Integrand size = 15, antiderivative size = 124 \[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=-\frac {1}{4} \sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )+\frac {1}{2} (a+b)^{3/2} \text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\frac {1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}-\frac {1}{6} \left (a+b \tanh ^4(x)\right )^{3/2} \]
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Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3751, 1262, 749, 829, 858, 223, 212, 739} \[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\frac {1}{2} (a+b)^{3/2} \text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\frac {1}{4} \sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )-\frac {1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac {1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)} \]
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Rule 212
Rule 223
Rule 739
Rule 749
Rule 829
Rule 858
Rule 1262
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x \left (a+b x^4\right )^{3/2}}{1-x^2} \, dx,x,\tanh (x)\right ) \\ & = \frac {1}{2} \text {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} \text {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 {\text {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 \text {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)) \text {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 \text {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)) \text {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) \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )+\frac {1}{2} (a+b)^{3/2} \text {arctanh}\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*}
Time = 4.55 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.34 \[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\frac {1}{12} \left (-6 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a+b \tanh ^4(x)}}\right )+6 (a+b)^{3/2} \text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )-\sqrt {a+b \tanh ^4(x)} \left (8 a+6 b+3 b \tanh ^2(x)+2 b \tanh ^4(x)\right )-\frac {3 \sqrt {a} \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b} \tanh ^2(x)}{\sqrt {a}}\right ) \sqrt {a+b \tanh ^4(x)}}{\sqrt {1+\frac {b \tanh ^4(x)}{a}}}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.90 (sec) , antiderivative size = 620, normalized size of antiderivative = 5.00
method | result | size |
derivativedivides | \(-\frac {b \tanh \left (x \right )^{4} \sqrt {a +b \tanh \left (x \right )^{4}}}{6}-\frac {b \tanh \left (x \right )^{2} \sqrt {a +b \tanh \left (x \right )^{4}}}{4}-\frac {2 \sqrt {a +b \tanh \left (x \right )^{4}}\, a}{3}-\frac {b \sqrt {a +b \tanh \left (x \right )^{4}}}{2}-\frac {\left (\frac {5}{3} a b +b^{2}\right ) \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}-\frac {3 \ln \left (2 \sqrt {b}\, \tanh \left (x \right )^{2}+2 \sqrt {a +b \tanh \left (x \right )^{4}}\right ) \sqrt {b}\, a}{4}-\frac {\ln \left (2 \sqrt {b}\, \tanh \left (x \right )^{2}+2 \sqrt {a +b \tanh \left (x \right )^{4}}\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}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}\, \sqrt {b}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}+\frac {a b \,\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{\sqrt {a +b}}+\frac {b^{2} \operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}-\frac {\left (-\frac {5}{3} a b -b^{2}\right ) \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}-\frac {i \left (\frac {7}{5} a b +b^{2}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}\, \sqrt {b}}\) | \(620\) |
default | \(-\frac {b \tanh \left (x \right )^{4} \sqrt {a +b \tanh \left (x \right )^{4}}}{6}-\frac {b \tanh \left (x \right )^{2} \sqrt {a +b \tanh \left (x \right )^{4}}}{4}-\frac {2 \sqrt {a +b \tanh \left (x \right )^{4}}\, a}{3}-\frac {b \sqrt {a +b \tanh \left (x \right )^{4}}}{2}-\frac {\left (\frac {5}{3} a b +b^{2}\right ) \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}-\frac {3 \ln \left (2 \sqrt {b}\, \tanh \left (x \right )^{2}+2 \sqrt {a +b \tanh \left (x \right )^{4}}\right ) \sqrt {b}\, a}{4}-\frac {\ln \left (2 \sqrt {b}\, \tanh \left (x \right )^{2}+2 \sqrt {a +b \tanh \left (x \right )^{4}}\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}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}\, \sqrt {b}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}+\frac {a b \,\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{\sqrt {a +b}}+\frac {b^{2} \operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}-\frac {\left (-\frac {5}{3} a b -b^{2}\right ) \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}}-\frac {i \left (\frac {7}{5} a b +b^{2}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tanh \left (x \right )^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (\tanh \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tanh \left (x \right )^{4}}\, \sqrt {b}}\) | \(620\) |
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Leaf count of result is larger than twice the leaf count of optimal. 2646 vs. \(2 (101) = 202\).
Time = 0.55 (sec) , antiderivative size = 11528, normalized size of antiderivative = 92.97 \[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\text {Too large to display} \]
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\[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\int \left (a + b \tanh ^{4}{\left (x \right )}\right )^{\frac {3}{2}} \tanh {\left (x \right )}\, dx \]
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\[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\int { {\left (b \tanh \left (x\right )^{4} + a\right )}^{\frac {3}{2}} \tanh \left (x\right ) \,d x } \]
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\[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\int { {\left (b \tanh \left (x\right )^{4} + a\right )}^{\frac {3}{2}} \tanh \left (x\right ) \,d x } \]
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Timed out. \[ \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx=\int \mathrm {tanh}\left (x\right )\,{\left (b\,{\mathrm {tanh}\left (x\right )}^4+a\right )}^{3/2} \,d x \]
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