Optimal. Leaf size=124 \[ -\frac {1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}+\frac {1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\frac {1}{4} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right ) \]
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Rubi [A] time = 0.24, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3670, 1248, 735, 815, 844, 217, 206, 725} \[ \frac {1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\frac {1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{4} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )-\frac {1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 735
Rule 815
Rule 844
Rule 1248
Rule 3670
Rubi steps
\begin {align*} \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx &=\operatorname {Subst}\left (\int \frac {x \left (a+b x^4\right )^{3/2}}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{1-x} \, dx,x,\tanh ^2(x)\right )\\ &=-\frac {1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-a-b x) \sqrt {a+b x^2}}{1-x} \, dx,x,\tanh ^2(x)\right )\\ &=-\frac {1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}-\frac {1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {\operatorname {Subst}\left (\int \frac {-a b (2 a+b)-b^2 (3 a+2 b) x}{(1-x) \sqrt {a+b x^2}} \, dx,x,\tanh ^2(x)\right )}{4 b}\\ &=-\frac {1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}-\frac {1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}+\frac {1}{2} (a+b)^2 \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x^2}} \, dx,x,\tanh ^2(x)\right )-\frac {1}{4} (b (3 a+2 b)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\tanh ^2(x)\right )\\ &=-\frac {1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}-\frac {1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{2} (a+b)^2 \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 )-\frac {1}{4} (b (3 a+2 b)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )\\ &=-\frac {1}{4} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )+\frac {1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\frac {1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}-\frac {1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 4.81, size = 166, normalized size = 1.34 \[ \frac {1}{12} \left (-\sqrt {a+b \tanh ^4(x)} \left (8 a+2 b \tanh ^4(x)+3 b \tanh ^2(x)+6 b\right )+6 (a+b)^{3/2} \tanh ^{-1}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-6 \sqrt {b} (a+b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )-\frac {3 \sqrt {a} \sqrt {b} \sqrt {a+b \tanh ^4(x)} \sinh ^{-1}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a}}\right )}{\sqrt {\frac {b \tanh ^4(x)}{a}+1}}\right ) \]
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 \[ \int {\left (b \tanh \relax (x)^{4} + a\right )}^{\frac {3}{2}} \tanh \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.24, size = 620, normalized size = 5.00 \[ -\frac {b \left (\tanh ^{4}\relax (x )\right ) \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}{6}-\frac {b \left (\tanh ^{2}\relax (x )\right ) \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}{4}-\frac {2 \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}\, a}{3}-\frac {b \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}{2}-\frac {\left (\frac {5}{3} a b +b^{2}\right ) \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}}}\, \EllipticF \left (\tanh \relax (x ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}-\frac {3 \ln \left (2 \sqrt {b}\, \left (\tanh ^{2}\relax (x )\right )+2 \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}\right ) \sqrt {b}\, a}{4}-\frac {\ln \left (2 \sqrt {b}\, \left (\tanh ^{2}\relax (x )\right )+2 \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}\right ) b^{\frac {3}{2}}}{2}-\frac {i \left (-\frac {7}{5} a b -b^{2}\right ) \sqrt {a}\, \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}}}\, \left (\EllipticF \left (\tanh \relax (x ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (\tanh \relax (x ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}\, \sqrt {b}}+\frac {a^{2} \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 {a b \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 )}{\sqrt {a +b}}+\frac {b^{2} \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 {\left (-\frac {5}{3} a b -b^{2}\right ) \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}}}\, \EllipticF \left (\tanh \relax (x ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}}-\frac {i \left (\frac {7}{5} a b +b^{2}\right ) \sqrt {a}\, \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}}}\, \left (\EllipticF \left (\tanh \relax (x ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (\tanh \relax (x ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\tanh ^{4}\relax (x )\right )}\, \sqrt {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 \[ \int {\left (b \tanh \relax (x)^{4} + a\right )}^{\frac {3}{2}} \tanh \relax (x)\,{d x} \]
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 \[ \int \mathrm {tanh}\relax (x)\,{\left (b\,{\mathrm {tanh}\relax (x)}^4+a\right )}^{3/2} \,d x \]
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 \[ \int \left (a + b \tanh ^{4}{\relax (x )}\right )^{\frac {3}{2}} \tanh {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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