Optimal. Leaf size=90 \[ \frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f} \]
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Rubi [A] time = 0.10, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3670, 444, 50, 63, 208} \[ \frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 444
Rule 3670
Rubi steps
\begin {align*} \int \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x \left (a+b x^2\right )^{3/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}+\frac {(a-b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}+\frac {(a-b)^2 \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}+\frac {(a-b)^2 \operatorname {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan ^2(e+f x)}\right )}{b f}\\ &=-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 80, normalized size = 0.89 \[ \frac {\sqrt {a+b \tan ^2(e+f x)} \left (4 a+b \tan ^2(e+f x)-3 b\right )-3 (a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 255, normalized size = 2.83 \[ \left [-\frac {3 \, {\left (a - b\right )}^{\frac {3}{2}} \log \left (-\frac {b^{2} \tan \left (f x + e\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 4 \, {\left (b \tan \left (f x + e\right )^{2} + 2 \, a - b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) - 4 \, {\left (b \tan \left (f x + e\right )^{2} + 4 \, a - 3 \, b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{12 \, f}, \frac {3 \, {\left (a - b\right )} \sqrt {-a + b} \arctan \left (\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{b \tan \left (f x + e\right )^{2} + 2 \, a - b}\right ) + 2 \, {\left (b \tan \left (f x + e\right )^{2} + 4 \, a - 3 \, b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{6 \, f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 114, normalized size = 1.27 \[ \frac {{\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {-a + b}}\right )}{\sqrt {-a + b} f} + \frac {{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} f^{2} + 3 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} a f^{2} - 3 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b f^{2}}{3 \, f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 181, normalized size = 2.01 \[ \frac {b \left (\tan ^{2}\left (f x +e \right )\right ) \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{3 f}+\frac {4 a \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{3 f}-\frac {b \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{f}+\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}-\frac {2 a b \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}+\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.99, size = 91, normalized size = 1.01 \[ \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{3\,f}+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (a-b\right )}{f}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,{\left (a-b\right )}^{3/2}}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^{3/2}}{f} \]
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 \[ \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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