Optimal. Leaf size=165 \[ -\frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{9/2} f}-\frac {32 a^3}{35 d^2 f (d \tan (e+f x))^{5/2}}-\frac {4 a^3}{3 d^3 f (d \tan (e+f x))^{3/2}}+\frac {4 a^3}{d^4 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{7 d f (d \tan (e+f x))^{7/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3646, 3709,
3610, 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^{9/2} f}+\frac {4 a^3}{d^4 f \sqrt {d \tan (e+f x)}}-\frac {4 a^3}{3 d^3 f (d \tan (e+f x))^{3/2}}-\frac {32 a^3}{35 d^2 f (d \tan (e+f x))^{5/2}}-\frac {2 \left (a^3 \tan (e+f x)+a^3\right )}{7 d f (d \tan (e+f x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
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))^{9/2}} \, dx &=-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{7 d f (d \tan (e+f x))^{7/2}}+\frac {2 \int \frac {8 a^3 d^2+7 a^3 d^2 \tan (e+f x)+a^3 d^2 \tan ^2(e+f x)}{(d \tan (e+f x))^{7/2}} \, dx}{7 d^3}\\ &=-\frac {32 a^3}{35 d^2 f (d \tan (e+f x))^{5/2}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{7 d f (d \tan (e+f x))^{7/2}}+\frac {2 \int \frac {7 a^3 d^3-7 a^3 d^3 \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx}{7 d^5}\\ &=-\frac {32 a^3}{35 d^2 f (d \tan (e+f x))^{5/2}}-\frac {4 a^3}{3 d^3 f (d \tan (e+f x))^{3/2}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{7 d f (d \tan (e+f x))^{7/2}}+\frac {2 \int \frac {-7 a^3 d^4-7 a^3 d^4 \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{7 d^7}\\ &=-\frac {32 a^3}{35 d^2 f (d \tan (e+f x))^{5/2}}-\frac {4 a^3}{3 d^3 f (d \tan (e+f x))^{3/2}}+\frac {4 a^3}{d^4 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{7 d f (d \tan (e+f x))^{7/2}}+\frac {2 \int \frac {-7 a^3 d^5+7 a^3 d^5 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{7 d^9}\\ &=-\frac {32 a^3}{35 d^2 f (d \tan (e+f x))^{5/2}}-\frac {4 a^3}{3 d^3 f (d \tan (e+f x))^{3/2}}+\frac {4 a^3}{d^4 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{7 d f (d \tan (e+f x))^{7/2}}-\frac {\left (28 a^6 d\right ) \text {Subst}\left (\int \frac {1}{-98 a^6 d^{10}+d x^2} \, dx,x,\frac {-7 a^3 d^5-7 a^3 d^5 \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^{9/2} f}-\frac {32 a^3}{35 d^2 f (d \tan (e+f x))^{5/2}}-\frac {4 a^3}{3 d^3 f (d \tan (e+f x))^{3/2}}+\frac {4 a^3}{d^4 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{7 d f (d \tan (e+f x))^{7/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.90, size = 175, normalized size = 1.06 \begin {gather*} -\frac {a^3 \cos (e+f x) (1+\cot (e+f x))^3 \left (10 \cos ^2(e+f x) \cot (e+f x) \, _2F_1\left (-\frac {7}{4},1;-\frac {3}{4};-\tan ^2(e+f x)\right )+42 \cos ^2(e+f x) \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\tan ^2(e+f x)\right )+70 \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\tan ^2(e+f x)\right ) \sin ^2(e+f x)+35 \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\tan ^2(e+f x)\right ) \sin (2 (e+f x))\right ) \sqrt {d \tan (e+f x)}}{35 d^5 f (\cos (e+f x)+\sin (e+f x))^3} \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. \(337\) vs.
\(2(140)=280\).
time = 0.20, size = 338, normalized size = 2.05
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (-\frac {d}{7 \left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}-\frac {3}{5 \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {2}{d^{2} \sqrt {d \tan \left (f x +e \right )}}-\frac {2}{3 d \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d^{2}}\right )}{f \,d^{2}}\) | \(338\) |
default | \(\frac {2 a^{3} \left (-\frac {d}{7 \left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}-\frac {3}{5 \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {2}{d^{2} \sqrt {d \tan \left (f x +e \right )}}-\frac {2}{3 d \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d^{2}}\right )}{f \,d^{2}}\) | \(338\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 168, normalized size = 1.02 \begin {gather*} -\frac {\frac {105 \, 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^{3}} - \frac {2 \, {\left (210 \, a^{3} d^{3} \tan \left (f x + e\right )^{3} - 70 \, a^{3} d^{3} \tan \left (f x + e\right )^{2} - 63 \, a^{3} d^{3} \tan \left (f x + e\right ) - 15 \, a^{3} d^{3}\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} d^{3}}}{105 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 304, normalized size = 1.84 \begin {gather*} \left [\frac {105 \, \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 )^{4} + 2 \, {\left (210 \, a^{3} \tan \left (f x + e\right )^{3} - 70 \, a^{3} \tan \left (f x + e\right )^{2} - 63 \, a^{3} \tan \left (f x + e\right ) - 15 \, a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}}{105 \, d^{5} f \tan \left (f x + e\right )^{4}}, \frac {2 \, {\left (105 \, \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 )^{4} + {\left (210 \, a^{3} \tan \left (f x + e\right )^{3} - 70 \, a^{3} \tan \left (f x + e\right )^{2} - 63 \, a^{3} \tan \left (f x + e\right ) - 15 \, a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}\right )}}{105 \, d^{5} f \tan \left (f x + e\right )^{4}}\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 {9}{2}}}\, dx + \int \frac {3 \tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {9}{2}}}\, dx + \int \frac {3 \tan ^{2}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {9}{2}}}\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {9}{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. 340 vs.
\(2 (147) = 294\).
time = 0.97, size = 340, normalized size = 2.06 \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^{6} 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^{6} 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^{6} 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^{6} f} + \frac {2 \, {\left (210 \, a^{3} d^{3} \tan \left (f x + e\right )^{3} - 70 \, a^{3} d^{3} \tan \left (f x + e\right )^{2} - 63 \, a^{3} d^{3} \tan \left (f x + e\right ) - 15 \, a^{3} d^{3}\right )}}{105 \, \sqrt {d \tan \left (f x + e\right )} d^{7} f \tan \left (f x + e\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.75, size = 130, normalized size = 0.79 \begin {gather*} -\frac {-4\,d\,a^3\,{\mathrm {tan}\left (e+f\,x\right )}^3+\frac {4\,d\,a^3\,{\mathrm {tan}\left (e+f\,x\right )}^2}{3}+\frac {6\,d\,a^3\,\mathrm {tan}\left (e+f\,x\right )}{5}+\frac {2\,d\,a^3}{7}}{d^2\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}}-\frac {2\,\sqrt {2}\,a^3\,\mathrm {atanh}\left (\frac {32\,\sqrt {2}\,a^6\,d^{9/2}\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{32\,a^6\,d^5\,f+32\,a^6\,d^5\,f\,\mathrm {tan}\left (e+f\,x\right )}\right )}{d^{9/2}\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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