Optimal. Leaf size=189 \[ -\frac {31 \text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 d^{3/2} f}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 d^{3/2} f}-\frac {27}{8 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {9}{8 a^3 d f \sqrt {d \tan (e+f x)} (1+\tan (e+f x))}+\frac {1}{4 a d f \sqrt {d \tan (e+f x)} (a+a \tan (e+f x))^2} \]
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Rubi [A]
time = 0.50, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3650, 3730,
3735, 3613, 214, 3715, 65, 211} \begin {gather*} -\frac {31 \text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 d^{3/2} f}-\frac {\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 d^{3/2} f}-\frac {27}{8 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {9}{8 a^3 d f (\tan (e+f x)+1) \sqrt {d \tan (e+f x)}}+\frac {1}{4 a d f (a \tan (e+f x)+a)^2 \sqrt {d \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 211
Rule 214
Rule 3613
Rule 3650
Rule 3715
Rule 3730
Rule 3735
Rubi steps
\begin {align*} \int \frac {1}{(d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^3} \, dx &=\frac {1}{4 a d f \sqrt {d \tan (e+f x)} (a+a \tan (e+f x))^2}+\frac {\int \frac {\frac {9 a^2 d}{2}-2 a^2 d \tan (e+f x)+\frac {5}{2} a^2 d \tan ^2(e+f x)}{(d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2} \, dx}{4 a^3 d}\\ &=\frac {9}{8 a^3 d f \sqrt {d \tan (e+f x)} (1+\tan (e+f x))}+\frac {1}{4 a d f \sqrt {d \tan (e+f x)} (a+a \tan (e+f x))^2}+\frac {\int \frac {\frac {27 a^4 d^2}{2}-4 a^4 d^2 \tan (e+f x)+\frac {27}{2} a^4 d^2 \tan ^2(e+f x)}{(d \tan (e+f x))^{3/2} (a+a \tan (e+f x))} \, dx}{8 a^6 d^2}\\ &=-\frac {27}{8 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {9}{8 a^3 d f \sqrt {d \tan (e+f x)} (1+\tan (e+f x))}+\frac {1}{4 a d f \sqrt {d \tan (e+f x)} (a+a \tan (e+f x))^2}-\frac {\int \frac {\frac {35 a^5 d^4}{4}+\frac {27}{4} a^5 d^4 \tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{4 a^7 d^5}\\ &=-\frac {27}{8 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {9}{8 a^3 d f \sqrt {d \tan (e+f x)} (1+\tan (e+f x))}+\frac {1}{4 a d f \sqrt {d \tan (e+f x)} (a+a \tan (e+f x))^2}-\frac {\int \frac {2 a^6 d^4-2 a^6 d^4 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{8 a^9 d^5}-\frac {31 \int \frac {1+\tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{16 a^2 d}\\ &=-\frac {27}{8 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {9}{8 a^3 d f \sqrt {d \tan (e+f x)} (1+\tan (e+f x))}+\frac {1}{4 a d f \sqrt {d \tan (e+f x)} (a+a \tan (e+f x))^2}-\frac {31 \text {Subst}\left (\int \frac {1}{\sqrt {d x} (a+a x)} \, dx,x,\tan (e+f x)\right )}{16 a^2 d f}+\frac {\left (a^3 d^3\right ) \text {Subst}\left (\int \frac {1}{-8 a^{12} d^8+d x^2} \, dx,x,\frac {2 a^6 d^4+2 a^6 d^4 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f}\\ &=-\frac {\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 d^{3/2} f}-\frac {27}{8 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {9}{8 a^3 d f \sqrt {d \tan (e+f x)} (1+\tan (e+f x))}+\frac {1}{4 a d f \sqrt {d \tan (e+f x)} (a+a \tan (e+f x))^2}-\frac {31 \text {Subst}\left (\int \frac {1}{a+\frac {a x^2}{d}} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{8 a^2 d^2 f}\\ &=-\frac {31 \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 d^{3/2} f}-\frac {\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 d^{3/2} f}-\frac {27}{8 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {9}{8 a^3 d f \sqrt {d \tan (e+f x)} (1+\tan (e+f x))}+\frac {1}{4 a d f \sqrt {d \tan (e+f x)} (a+a \tan (e+f x))^2}\\ \end {align*}
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Mathematica [A]
time = 1.48, size = 173, normalized size = 0.92 \begin {gather*} \frac {\left (-62 \text {ArcTan}\left (\sqrt {\tan (e+f x)}\right )+2 \sqrt {2} \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 )-\frac {(90+90 \cos (2 (e+f x))+75 \cot (e+f x)-11 \cos (3 (e+f x)) \csc (e+f x)) \sqrt {\tan (e+f x)}}{2 (\cos (e+f x)+\sin (e+f x))^2}\right ) \tan ^{\frac {3}{2}}(e+f x)}{16 a^3 f (d \tan (e+f x))^{3/2}} \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. \(363\) vs.
