Optimal. Leaf size=74 \[ \frac {\sqrt {2} a \tanh ^{-1}\left (\frac {\sqrt {d} \tan (e+f x)+\sqrt {d}}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 a}{d f \sqrt {d \tan (e+f x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3529, 3532, 208} \[ \frac {\sqrt {2} a \tanh ^{-1}\left (\frac {\sqrt {d} \tan (e+f x)+\sqrt {d}}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 a}{d f \sqrt {d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 3529
Rule 3532
Rubi steps
\begin {align*} \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx &=-\frac {2 a}{d f \sqrt {d \tan (e+f x)}}+\frac {\int \frac {a d-a d \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{d^2}\\ &=-\frac {2 a}{d f \sqrt {d \tan (e+f x)}}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-2 a^2 d^2+d x^2} \, dx,x,\frac {a d+a d \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f}\\ &=\frac {\sqrt {2} a \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 a}{d f \sqrt {d \tan (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 64, normalized size = 0.86 \[ -\frac {(1+i) a \left (\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-i \tan (e+f x)\right )-i \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};i \tan (e+f x)\right )\right )}{d f \sqrt {d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 191, normalized size = 2.58 \[ \left [\frac {\sqrt {2} a \sqrt {d} \log \left (\frac {\tan \left (f x + e\right )^{2} + \frac {2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) + 1\right )}}{\sqrt {d}} + 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right ) - 4 \, \sqrt {d \tan \left (f x + e\right )} a}{2 \, d^{2} f \tan \left (f x + e\right )}, -\frac {\sqrt {2} a d \sqrt {-\frac {1}{d}} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-\frac {1}{d}} {\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) + 2 \, \sqrt {d \tan \left (f x + e\right )} a}{d^{2} f \tan \left (f x + e\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.58, size = 253, normalized size = 3.42 \[ -\frac {\frac {8 \, a}{\sqrt {d \tan \left (f x + e\right )} f} - \frac {2 \, \sqrt {2} {\left (a d \sqrt {{\left | d \right |}} - a {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d^{2} f} - \frac {2 \, \sqrt {2} {\left (a d \sqrt {{\left | d \right |}} - a {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d^{2} f} - \frac {\sqrt {2} {\left (a d \sqrt {{\left | d \right |}} + a {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{d^{2} f} + \frac {\sqrt {2} {\left (a d \sqrt {{\left | d \right |}} + a {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{d^{2} f}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 355, normalized size = 4.80 \[ \frac {a \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )}{4 f \,d^{2}}+\frac {a \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 f \,d^{2}}-\frac {a \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 f \,d^{2}}-\frac {a \sqrt {2}\, \ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )}{4 f d \left (d^{2}\right )^{\frac {1}{4}}}-\frac {a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 f d \left (d^{2}\right )^{\frac {1}{4}}}+\frac {a \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 f d \left (d^{2}\right )^{\frac {1}{4}}}-\frac {2 a}{d f \sqrt {d \tan \left (f x +e \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 97, normalized size = 1.31 \[ \frac {a {\left (\frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {4 \, a}{\sqrt {d \tan \left (f x + e\right )}}}{2 \, d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.58, size = 84, normalized size = 1.14 \[ -\frac {2\,a}{d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,\left (-1-\mathrm {i}\right )}{d^{3/2}\,f}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,\left (1-\mathrm {i}\right )}{d^{3/2}\,f} \]
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 \[ a \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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