Optimal. Leaf size=160 \[ \frac {2 \sqrt {2} a^3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{f}-\frac {4 a^3 d \sqrt {d \tan (e+f x)}}{f}+\frac {4 a^3 (d \tan (e+f x))^{3/2}}{3 f}+\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}+\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+a^3 \tan (e+f x)\right )}{7 d f} \]
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Rubi [A]
time = 0.17, antiderivative size = 160, 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 = {3647, 3711,
3609, 3613, 214} \begin {gather*} \frac {2 \sqrt {2} a^3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \tan (e+f x)+\sqrt {d}}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{f}+\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}+\frac {4 a^3 (d \tan (e+f x))^{3/2}}{3 f}-\frac {4 a^3 d \sqrt {d \tan (e+f x)}}{f}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{5/2}}{7 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 3609
Rule 3613
Rule 3647
Rule 3711
Rubi steps
\begin {align*} \int (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^3 \, dx &=\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+a^3 \tan (e+f x)\right )}{7 d f}+\frac {2 \int (d \tan (e+f x))^{3/2} \left (a^3 d+7 a^3 d \tan (e+f x)+8 a^3 d \tan ^2(e+f x)\right ) \, dx}{7 d}\\ &=\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}+\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+a^3 \tan (e+f x)\right )}{7 d f}+\frac {2 \int (d \tan (e+f x))^{3/2} \left (-7 a^3 d+7 a^3 d \tan (e+f x)\right ) \, dx}{7 d}\\ &=\frac {4 a^3 (d \tan (e+f x))^{3/2}}{3 f}+\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}+\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+a^3 \tan (e+f x)\right )}{7 d f}+\frac {2 \int \sqrt {d \tan (e+f x)} \left (-7 a^3 d^2-7 a^3 d^2 \tan (e+f x)\right ) \, dx}{7 d}\\ &=-\frac {4 a^3 d \sqrt {d \tan (e+f x)}}{f}+\frac {4 a^3 (d \tan (e+f x))^{3/2}}{3 f}+\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}+\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+a^3 \tan (e+f x)\right )}{7 d f}+\frac {2 \int \frac {7 a^3 d^3-7 a^3 d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{7 d}\\ &=-\frac {4 a^3 d \sqrt {d \tan (e+f x)}}{f}+\frac {4 a^3 (d \tan (e+f x))^{3/2}}{3 f}+\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}+\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+a^3 \tan (e+f x)\right )}{7 d f}-\frac {\left (28 a^6 d^5\right ) \text {Subst}\left (\int \frac {1}{-98 a^6 d^6+d x^2} \, dx,x,\frac {7 a^3 d^3+7 a^3 d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f}\\ &=\frac {2 \sqrt {2} a^3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{f}-\frac {4 a^3 d \sqrt {d \tan (e+f x)}}{f}+\frac {4 a^3 (d \tan (e+f x))^{3/2}}{3 f}+\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}+\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+a^3 \tan (e+f x)\right )}{7 d f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 2.78, size = 332, normalized size = 2.08 \begin {gather*} -\frac {a^3 \cos (e+f x) (d \tan (e+f x))^{3/2} (1+\tan (e+f x))^3 \left (210 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x)-210 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x)+105 \sqrt {2} \cos ^2(e+f x) \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-105 \sqrt {2} \cos ^2(e+f x) \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )+840 \cos ^2(e+f x) \sqrt {\tan (e+f x)}-280 \cos ^2(e+f x) \tan ^{\frac {3}{2}}(e+f x)+280 \cos ^2(e+f x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)-60 \sin ^2(e+f x) \tan ^{\frac {3}{2}}(e+f x)-126 \sin (2 (e+f x)) \tan ^{\frac {3}{2}}(e+f x)\right )}{210 f (\cos (e+f x)+\sin (e+f x))^3 \tan ^{\frac {3}{2}}(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. \(338\) vs.
\(2(135)=270\).
time = 0.36, size = 339, normalized size = 2.12
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {3 d \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 d^{3} \sqrt {d \tan \left (f x +e \right )}+2 d^{4} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(339\) |
default | \(\frac {2 a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {3 d \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 d^{3} \sqrt {d \tan \left (f x +e \right )}+2 d^{4} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(339\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 171, normalized size = 1.07 \begin {gather*} \frac {105 \, a^{3} d^{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 )} + \frac {2 \, {\left (15 \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} a^{3} + 63 \, \left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} a^{3} d + 70 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{3} d^{2} - 210 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{3}\right )}}{d}}{105 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.35, size = 273, normalized size = 1.71 \begin {gather*} \left [\frac {105 \, \sqrt {2} a^{3} d^{\frac {3}{2}} \log \left (\frac {d \tan \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} {\left (\tan \left (f x + e\right ) + 1\right )} + 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, {\left (15 \, a^{3} d \tan \left (f x + e\right )^{3} + 63 \, a^{3} d \tan \left (f x + e\right )^{2} + 70 \, a^{3} d \tan \left (f x + e\right ) - 210 \, a^{3} d\right )} \sqrt {d \tan \left (f x + e\right )}}{105 \, f}, -\frac {2 \, {\left (105 \, \sqrt {2} a^{3} \sqrt {-d} d \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} {\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, d \tan \left (f x + e\right )}\right ) - {\left (15 \, a^{3} d \tan \left (f x + e\right )^{3} + 63 \, a^{3} d \tan \left (f x + e\right )^{2} + 70 \, a^{3} d \tan \left (f x + e\right ) - 210 \, a^{3} d\right )} \sqrt {d \tan \left (f x + e\right )}\right )}}{105 \, f}\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 \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx + \int 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )}\, dx + \int 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{3}{\left (e + f x \right )}\, 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. 379 vs.
\(2 (142) = 284\).
time = 0.86, size = 379, normalized size = 2.37 \begin {gather*} \frac {1}{210} \, d {\left (\frac {105 \, \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 )}{d f} - \frac {105 \, \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 )}{d f} + \frac {210 \, {\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 f} + \frac {210 \, {\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 f} + \frac {4 \, {\left (15 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{21} f^{6} \tan \left (f x + e\right )^{3} + 63 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{21} f^{6} \tan \left (f x + e\right )^{2} + 70 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{21} f^{6} \tan \left (f x + e\right ) - 210 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{21} f^{6}\right )}}{d^{21} f^{7}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.15, size = 143, normalized size = 0.89 \begin {gather*} \frac {4\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{3\,f}+\frac {6\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{5\,d\,f}+\frac {2\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}}{7\,d^2\,f}-\frac {4\,a^3\,d\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{f}-\frac {\sqrt {2}\,a^3\,d^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,a^6\,d^{9/2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,32{}\mathrm {i}}{32\,a^6\,d^5+32\,a^6\,d^5\,\mathrm {tan}\left (e+f\,x\right )}\right )\,2{}\mathrm {i}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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