Optimal. Leaf size=244 \[ \frac {\sqrt {2} a^2 \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}-\frac {\sqrt {2} a^2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{f}+\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\frac {a^2 \sqrt {d} \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} f}-\frac {a^2 \sqrt {d} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} f} \]
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Rubi [A] time = 0.22, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3543, 12, 16, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {\sqrt {2} a^2 \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}-\frac {\sqrt {2} a^2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{f}+\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\frac {a^2 \sqrt {d} \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} f}-\frac {a^2 \sqrt {d} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} f} \]
Antiderivative was successfully verified.
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Rule 12
Rule 16
Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3473
Rule 3476
Rule 3543
Rubi steps
\begin {align*} \int \sqrt {d \tan (e+f x)} (a+a \tan (e+f x))^2 \, dx &=\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\int 2 a^2 \tan (e+f x) \sqrt {d \tan (e+f x)} \, dx\\ &=\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\left (2 a^2\right ) \int \tan (e+f x) \sqrt {d \tan (e+f x)} \, dx\\ &=\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac {\left (2 a^2\right ) \int (d \tan (e+f x))^{3/2} \, dx}{d}\\ &=\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}-\left (2 a^2 d\right ) \int \frac {1}{\sqrt {d \tan (e+f x)}} \, dx\\ &=\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}-\frac {\left (2 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (d^2+x^2\right )} \, dx,x,d \tan (e+f x)\right )}{f}\\ &=\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}-\frac {\left (4 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{f}\\ &=\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}-\frac {\left (2 a^2 d\right ) \operatorname {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{f}-\frac {\left (2 a^2 d\right ) \operatorname {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{f}\\ &=\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac {\left (a^2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} f}+\frac {\left (a^2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} f}-\frac {\left (a^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{f}-\frac {\left (a^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{f}\\ &=\frac {a^2 \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} f}-\frac {a^2 \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} f}+\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}-\frac {\left (\sqrt {2} a^2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {\left (\sqrt {2} a^2 \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}\\ &=\frac {\sqrt {2} a^2 \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}-\frac {\sqrt {2} a^2 \sqrt {d} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {a^2 \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} f}-\frac {a^2 \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} f}+\frac {4 a^2 \sqrt {d \tan (e+f x)}}{f}+\frac {2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 175, normalized size = 0.72 \[ \frac {a^2 \sqrt {d \tan (e+f x)} \left (6 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-6 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )+4 \tan ^{\frac {3}{2}}(e+f x)+24 \sqrt {\tan (e+f x)}+3 \sqrt {2} \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )-3 \sqrt {2} \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )\right )}{6 f \sqrt {\tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 661, normalized size = 2.71 \[ \frac {12 \, \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {1}{4}} f \arctan \left (-\frac {a^{8} d^{2} + \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {3}{4}} a^{2} f^{3} \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} - \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {3}{4}} f^{3} \sqrt {\frac {a^{4} d \sin \left (f x + e\right ) + \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {1}{4}} a^{2} f \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {\frac {a^{8} d^{2}}{f^{4}}} f^{2} \cos \left (f x + e\right )}{\cos \left (f x + e\right )}}}{a^{8} d^{2}}\right ) \cos \left (f x + e\right ) + 12 \, \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {1}{4}} f \arctan \left (\frac {a^{8} d^{2} - \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {3}{4}} a^{2} f^{3} \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} + \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {3}{4}} f^{3} \sqrt {\frac {a^{4} d \sin \left (f x + e\right ) - \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {1}{4}} a^{2} f \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {\frac {a^{8} d^{2}}{f^{4}}} f^{2} \cos \left (f x + e\right )}{\cos \left (f x + e\right )}}}{a^{8} d^{2}}\right ) \cos \left (f x + e\right ) - 3 \, \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {1}{4}} f \cos \left (f x + e\right ) \log \left (\frac {a^{4} d \sin \left (f x + e\right ) + \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {1}{4}} a^{2} f \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {\frac {a^{8} d^{2}}{f^{4}}} f^{2} \cos \left (f x + e\right )}{\cos \left (f x + e\right )}\right ) + 3 \, \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {1}{4}} f \cos \left (f x + e\right ) \log \left (\frac {a^{4} d \sin \left (f x + e\right ) - \sqrt {2} \left (\frac {a^{8} d^{2}}{f^{4}}\right )^{\frac {1}{4}} a^{2} f \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {\frac {a^{8} d^{2}}{f^{4}}} f^{2} \cos \left (f x + e\right )}{\cos \left (f x + e\right )}\right ) + 4 \, {\left (6 \, a^{2} \cos \left (f x + e\right ) + a^{2} \sin \left (f x + e\right )\right )} \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{6 \, f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 249, normalized size = 1.02 \[ -\frac {\sqrt {2} a^{2} \sqrt {{\left | d \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{f} - \frac {\sqrt {2} a^{2} \sqrt {{\left | d \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{f} - \frac {\sqrt {2} a^{2} \sqrt {{\left | d \right |}} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{2 \, f} + \frac {\sqrt {2} a^{2} \sqrt {{\left | d \right |}} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{2 \, f} + \frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right )} a^{2} d^{3} f^{2} \tan \left (f x + e\right ) + 6 \, \sqrt {d \tan \left (f x + e\right )} a^{2} d^{3} f^{2}\right )}}{3 \, d^{3} f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 204, normalized size = 0.84 \[ \frac {2 a^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 d f}+\frac {4 a^{2} \sqrt {d \tan \left (f x +e \right )}}{f}-\frac {a^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )}{2 f}+\frac {a^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{f}-\frac {a^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 192, normalized size = 0.79 \[ \frac {4 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{2} + 24 \, \sqrt {d \tan \left (f x + e\right )} a^{2} d - 3 \, {\left (2 \, \sqrt {2} d^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right ) + 2 \, \sqrt {2} d^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right ) + \sqrt {2} d^{\frac {3}{2}} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right ) - \sqrt {2} d^{\frac {3}{2}} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )\right )} a^{2}}{6 \, d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.38, size = 104, normalized size = 0.43 \[ \frac {4\,a^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{f}+\frac {2\,a^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{3\,d\,f}+\frac {{\left (-1\right )}^{1/4}\,a^2\,\sqrt {d}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,2{}\mathrm {i}}{f}+\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,\sqrt {d}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}{\sqrt {d}}\right )}{f} \]
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 \[ a^{2} \left (\int \sqrt {d \tan {\left (e + f x \right )}}\, dx + \int 2 \sqrt {d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\, dx + \int \sqrt {d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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