5} f} - \frac {2 \, {\left (10 \, a^{3} d^{2} \tan \left (f x + e\right )^{2} + 5 \, a^{3} d^{2} \tan \left (f x + e\right ) + a^{3} d^{2}\right )}}{5 \, \sqrt {d \tan \left (f x + e\right )} d^{5} f \tan \left (f x + e\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.08, size = 128, normalized size = 0.91 \begin {gather*} -\frac {4\,d\,a^3\,{\mathrm {tan}\left (e+f\,x\right )}^2+2\,d\,a^3\,\mathrm {tan}\left (e+f\,x\right )+\frac {2\,d\,a^3}{5}}{d^2\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}-\frac {\sqrt {2}\,a^3\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}+\frac {\sqrt {2}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{2\,d^{3/2}}\right )\right )}{d^{7/2}\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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