Optimal. Leaf size=189 \[ -\frac {31 d^{9/2} \text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 f}+\frac {d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 f}+\frac {27 d^4 \sqrt {d \tan (e+f x)}}{8 a^3 f}-\frac {9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac {d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2} \]
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Rubi [A]
time = 0.49, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3646, 3726,
3728, 3735, 3613, 214, 3715, 65, 211} \begin {gather*} -\frac {31 d^{9/2} \text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 f}+\frac {d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d} \tan (e+f x)+\sqrt {d}}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 f}+\frac {27 d^4 \sqrt {d \tan (e+f x)}}{8 a^3 f}-\frac {9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (\tan (e+f x)+1)}-\frac {d^2 (d \tan (e+f x))^{5/2}}{4 a f (a \tan (e+f x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 211
Rule 214
Rule 3613
Rule 3646
Rule 3715
Rule 3726
Rule 3728
Rule 3735
Rubi steps
\begin {align*} \int \frac {(d \tan (e+f x))^{9/2}}{(a+a \tan (e+f x))^3} \, dx &=-\frac {d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}+\frac {\int \frac {(d \tan (e+f x))^{3/2} \left (\frac {5 a^2 d^3}{2}-2 a^2 d^3 \tan (e+f x)+\frac {9}{2} a^2 d^3 \tan ^2(e+f x)\right )}{(a+a \tan (e+f x))^2} \, dx}{4 a^3}\\ &=-\frac {9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac {d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}+\frac {\int \frac {\sqrt {d \tan (e+f x)} \left (\frac {27 a^4 d^4}{2}-4 a^4 d^4 \tan (e+f x)+\frac {27}{2} a^4 d^4 \tan ^2(e+f x)\right )}{a+a \tan (e+f x)} \, dx}{8 a^6}\\ &=\frac {27 d^4 \sqrt {d \tan (e+f x)}}{8 a^3 f}-\frac {9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac {d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}+\frac {\int \frac {-\frac {27}{4} a^5 d^5-\frac {35}{4} a^5 d^5 \tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{4 a^7}\\ &=\frac {27 d^4 \sqrt {d \tan (e+f x)}}{8 a^3 f}-\frac {9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac {d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}+\frac {\int \frac {2 a^6 d^5-2 a^6 d^5 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{8 a^9}-\frac {\left (31 d^5\right ) \int \frac {1+\tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{16 a^2}\\ &=\frac {27 d^4 \sqrt {d \tan (e+f x)}}{8 a^3 f}-\frac {9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac {d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}-\frac {\left (31 d^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d x} (a+a x)} \, dx,x,\tan (e+f x)\right )}{16 a^2 f}-\frac {\left (a^3 d^{10}\right ) \text {Subst}\left (\int \frac {1}{-8 a^{12} d^{10}+d x^2} \, dx,x,\frac {2 a^6 d^5+2 a^6 d^5 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f}\\ &=\frac {d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 f}+\frac {27 d^4 \sqrt {d \tan (e+f x)}}{8 a^3 f}-\frac {9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac {d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}-\frac {\left (31 d^4\right ) \text {Subst}\left (\int \frac {1}{a+\frac {a x^2}{d}} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{8 a^2 f}\\ &=-\frac {31 d^{9/2} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 f}+\frac {d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 f}+\frac {27 d^4 \sqrt {d \tan (e+f x)}}{8 a^3 f}-\frac {9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (1+\tan (e+f x))}-\frac {d^2 (d \tan (e+f x))^{5/2}}{4 a f (a+a \tan (e+f x))^2}\\ \end {align*}
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Mathematica [A]
time = 6.26, size = 346, normalized size = 1.83 \begin {gather*} \frac {\cot (e+f x) \csc ^3(e+f x) (\cos (e+f x)+\sin (e+f x))^3 \left (\frac {7}{2}-\frac {1}{8 (\cos (e+f x)+\sin (e+f x))^2}-\frac {11 \sin (e+f x)}{8 (\cos (e+f x)+\sin (e+f x))}\right ) (d \tan (e+f x))^{9/2}}{f (a+a \tan (e+f x))^3}+\frac {\sec ^3(e+f x) (\cos (e+f x)+\sin (e+f x))^3 (d \tan (e+f x))^{9/2} \left (-\frac {62 \text {ArcTan}\left (\sqrt {\tan (e+f x)}\right ) \csc (e+f x) \sec ^3(e+f x) (1+\tan (e+f x))}{(1+\cot (e+f x)) \left (1+\tan ^2(e+f x)\right )^2}+\frac {2 \sqrt {2} \cos (2 (e+f x)) \csc (e+f x) \left (-\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 ) \sec ^3(e+f x)}{(1+\cot (e+f x)) (1-\tan (e+f x)) \left (1+\tan ^2(e+f x)\right )}\right )}{16 f \tan ^{\frac {9}{2}}(e+f x) (a+a \tan (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. \(354\) vs.
