Optimal. Leaf size=116 \[ \frac {\left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b f}-\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f} \]
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Rubi [A] time = 0.15, antiderivative size = 116, 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 = {3670, 446, 80, 50, 63, 208} \[ \frac {\left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b f}-\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac {(a-b) \sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 208
Rule 446
Rule 3670
Rubi steps
\begin {align*} \int \tan ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3 \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 {x (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 )^{5/2}}{5 b 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 {\left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b 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 {\left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b 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 {\left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b 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}+\frac {\left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 b f}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 112, normalized size = 0.97 \[ \frac {\sqrt {a+b \tan ^2(e+f x)} \left (3 a^2+b (6 a-5 b) \tan ^2(e+f x)-20 a b+3 b^2 \tan ^4(e+f x)+15 b^2\right )+15 b (a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{15 b f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 334, normalized size = 2.88 \[ \left [-\frac {15 \, {\left (a b - b^{2}\right )} \sqrt {a - b} \log \left (-\frac {b^{2} \tan \left (f x + e\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} - 4 \, {\left (b \tan \left (f x + e\right )^{2} + 2 \, a - b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) - 4 \, {\left (3 \, b^{2} \tan \left (f x + e\right )^{4} + {\left (6 \, a b - 5 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2} - 20 \, a b + 15 \, b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{60 \, b f}, -\frac {15 \, {\left (a b - b^{2}\right )} \sqrt {-a + b} \arctan \left (\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{b \tan \left (f x + e\right )^{2} + 2 \, a - b}\right ) - 2 \, {\left (3 \, b^{2} \tan \left (f x + e\right )^{4} + {\left (6 \, a b - 5 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2} - 20 \, a b + 15 \, b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{30 \, b f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 150, normalized size = 1.29 \[ -\frac {{\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {-a + b}}\right )}{\sqrt {-a + b} f} + \frac {3 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}} b^{4} f^{4} - 5 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} b^{5} f^{4} - 15 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} a b^{5} f^{4} + 15 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b^{6} f^{4}}{15 \, b^{5} f^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 204, normalized size = 1.76 \[ \frac {\left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{5 b f}-\frac {b \left (\tan ^{2}\left (f x +e \right )\right ) \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{3 f}-\frac {4 a \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{3 f}+\frac {b \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{f}-\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}+\frac {2 a b \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}-\frac {a^{2} \arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}} \]
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 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 22.57, size = 156, normalized size = 1.34 \[ \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{5/2}}{5\,b\,f}-\left (\frac {a}{3\,b\,f}-\frac {a-b}{3\,b\,f}\right )\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}-\left (\frac {a}{b\,f}-\frac {a-b}{b\,f}\right )\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (a-b\right )-\frac {\mathrm {atan}\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,{\left (a-b\right )}^{3/2}\,1{}\mathrm {i}}{a^2-2\,a\,b+b^2}\right )\,{\left (a-b\right )}^{3/2}\,1{}\mathrm {i}}{f} \]
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 \[ \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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