Optimal. Leaf size=109 \[ -\frac {(b-2 a) \tan ^2(x)+3 a}{6 a (a+b)^2 \sqrt {a+b \tan ^4(x)}}-\frac {1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3670, 1252, 823, 12, 725, 206} \[ -\frac {(b-2 a) \tan ^2(x)+3 a}{6 a (a+b)^2 \sqrt {a+b \tan ^4(x)}}-\frac {1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 725
Rule 823
Rule 1252
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tan ^3(x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^2\right ) \left (a+b x^4\right )^{5/2}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(1+x) \left (a+b x^2\right )^{5/2}} \, dx,x,\tan ^2(x)\right )\\ &=-\frac {1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {a b-2 a b x}{(1+x) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan ^2(x)\right )}{6 a b (a+b)}\\ &=-\frac {1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac {3 a-(2 a-b) \tan ^2(x)}{6 a (a+b)^2 \sqrt {a+b \tan ^4(x)}}+\frac {\operatorname {Subst}\left (\int -\frac {3 a^2 b^2}{(1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )}{6 a^2 b^2 (a+b)^2}\\ &=-\frac {1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac {3 a-(2 a-b) \tan ^2(x)}{6 a (a+b)^2 \sqrt {a+b \tan ^4(x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )}{2 (a+b)^2}\\ &=-\frac {1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac {3 a-(2 a-b) \tan ^2(x)}{6 a (a+b)^2 \sqrt {a+b \tan ^4(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {a-b \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^2}\\ &=\frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {1-\tan ^2(x)}{6 (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac {3 a-(2 a-b) \tan ^2(x)}{6 a (a+b)^2 \sqrt {a+b \tan ^4(x)}}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 104, normalized size = 0.95 \[ \frac {1}{6} \left (\frac {3 a^2 \tan ^2(x)+b (2 a-b) \tan ^6(x)-3 a b \tan ^4(x)-a (4 a+b)}{a (a+b)^2 \left (a+b \tan ^4(x)\right )^{3/2}}+\frac {3 \tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{5/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 556, normalized size = 5.10 \[ \left [\frac {3 \, {\left (a b^{2} \tan \relax (x)^{8} + 2 \, a^{2} b \tan \relax (x)^{4} + a^{3}\right )} \sqrt {a + b} \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \relax (x)^{4} - 2 \, a b \tan \relax (x)^{2} - 2 \, \sqrt {b \tan \relax (x)^{4} + a} {\left (b \tan \relax (x)^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \relax (x)^{4} + 2 \, \tan \relax (x)^{2} + 1}\right ) + 2 \, {\left ({\left (2 \, a^{2} b + a b^{2} - b^{3}\right )} \tan \relax (x)^{6} - 3 \, {\left (a^{2} b + a b^{2}\right )} \tan \relax (x)^{4} - 4 \, a^{3} - 5 \, a^{2} b - a b^{2} + 3 \, {\left (a^{3} + a^{2} b\right )} \tan \relax (x)^{2}\right )} \sqrt {b \tan \relax (x)^{4} + a}}{12 \, {\left ({\left (a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right )} \tan \relax (x)^{8} + a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3} + 2 \, {\left (a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right )} \tan \relax (x)^{4}\right )}}, \frac {3 \, {\left (a b^{2} \tan \relax (x)^{8} + 2 \, a^{2} b \tan \relax (x)^{4} + a^{3}\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \relax (x)^{4} + a} {\left (b \tan \relax (x)^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \relax (x)^{4} + a^{2} + a b}\right ) + {\left ({\left (2 \, a^{2} b + a b^{2} - b^{3}\right )} \tan \relax (x)^{6} - 3 \, {\left (a^{2} b + a b^{2}\right )} \tan \relax (x)^{4} - 4 \, a^{3} - 5 \, a^{2} b - a b^{2} + 3 \, {\left (a^{3} + a^{2} b\right )} \tan \relax (x)^{2}\right )} \sqrt {b \tan \relax (x)^{4} + a}}{6 \, {\left ({\left (a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right )} \tan \relax (x)^{8} + a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3} + 2 \, {\left (a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right )} \tan \relax (x)^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.69, size = 597, normalized size = 5.48 \[ \frac {{\left ({\left (\frac {{\left (2 \, a^{7} b^{2} + 11 \, a^{6} b^{3} + 24 \, a^{5} b^{4} + 25 \, a^{4} b^{5} + 10 \, a^{3} b^{6} - 3 \, a^{2} b^{7} - 4 \, a b^{8} - b^{9}\right )} \tan \relax (x)^{2}}{a^{9} b + 8 \, a^{8} b^{2} + 28 \, a^{7} b^{3} + 56 \, a^{6} b^{4} + 70 \, a^{5} b^{5} + 56 \, a^{4} b^{6} + 28 \, a^{3} b^{7} + 8 \, a^{2} b^{8} + a b^{9}} - \frac {3 \, {\left (a^{7} b^{2} + 6 \, a^{6} b^{3} + 15 \, a^{5} b^{4} + 20 \, a^{4} b^{5} + 15 \, a^{3} b^{6} + 6 \, a^{2} b^{7} + a b^{8}\right )}}{a^{9} b + 8 \, a^{8} b^{2} + 28 \, a^{7} b^{3} + 56 \, a^{6} b^{4} + 70 \, a^{5} b^{5} + 56 \, a^{4} b^{6} + 28 \, a^{3} b^{7} + 8 \, a^{2} b^{8} + a b^{9}}\right )} \tan \relax (x)^{2} + \frac {3 \, {\left (a^{8} b + 6 \, a^{7} b^{2} + 15 \, a^{6} b^{3} + 20 \, a^{5} b^{4} + 15 \, a^{4} b^{5} + 6 \, a^{3} b^{6} + a^{2} b^{7}\right )}}{a^{9} b + 8 \, a^{8} b^{2} + 28 \, a^{7} b^{3} + 56 \, a^{6} b^{4} + 70 \, a^{5} b^{5} + 56 \, a^{4} b^{6} + 28 \, a^{3} b^{7} + 8 \, a^{2} b^{8} + a b^{9}}\right )} \tan \relax (x)^{2} - \frac {4 \, a^{8} b + 25 \, a^{7} b^{2} + 66 \, a^{6} b^{3} + 95 \, a^{5} b^{4} + 80 \, a^{4} b^{5} + 39 \, a^{3} b^{6} + 10 \, a^{2} b^{7} + a b^{8}}{a^{9} b + 8 \, a^{8} b^{2} + 28 \, a^{7} b^{3} + 56 \, a^{6} b^{4} + 70 \, a^{5} b^{5} + 56 \, a^{4} b^{6} + 28 \, a^{3} b^{7} + 8 \, a^{2} b^{8} + a b^{9}}}{6 \, {\left (b \tan \relax (x)^{4} + a\right )}^{\frac {3}{2}}} + \frac {\arctan \left (\frac {\sqrt {b} \tan \relax (x)^{2} - \sqrt {b \tan \relax (x)^{4} + a} + \sqrt {b}}{\sqrt {-a - b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a - b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 654, normalized size = 6.00 \[ \frac {\sqrt {a +b \left (\tan ^{4}\relax (x )\right )}\, \left (\tan ^{2}\relax (x )\right ) \left (2 b \left (\tan ^{4}\relax (x )\right )+3 a \right )}{6 a^{2} \left (\left (\tan ^{8}\relax (x )\right ) b^{2}+2 \left (\tan ^{4}\relax (x )\right ) a b +a^{2}\right )}-\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {\left (\tan ^{2}\relax (x )-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\tan ^{2}\relax (x )-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right )^{2} a^{2} \left (\tan ^{2}\relax (x )-\frac {\sqrt {-a b}}{b}\right )}+\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {\left (\tan ^{2}\relax (x )+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\tan ^{2}\relax (x )+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right )^{2} a^{2} \left (\tan ^{2}\relax (x )+\frac {\sqrt {-a b}}{b}\right )}-\frac {\sqrt {\left (\tan ^{2}\relax (x )+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\tan ^{2}\relax (x )+\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}-b \right ) a \sqrt {-a b}\, \left (\tan ^{2}\relax (x )+\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {\sqrt {\left (\tan ^{2}\relax (x )+\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (\tan ^{2}\relax (x )+\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}-b \right ) a^{2} \left (\tan ^{2}\relax (x )+\frac {\sqrt {-a b}}{b}\right )}+\frac {b^{2} \ln \left (\frac {2 a +2 b -2 \left (1+\tan ^{2}\relax (x )\right ) b +2 \sqrt {a +b}\, \sqrt {\left (1+\tan ^{2}\relax (x )\right )^{2} b -2 \left (1+\tan ^{2}\relax (x )\right ) b +a +b}}{1+\tan ^{2}\relax (x )}\right )}{2 \left (\sqrt {-a b}+b \right )^{2} \left (\sqrt {-a b}-b \right )^{2} \sqrt {a +b}}-\frac {\sqrt {\left (\tan ^{2}\relax (x )-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\tan ^{2}\relax (x )-\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}+b \right ) a \sqrt {-a b}\, \left (\tan ^{2}\relax (x )-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (\tan ^{2}\relax (x )-\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (\tan ^{2}\relax (x )-\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}+b \right ) a^{2} \left (\tan ^{2}\relax (x )-\frac {\sqrt {-a b}}{b}\right )} \]
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 \[ \int \frac {\tan \relax (x)^{3}}{{\left (b \tan \relax (x)^{4} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tan}\relax (x)}^3}{{\left (b\,{\mathrm {tan}\relax (x)}^4+a\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\tan ^{3}{\relax (x )}}{\left (a + b \tan ^{4}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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