\(2(156)=312\).
time = 0.19, size = 355, normalized size = 1.88
method | result | size |
derivativedivides | \(\frac {2 d^{4} \left (\sqrt {d \tan \left (f x +e \right )}-\frac {d \left (\frac {-\frac {13 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{4}-\frac {11 d \sqrt {d \tan \left (f x +e \right )}}{4}}{\left (d \tan \left (f x +e \right )+d \right )^{2}}+\frac {31 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{4 \sqrt {d}}\right )}{4}+\frac {d \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{4}\right )}{f \,a^{3}}\) | \(355\) |
default | \(\frac {2 d^{4} \left (\sqrt {d \tan \left (f x +e \right )}-\frac {d \left (\frac {-\frac {13 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{4}-\frac {11 d \sqrt {d \tan \left (f x +e \right )}}{4}}{\left (d \tan \left (f x +e \right )+d \right )^{2}}+\frac {31 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{4 \sqrt {d}}\right )}{4}+\frac {d \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{4}\right )}{f \,a^{3}}\) | \(355\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 211, normalized size = 1.12 \begin {gather*} \frac {\frac {d^{6} {\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 )}}{a^{3}} - \frac {31 \, d^{\frac {11}{2}} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{3}} + \frac {16 \, \sqrt {d \tan \left (f x + e\right )} d^{5}}{a^{3}} + \frac {13 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} d^{6} + 11 \, \sqrt {d \tan \left (f x + e\right )} d^{7}}{a^{3} d^{2} \tan \left (f x + e\right )^{2} + 2 \, a^{3} d^{2} \tan \left (f x + e\right ) + a^{3} d^{2}}}{8 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.22, size = 502, normalized size = 2.66 \begin {gather*} \left [-\frac {4 \, {\left (\sqrt {2} d^{4} \tan \left (f x + e\right )^{2} + 2 \, \sqrt {2} d^{4} \tan \left (f x + e\right ) + \sqrt {2} d^{4}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )} {\left (\sqrt {2} \tan \left (f x + e\right ) + \sqrt {2}\right )} \sqrt {-d}}{2 \, d \tan \left (f x + e\right )}\right ) - 31 \, {\left (d^{4} \tan \left (f x + e\right )^{2} + 2 \, d^{4} \tan \left (f x + e\right ) + d^{4}\right )} \sqrt {-d} \log \left (\frac {d \tan \left (f x + e\right ) - 2 \, \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} - d}{\tan \left (f x + e\right ) + 1}\right ) - 2 \, {\left (16 \, d^{4} \tan \left (f x + e\right )^{2} + 45 \, d^{4} \tan \left (f x + e\right ) + 27 \, d^{4}\right )} \sqrt {d \tan \left (f x + e\right )}}{16 \, {\left (a^{3} f \tan \left (f x + e\right )^{2} + 2 \, a^{3} f \tan \left (f x + e\right ) + a^{3} f\right )}}, -\frac {31 \, {\left (d^{4} \tan \left (f x + e\right )^{2} + 2 \, d^{4} \tan \left (f x + e\right ) + d^{4}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right ) - {\left (\sqrt {2} d^{4} \tan \left (f x + e\right )^{2} + 2 \, \sqrt {2} d^{4} \tan \left (f x + e\right ) + \sqrt {2} d^{4}\right )} \sqrt {d} \log \left (\frac {d \tan \left (f x + e\right )^{2} + 2 \, \sqrt {d \tan \left (f x + e\right )} {\left (\sqrt {2} \tan \left (f x + e\right ) + \sqrt {2}\right )} \sqrt {d} + 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left (16 \, d^{4} \tan \left (f x + e\right )^{2} + 45 \, d^{4} \tan \left (f x + e\right ) + 27 \, d^{4}\right )} \sqrt {d \tan \left (f x + e\right )}}{8 \, {\left (a^{3} f \tan \left (f x + e\right )^{2} + 2 \, a^{3} f \tan \left (f x + e\right ) + a^{3} f\right )}}\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*} \frac {\int \frac {\left (d \tan {\left (e + f x \right )}\right )^{\frac {9}{2}}}{\tan ^{3}{\left (e + f x \right )} + 3 \tan ^{2}{\left (e + f x \right )} + 3 \tan {\left (e + f x \right )} + 1}\, dx}{a^{3}} \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. 345 vs.
\(2 (164) = 328\).
time = 0.88, size = 345, normalized size = 1.83 \begin {gather*} \frac {1}{16} \, d^{4} {\left (\frac {2 \, \sqrt {2} {\left (d \sqrt {{\left | d \right |}} - {\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 )}{a^{3} d f} + \frac {2 \, \sqrt {2} {\left (d \sqrt {{\left | d \right |}} - {\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 )}{a^{3} d f} - \frac {62 \, \sqrt {d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{3} f} + \frac {\sqrt {2} {\left (d \sqrt {{\left | d \right |}} + {\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 )}{a^{3} d f} - \frac {\sqrt {2} {\left (d \sqrt {{\left | d \right |}} + {\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 )}{a^{3} d f} + \frac {32 \, \sqrt {d \tan \left (f x + e\right )}}{a^{3} f} + \frac {2 \, {\left (13 \, \sqrt {d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right ) + 11 \, \sqrt {d \tan \left (f x + e\right )} d^{2}\right )}}{{\left (d \tan \left (f x + e\right ) + d\right )}^{2} a^{3} f}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.95, size = 176, normalized size = 0.93 \begin {gather*} \frac {\frac {11\,d^6\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{8}+\frac {13\,d^5\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{8}}{f\,a^3\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2+2\,f\,a^3\,d^2\,\mathrm {tan}\left (e+f\,x\right )+f\,a^3\,d^2}+\frac {2\,d^4\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{a^3\,f}-\frac {31\,d^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{8\,a^3\,f}-\frac {\sqrt {2}\,d^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,d^{49/2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,969{}\mathrm {i}}{32\,\left (\frac {969\,d^{25}\,\mathrm {tan}\left (e+f\,x\right )}{32}+\frac {969\,d^{25}}{32}\right )}\right )\,1{}\mathrm {i}}{4\,a^3\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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