\(2(156)=312\).
time = 0.19, size = 364, normalized size = 1.93
method | result | size |
derivativedivides | \(\frac {2 d^{4} \left (-\frac {1}{d^{5} \sqrt {d \tan \left (f x +e \right )}}-\frac {\frac {\frac {11 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{4}+\frac {13 d \sqrt {d \tan \left (f x +e \right )}}{4}}{\left (d \tan \left (f x +e \right )+d \right )^{2}}+\frac {31 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{4 \sqrt {d}}}{4 d^{5}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}}{4 d^{5}}\right )}{f \,a^{3}}\) | \(364\) |
default | \(\frac {2 d^{4} \left (-\frac {1}{d^{5} \sqrt {d \tan \left (f x +e \right )}}-\frac {\frac {\frac {11 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{4}+\frac {13 d \sqrt {d \tan \left (f x +e \right )}}{4}}{\left (d \tan \left (f x +e \right )+d \right )^{2}}+\frac {31 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{4 \sqrt {d}}}{4 d^{5}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}}{4 d^{5}}\right )}{f \,a^{3}}\) | \(364\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 200, normalized size = 1.06 \begin {gather*} -\frac {\frac {27 \, d^{2} \tan \left (f x + e\right )^{2} + 45 \, d^{2} \tan \left (f x + e\right ) + 16 \, d^{2}}{\left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} a^{3} + 2 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{3} d + \sqrt {d \tan \left (f x + e\right )} a^{3} d^{2}} + \frac {\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}}}{a^{3}} + \frac {31 \, \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{3} \sqrt {d}}}{8 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 493, normalized size = 2.61 \begin {gather*} \left [\frac {4 \, \sqrt {2} {\left (\tan \left (f x + e\right )^{3} + 2 \, \tan \left (f x + e\right )^{2} + \tan \left (f x + e\right )\right )} \sqrt {-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 ) - 31 \, {\left (\tan \left (f x + e\right )^{3} + 2 \, \tan \left (f x + e\right )^{2} + \tan \left (f x + e\right )\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 \, \sqrt {d \tan \left (f x + e\right )} {\left (27 \, \tan \left (f x + e\right )^{2} + 45 \, \tan \left (f x + e\right ) + 16\right )}}{16 \, {\left (a^{3} d^{2} f \tan \left (f x + e\right )^{3} + 2 \, a^{3} d^{2} f \tan \left (f x + e\right )^{2} + a^{3} d^{2} f \tan \left (f x + e\right )\right )}}, \frac {\sqrt {2} {\left (\tan \left (f x + e\right )^{3} + 2 \, \tan \left (f x + e\right )^{2} + \tan \left (f x + e\right )\right )} \sqrt {d} \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 ) - 31 \, {\left (\tan \left (f x + e\right )^{3} + 2 \, \tan \left (f x + e\right )^{2} + \tan \left (f x + e\right )\right )} \sqrt {d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right ) - \sqrt {d \tan \left (f x + e\right )} {\left (27 \, \tan \left (f x + e\right )^{2} + 45 \, \tan \left (f x + e\right ) + 16\right )}}{8 \, {\left (a^{3} d^{2} f \tan \left (f x + e\right )^{3} + 2 \, a^{3} d^{2} f \tan \left (f x + e\right )^{2} + a^{3} d^{2} f \tan \left (f x + e\right )\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 {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{3}{\left (e + f x \right )} + 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )} + 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )} + \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, 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. 341 vs.
\(2 (164) = 328\).
time = 0.86, size = 341, normalized size = 1.80 \begin {gather*} -\frac {\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^{2} 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^{2} f} + \frac {62 \, \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{3} \sqrt {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^{2} 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^{2} f} + \frac {32}{\sqrt {d \tan \left (f x + e\right )} a^{3} f} + \frac {2 \, {\left (11 \, \sqrt {d \tan \left (f x + e\right )} d \tan \left (f x + e\right ) + 13 \, \sqrt {d \tan \left (f x + e\right )} d\right )}}{{\left (d \tan \left (f x + e\right ) + d\right )}^{2} a^{3} f}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.82, size = 176, normalized size = 0.93 \begin {gather*} -\frac {\frac {27\,d\,{\mathrm {tan}\left (e+f\,x\right )}^2}{8}+\frac {45\,d\,\mathrm {tan}\left (e+f\,x\right )}{8}+2\,d}{a^3\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}+2\,a^3\,d\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}+a^3\,d^2\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}-\frac {31\,\mathrm {atan}\left (\frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{8\,a^3\,d^{3/2}\,f}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {63504384\,\sqrt {2}\,a^9\,d^{15/2}\,f^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{63504384\,a^9\,d^8\,f^3+63504384\,a^9\,d^8\,f^3\,\mathrm {tan}\left (e+f\,x\right )}\right )}{4\,a^3\,d^{3/2}\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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