Optimal. Leaf size=161 \[ \frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 f}-\frac {3 \sqrt {d \tan (e+f x)}}{8 f \left (a^3 \tan (e+f x)+a^3\right )}+\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \tan (e+f x)+\sqrt {d}}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 f}-\frac {\sqrt {d \tan (e+f x)}}{4 a f (a \tan (e+f x)+a)^2} \]
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Rubi [A] time = 0.54, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3568, 3649, 3654, 3532, 208, 3634, 63, 205} \[ \frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 f}-\frac {3 \sqrt {d \tan (e+f x)}}{8 f \left (a^3 \tan (e+f x)+a^3\right )}+\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \tan (e+f x)+\sqrt {d}}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 f}-\frac {\sqrt {d \tan (e+f x)}}{4 a f (a \tan (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 208
Rule 3532
Rule 3568
Rule 3634
Rule 3649
Rule 3654
Rubi steps
\begin {align*} \int \frac {\sqrt {d \tan (e+f x)}}{(a+a \tan (e+f x))^3} \, dx &=-\frac {\sqrt {d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}-\frac {\int \frac {-\frac {a d}{2}-2 a d \tan (e+f x)+\frac {3}{2} a d \tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))^2} \, dx}{4 a^2}\\ &=-\frac {\sqrt {d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}-\frac {3 \sqrt {d \tan (e+f x)}}{8 f \left (a^3+a^3 \tan (e+f x)\right )}-\frac {\int \frac {-\frac {5}{2} a^3 d^2+\frac {3}{2} a^3 d^2 \tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{8 a^5 d}\\ &=-\frac {\sqrt {d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}-\frac {3 \sqrt {d \tan (e+f x)}}{8 f \left (a^3+a^3 \tan (e+f x)\right )}-\frac {\int \frac {-4 a^4 d^2+4 a^4 d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{16 a^7 d}+\frac {d \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 {\sqrt {d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}-\frac {3 \sqrt {d \tan (e+f x)}}{8 f \left (a^3+a^3 \tan (e+f x)\right )}+\frac {d \operatorname {Subst}\left (\int \frac {1}{\sqrt {d x} (a+a x)} \, dx,x,\tan (e+f x)\right )}{16 a^2 f}+\frac {\left (2 a d^3\right ) \operatorname {Subst}\left (\int \frac {1}{-32 a^8 d^4+d x^2} \, dx,x,\frac {-4 a^4 d^2-4 a^4 d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f}\\ &=\frac {\sqrt {d} \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 {\sqrt {d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}-\frac {3 \sqrt {d \tan (e+f x)}}{8 f \left (a^3+a^3 \tan (e+f x)\right )}+\frac {\operatorname {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 {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 f}+\frac {\sqrt {d} \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 {\sqrt {d \tan (e+f x)}}{4 a f (a+a \tan (e+f x))^2}-\frac {3 \sqrt {d \tan (e+f x)}}{8 f \left (a^3+a^3 \tan (e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.86, size = 253, normalized size = 1.57 \[ -\frac {\sqrt {d \tan (e+f x)} \left (5 \sqrt {\tan (e+f x)}+2 \sqrt {2} \log \left (-\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}-1\right )-2 \sqrt {2} \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )-2 (\sin (2 (e+f x))+1) \tan ^{-1}\left (\sqrt {\tan (e+f x)}\right )+5 \cos (2 (e+f x)) \sqrt {\tan (e+f x)}+\sin (2 (e+f x)) \left (3 \sqrt {\tan (e+f x)}+2 \sqrt {2} \left (\log \left (-\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}-1\right )-\log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )\right )\right )\right )}{16 a^3 f \sqrt {\tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 392, normalized size = 2.43 \[ \left [-\frac {4 \, {\left (\sqrt {2} \tan \left (f x + e\right )^{2} + 2 \, \sqrt {2} \tan \left (f x + e\right ) + \sqrt {2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )} {\left (\sqrt {2} \tan \left (f x + e\right ) + \sqrt {2}\right )} \sqrt {-d}}{2 \, d \tan \left (f x + e\right )}\right ) - {\left (\tan \left (f x + e\right )^{2} + 2 \, \tan \left (f x + e\right ) + 1\right )} \sqrt {-d} \log \left (\frac {d \tan \left (f x + e\right ) + 2 \, \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} - d}{\tan \left (f x + e\right ) + 1}\right ) + 2 \, \sqrt {d \tan \left (f x + e\right )} {\left (3 \, \tan \left (f x + e\right ) + 5\right )}}{16 \, {\left (a^{3} f \tan \left (f x + e\right )^{2} + 2 \, a^{3} f \tan \left (f x + e\right ) + a^{3} f\right )}}, \frac {{\left (\tan \left (f x + e\right )^{2} + 2 \, \tan \left (f x + e\right ) + 1\right )} \sqrt {d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right ) + {\left (\sqrt {2} \tan \left (f x + e\right )^{2} + 2 \, \sqrt {2} \tan \left (f x + e\right ) + \sqrt {2}\right )} \sqrt {d} \log \left (\frac {d \tan \left (f x + e\right )^{2} + 2 \, \sqrt {d \tan \left (f x + e\right )} {\left (\sqrt {2} \tan \left (f x + e\right ) + \sqrt {2}\right )} \sqrt {d} + 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) - \sqrt {d \tan \left (f x + e\right )} {\left (3 \, \tan \left (f x + e\right ) + 5\right )}}{8 \, {\left (a^{3} f \tan \left (f x + e\right )^{2} + 2 \, a^{3} f \tan \left (f x + e\right ) + a^{3} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.63, size = 314, normalized size = 1.95 \[ \frac {\frac {2 \, \sqrt {2} {\left (d \sqrt {{\left | d \right |}} - {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{a^{3} f} + \frac {2 \, \sqrt {2} {\left (d \sqrt {{\left | d \right |}} - {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{a^{3} f} + \frac {2 \, d^{\frac {3}{2}} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{3} f} + \frac {\sqrt {2} {\left (d \sqrt {{\left | d \right |}} + {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{a^{3} f} - \frac {\sqrt {2} {\left (d \sqrt {{\left | d \right |}} + {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{a^{3} f} - \frac {2 \, {\left (3 \, \sqrt {d \tan \left (f x + e\right )} d^{3} \tan \left (f x + e\right ) + 5 \, \sqrt {d \tan \left (f x + e\right )} d^{3}\right )}}{{\left (d \tan \left (f x + e\right ) + d\right )}^{2} a^{3} f}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 423, normalized size = 2.63 \[ \frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )}{16 f \,a^{3}}+\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{8 f \,a^{3}}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{8 f \,a^{3}}-\frac {d \sqrt {2}\, \ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )}{16 f \,a^{3} \left (d^{2}\right )^{\frac {1}{4}}}-\frac {d \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{8 f \,a^{3} \left (d^{2}\right )^{\frac {1}{4}}}+\frac {d \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{8 f \,a^{3} \left (d^{2}\right )^{\frac {1}{4}}}-\frac {3 d \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{8 f \,a^{3} \left (d \tan \left (f x +e \right )+d \right )^{2}}-\frac {5 d^{2} \sqrt {d \tan \left (f x +e \right )}}{8 f \,a^{3} \left (d \tan \left (f x +e \right )+d \right )^{2}}+\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right ) \sqrt {d}}{8 a^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 184, normalized size = 1.14 \[ -\frac {\frac {3 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} d^{2} + 5 \, \sqrt {d \tan \left (f x + e\right )} d^{3}}{a^{3} d^{2} \tan \left (f x + e\right )^{2} + 2 \, a^{3} d^{2} \tan \left (f x + e\right ) + a^{3} d^{2}} - \frac {d^{2} {\left (\frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )}}{a^{3}} - \frac {d^{\frac {3}{2}} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{3}}}{8 \, d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 152, normalized size = 0.94 \[ \frac {\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{8\,a^3\,f}-\frac {\frac {3\,d\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{8}+\frac {5\,d^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{8}}{f\,a^3\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2+2\,f\,a^3\,d^2\,\mathrm {tan}\left (e+f\,x\right )+f\,a^3\,d^2}+\frac {\sqrt {2}\,\sqrt {d}\,\mathrm {atanh}\left (\frac {9\,\sqrt {2}\,d^{17/2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{32\,\left (\frac {9\,d^9\,\mathrm {tan}\left (e+f\,x\right )}{32}+\frac {9\,d^9}{32}\right )}\right )}{4\,a^3\,f} \]
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 \[ \frac {\int \frac {\sqrt {d \tan {\left (e + f x \right )}}}{\tan ^{3}{\left (e + f x \right )} + 3 \tan ^{2}{\left (e + f x \right )} + 3 \tan {\left (e + f x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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