Optimal. Leaf size=111 \[ -\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{a f}+\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a f}+\frac {2 d^2 \sqrt {d \tan (e+f x)}}{a f} \]
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Rubi [A] time = 0.41, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3566, 3653, 3532, 205, 3634, 63} \[ -\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{a f}+\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a f}+\frac {2 d^2 \sqrt {d \tan (e+f x)}}{a f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 3532
Rule 3566
Rule 3634
Rule 3653
Rubi steps
\begin {align*} \int \frac {(d \tan (e+f x))^{5/2}}{a+a \tan (e+f x)} \, dx &=\frac {2 d^2 \sqrt {d \tan (e+f x)}}{a f}+\frac {2 \int \frac {-\frac {a d^3}{2}-\frac {1}{2} a d^3 \tan (e+f x)-\frac {1}{2} a d^3 \tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{a}\\ &=\frac {2 d^2 \sqrt {d \tan (e+f x)}}{a f}+\frac {\int \frac {-\frac {1}{2} a^2 d^3-\frac {1}{2} a^2 d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{a^3}-\frac {1}{2} d^3 \int \frac {1+\tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx\\ &=\frac {2 d^2 \sqrt {d \tan (e+f x)}}{a f}-\frac {d^3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {d x} (a+a x)} \, dx,x,\tan (e+f x)\right )}{2 f}-\frac {\left (a d^6\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {a^4 d^6}{2}+d x^2} \, dx,x,\frac {-\frac {1}{2} a^2 d^3+\frac {1}{2} a^2 d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{2 f}\\ &=\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a f}+\frac {2 d^2 \sqrt {d \tan (e+f x)}}{a f}-\frac {d^2 \operatorname {Subst}\left (\int \frac {1}{a+\frac {a x^2}{d}} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{f}\\ &=-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{a f}+\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a f}+\frac {2 d^2 \sqrt {d \tan (e+f x)}}{a f}\\ \end {align*}
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Mathematica [A] time = 0.92, size = 110, normalized size = 0.99 \[ \frac {\left (\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )-2 \tan ^{-1}\left (\sqrt {\tan (e+f x)}\right )+4 \sqrt {\tan (e+f x)}\right ) (d \tan (e+f x))^{5/2}}{2 a f \tan ^{\frac {5}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 247, normalized size = 2.23 \[ \left [\frac {\sqrt {2} \sqrt {-d} d^{2} \log \left (\frac {d \tan \left (f x + e\right )^{2} - 2 \, \sqrt {d \tan \left (f x + e\right )} {\left (\sqrt {2} \tan \left (f x + e\right ) - \sqrt {2}\right )} \sqrt {-d} - 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, \sqrt {-d} d^{2} \log \left (\frac {d \tan \left (f x + e\right ) - 2 \, \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} - d}{\tan \left (f x + e\right ) + 1}\right ) + 8 \, \sqrt {d \tan \left (f x + e\right )} d^{2}}{4 \, a f}, -\frac {\sqrt {2} d^{\frac {5}{2}} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )} {\left (\sqrt {2} \tan \left (f x + e\right ) - \sqrt {2}\right )}}{2 \, \sqrt {d} \tan \left (f x + e\right )}\right ) + 2 \, d^{\frac {5}{2}} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right ) - 4 \, \sqrt {d \tan \left (f x + e\right )} d^{2}}{2 \, a f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.06, size = 284, normalized size = 2.56 \[ -\frac {1}{8} \, d^{2} {\left (\frac {2 \, \sqrt {2} {\left (d \sqrt {{\left | d \right |}} + {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{a d f} + \frac {2 \, \sqrt {2} {\left (d \sqrt {{\left | d \right |}} + {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{a d f} + \frac {8 \, \sqrt {d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a f} + \frac {\sqrt {2} {\left (d \sqrt {{\left | d \right |}} - {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{a d f} - \frac {\sqrt {2} {\left (d \sqrt {{\left | d \right |}} - {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{a d f} - \frac {16 \, \sqrt {d \tan \left (f x + e\right )}}{a f}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 395, normalized size = 3.56 \[ \frac {2 d^{2} \sqrt {d \tan \left (f x +e \right )}}{a f}-\frac {d^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )}{8 f a}-\frac {d^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{4 f a}+\frac {d^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{4 f a}-\frac {d^{3} \sqrt {2}\, \ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )}{8 f a \left (d^{2}\right )^{\frac {1}{4}}}-\frac {d^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{4 f a \left (d^{2}\right )^{\frac {1}{4}}}+\frac {d^{3} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{4 f a \left (d^{2}\right )^{\frac {1}{4}}}-\frac {d^{\frac {5}{2}} \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 130, normalized size = 1.17 \[ -\frac {\frac {d^{4} {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}}\right )}}{a} + \frac {2 \, d^{\frac {7}{2}} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a} - \frac {4 \, \sqrt {d \tan \left (f x + e\right )} d^{3}}{a}}{2 \, d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.45, size = 124, normalized size = 1.12 \[ \frac {2\,d^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{a\,f}-\frac {d^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{a\,f}-\frac {\sqrt {2}\,d^{5/2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}+\frac {\sqrt {2}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{2\,d^{3/2}}\right )\right )}{4\,a\,f} \]
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 \[ \frac {\int \frac {\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}{\tan {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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