Optimal. Leaf size=118 \[ -\frac {3 a^2-b (5 a+2 b) \tanh ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tanh ^4(x)}}-\frac {a-b \tanh ^2(x)}{6 a (a+b) \left (a+b \tanh ^4(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{2 (a+b)^{5/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3670, 1248, 741, 823, 12, 725, 206} \[ -\frac {3 a^2-b (5 a+2 b) \tanh ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tanh ^4(x)}}+\frac {\tanh ^{-1}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a-b \tanh ^2(x)}{6 a (a+b) \left (a+b \tanh ^4(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 725
Rule 741
Rule 823
Rule 1248
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{\left (a+b \tanh ^4(x)\right )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\left (1-x^2\right ) \left (a+b x^4\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1-x) \left (a+b x^2\right )^{5/2}} \, dx,x,\tanh ^2(x)\right )\\ &=-\frac {a-b \tanh ^2(x)}{6 a (a+b) \left (a+b \tanh ^4(x)\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {-3 a-2 b+2 b x}{(1-x) \left (a+b x^2\right )^{3/2}} \, dx,x,\tanh ^2(x)\right )}{6 a (a+b)}\\ &=-\frac {a-b \tanh ^2(x)}{6 a (a+b) \left (a+b \tanh ^4(x)\right )^{3/2}}-\frac {3 a^2-b (5 a+2 b) \tanh ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tanh ^4(x)}}+\frac {\operatorname {Subst}\left (\int \frac {3 a^2 b}{(1-x) \sqrt {a+b x^2}} \, dx,x,\tanh ^2(x)\right )}{6 a^2 b (a+b)^2}\\ &=-\frac {a-b \tanh ^2(x)}{6 a (a+b) \left (a+b \tanh ^4(x)\right )^{3/2}}-\frac {3 a^2-b (5 a+2 b) \tanh ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tanh ^4(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x^2}} \, dx,x,\tanh ^2(x)\right )}{2 (a+b)^2}\\ &=-\frac {a-b \tanh ^2(x)}{6 a (a+b) \left (a+b \tanh ^4(x)\right )^{3/2}}-\frac {3 a^2-b (5 a+2 b) \tanh ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tanh ^4(x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {-a-b \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )}{2 (a+b)^2}\\ &=\frac {\tanh ^{-1}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a-b \tanh ^2(x)}{6 a (a+b) \left (a+b \tanh ^4(x)\right )^{3/2}}-\frac {3 a^2-b (5 a+2 b) \tanh ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tanh ^4(x)}}\\ \end {align*}
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Mathematica [A] time = 0.85, size = 113, normalized size = 0.96 \[ \frac {1}{6} \left (\frac {-3 a^2 b \tanh ^4(x)-a^2 (4 a+b)+b^2 (5 a+2 b) \tanh ^6(x)+3 a b (2 a+b) \tanh ^2(x)}{a^2 (a+b)^2 \left (a+b \tanh ^4(x)\right )^{3/2}}+\frac {3 \tanh ^{-1}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{(a+b)^{5/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \relax (x)}{{\left (b \tanh \relax (x)^{4} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 637, normalized size = 5.40 \[ -\frac {\left (-\frac {\tanh ^{3}\relax (x )}{6 a \left (a +b \right ) b}-\frac {\tanh ^{2}\relax (x )}{6 a \left (a +b \right ) b}-\frac {\tanh \relax (x )}{6 a \left (a +b \right ) b}+\frac {1}{6 \left (a +b \right ) b^{2}}\right ) \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}{2 \left (\tanh ^{4}\relax (x )+\frac {a}{b}\right )^{2}}+\frac {b \left (\frac {\left (3 a +b \right ) \left (\tanh ^{3}\relax (x )\right )}{8 a^{2} \left (a +b \right )^{2}}+\frac {\left (5 a +2 b \right ) \left (\tanh ^{2}\relax (x )\right )}{12 a^{2} \left (a +b \right )^{2}}+\frac {\left (11 a +5 b \right ) \tanh \relax (x )}{24 a^{2} \left (a +b \right )^{2}}-\frac {1}{4 \left (a +b \right )^{2} b}\right )}{\sqrt {\left (\tanh ^{4}\relax (x )+\frac {a}{b}\right ) b}}-\frac {-\frac {\arctanh \left (\frac {2 b \left (\tanh ^{2}\relax (x )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}\right )}{2 \sqrt {a +b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\tanh ^{2}\relax (x )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\tanh ^{2}\relax (x )\right )}{\sqrt {a}}}\, \EllipticPi \left (\tanh \relax (x ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}}{2 \left (a +b \right )^{2}}-\frac {\left (\frac {\tanh ^{3}\relax (x )}{6 a \left (a +b \right ) b}-\frac {\tanh ^{2}\relax (x )}{6 a \left (a +b \right ) b}+\frac {\tanh \relax (x )}{6 a \left (a +b \right ) b}+\frac {1}{6 \left (a +b \right ) b^{2}}\right ) \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}{2 \left (\tanh ^{4}\relax (x )+\frac {a}{b}\right )^{2}}+\frac {b \left (-\frac {\left (3 a +b \right ) \left (\tanh ^{3}\relax (x )\right )}{8 a^{2} \left (a +b \right )^{2}}+\frac {\left (5 a +2 b \right ) \left (\tanh ^{2}\relax (x )\right )}{12 a^{2} \left (a +b \right )^{2}}-\frac {\left (11 a +5 b \right ) \tanh \relax (x )}{24 a^{2} \left (a +b \right )^{2}}-\frac {1}{4 \left (a +b \right )^{2} b}\right )}{\sqrt {\left (\tanh ^{4}\relax (x )+\frac {a}{b}\right ) b}}-\frac {-\frac {\arctanh \left (\frac {2 b \left (\tanh ^{2}\relax (x )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}\right )}{2 \sqrt {a +b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\tanh ^{2}\relax (x )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\tanh ^{2}\relax (x )\right )}{\sqrt {a}}}\, \EllipticPi \left (\tanh \relax (x ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}}{2 \left (a +b \right )^{2}} \]
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 \[ \int \frac {\tanh \relax (x)}{{\left (b \tanh \relax (x)^{4} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\mathrm {tanh}\relax (x)}{{\left (b\,{\mathrm {tanh}\relax (x)}^4+a\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\tanh {\relax (x )}}{\left (a + b \tanh ^{4}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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