Optimal. Leaf size=135 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{a d^{5/2} f}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a d^{5/2} f}-\frac {2}{3 a d f (d \tan (e+f x))^{3/2}}+\frac {2}{a d^2 f \sqrt {d \tan (e+f x)}} \]
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Rubi [A]
time = 0.33, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3650, 3730,
12, 16, 3654, 3613, 214, 3715, 65, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{a d^{5/2} f}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d} \tan (e+f x)+\sqrt {d}}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a d^{5/2} f}+\frac {2}{a d^2 f \sqrt {d \tan (e+f x)}}-\frac {2}{3 a d f (d \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 16
Rule 65
Rule 211
Rule 214
Rule 3613
Rule 3650
Rule 3654
Rule 3715
Rule 3730
Rubi steps
\begin {align*} \int \frac {1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))} \, dx &=-\frac {2}{3 a d f (d \tan (e+f x))^{3/2}}-\frac {2 \int \frac {\frac {3 a d^2}{2}+\frac {3}{2} a d^2 \tan (e+f x)+\frac {3}{2} a d^2 \tan ^2(e+f x)}{(d \tan (e+f x))^{3/2} (a+a \tan (e+f x))} \, dx}{3 a d^3}\\ &=-\frac {2}{3 a d f (d \tan (e+f x))^{3/2}}+\frac {2}{a d^2 f \sqrt {d \tan (e+f x)}}+\frac {4 \int \frac {3 a^2 d^4 \tan ^2(e+f x)}{4 \sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{3 a^2 d^6}\\ &=-\frac {2}{3 a d f (d \tan (e+f x))^{3/2}}+\frac {2}{a d^2 f \sqrt {d \tan (e+f x)}}+\frac {\int \frac {\tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{d^2}\\ &=-\frac {2}{3 a d f (d \tan (e+f x))^{3/2}}+\frac {2}{a d^2 f \sqrt {d \tan (e+f x)}}+\frac {\int \frac {(d \tan (e+f x))^{3/2}}{a+a \tan (e+f x)} \, dx}{d^4}\\ &=-\frac {2}{3 a d f (d \tan (e+f x))^{3/2}}+\frac {2}{a d^2 f \sqrt {d \tan (e+f x)}}+\frac {\int \frac {-a d^2+a d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{2 a^2 d^4}+\frac {\int \frac {1+\tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{2 d^2}\\ &=-\frac {2}{3 a d f (d \tan (e+f x))^{3/2}}+\frac {2}{a d^2 f \sqrt {d \tan (e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{-2 a^2 d^4+d x^2} \, dx,x,\frac {-a d^2-a d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {d x} (a+a x)} \, dx,x,\tan (e+f x)\right )}{2 d^2 f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a d^{5/2} f}-\frac {2}{3 a d f (d \tan (e+f x))^{3/2}}+\frac {2}{a d^2 f \sqrt {d \tan (e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{a+\frac {a x^2}{d}} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{d^3 f}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{a d^{5/2} f}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a d^{5/2} f}-\frac {2}{3 a d f (d \tan (e+f x))^{3/2}}+\frac {2}{a d^2 f \sqrt {d \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.47, size = 130, normalized size = 0.96 \begin {gather*} \frac {-8+24 \tan (e+f x)+12 \text {ArcTan}\left (\sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)+3 \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 ) \tan ^{\frac {3}{2}}(e+f x)}{12 a d 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. \(332\) vs.
\(2(112)=224\).
time = 0.16, size = 333, normalized size = 2.47
method | result | size |
derivativedivides | \(\frac {2 d^{2} \left (-\frac {1}{3 d^{3} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {1}{d^{4} \sqrt {d \tan \left (f x +e \right )}}+\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 d^{\frac {9}{2}}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}}{2 d^{4}}\right )}{f a}\) | \(333\) |
default | \(\frac {2 d^{2} \left (-\frac {1}{3 d^{3} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {1}{d^{4} \sqrt {d \tan \left (f x +e \right )}}+\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 d^{\frac {9}{2}}}+\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}}{2 d^{4}}\right )}{f a}\) | \(333\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 151, normalized size = 1.12 \begin {gather*} -\frac {\frac {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 )}}{a d} - \frac {12 \, \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a d^{\frac {3}{2}}} - \frac {8 \, {\left (3 \, d \tan \left (f x + e\right ) - d\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a d}}{12 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.98, size = 326, normalized size = 2.41 \begin {gather*} \left [\frac {3 \, \sqrt {2} \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 ) \tan \left (f x + e\right )^{2} - 3 \, \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 ) \tan \left (f x + e\right )^{2} + 4 \, \sqrt {d \tan \left (f x + e\right )} {\left (3 \, \tan \left (f x + e\right ) - 1\right )}}{6 \, a d^{3} f \tan \left (f x + e\right )^{2}}, \frac {3 \, \sqrt {2} \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 ) \tan \left (f x + e\right )^{2} + 12 \, \sqrt {d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right ) \tan \left (f x + e\right )^{2} + 8 \, \sqrt {d \tan \left (f x + e\right )} {\left (3 \, \tan \left (f x + e\right ) - 1\right )}}{12 \, a d^{3} f \tan \left (f x + e\right )^{2}}\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 {5}{2}} \tan {\left (e + f x \right )} + \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx}{a} \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. 305 vs.
\(2 (117) = 234\).
time = 0.75, size = 305, normalized size = 2.26 \begin {gather*} -\frac {\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 )}{4 \, a d^{4} f} - \frac {\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 )}{4 \, a d^{4} f} + \frac {\arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a d^{\frac {5}{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 )}{8 \, a d^{4} 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 )}{8 \, a d^{4} f} + \frac {2 \, {\left (3 \, d \tan \left (f x + e\right ) - d\right )}}{3 \, \sqrt {d \tan \left (f x + e\right )} a d^{3} f \tan \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.82, size = 130, normalized size = 0.96 \begin {gather*} \frac {\frac {2\,\mathrm {tan}\left (e+f\,x\right )}{d}-\frac {2}{3\,d}}{a\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}+\frac {\mathrm {atan}\left (\frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{a\,d^{5/2}\,f}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {12\,\sqrt {2}\,a^3\,d^{21/2}\,f^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{12\,a^3\,d^{11}\,f^3+12\,a^3\,d^{11}\,f^3\,\mathrm {tan}\left (e+f\,x\right )}\right )}{2\,a\,d^{5/2}\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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