Optimal. Leaf size=118 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{a^{5/2}}-\frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{b^{5/2}}-\frac {(a+b) \tanh ^3(x)}{3 a b \left (a-b \tanh ^2(x)+b\right )^{3/2}} \]
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Rubi [A] time = 0.34, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {4141, 1975, 470, 578, 523, 217, 203, 377, 206} \[ -\frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{a^{5/2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{b^{5/2}}-\frac {(a+b) \tanh ^3(x)}{3 a b \left (a-b \tanh ^2(x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 217
Rule 377
Rule 470
Rule 523
Rule 578
Rule 1975
Rule 4141
Rubi steps
\begin {align*} \int \frac {\tanh ^6(x)}{\left (a+b \text {sech}^2(x)\right )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right ) \left (a+b-b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {(a+b) \tanh ^3(x)}{3 a b \left (a+b-b \tanh ^2(x)\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 (a+b)-3 a x^2\right )}{\left (1-x^2\right ) \left (a+b-b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{3 a b}\\ &=-\frac {(a+b) \tanh ^3(x)}{3 a b \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}-\frac {\operatorname {Subst}\left (\int \frac {3 \left (a^2-b^2\right )-3 a^2 x^2}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )}{3 a^2 b^2}\\ &=-\frac {(a+b) \tanh ^3(x)}{3 a b \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )}{a^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )}{b^2}\\ &=-\frac {(a+b) \tanh ^3(x)}{3 a b \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{a^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{b^2}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{b^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{a^{5/2}}-\frac {(a+b) \tanh ^3(x)}{3 a b \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 178, normalized size = 1.51 \[ \frac {\text {sech}^5(x) \left (\frac {\sqrt {2} (a \cosh (2 x)+a+2 b)^{5/2} \left (b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sinh (x)}{\sqrt {a \cosh (2 x)+a+2 b}}\right )-a^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sinh (x)}{\sqrt {a \cosh (2 x)+a+2 b}}\right )\right )}{a^{5/2} b^{5/2}}+\frac {2 (a+b) \sinh (x) \left (3 a^2+a (3 a-4 b) \cosh (2 x)+4 a b-6 b^2\right ) (a \cosh (2 x)+a+2 b)}{3 a^2 b^2}\right )}{8 \left (a+b \text {sech}^2(x)\right )^{5/2}} \]
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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{6}\relax (x )}{\left (a +b \mathrm {sech}\relax (x )^{2}\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {\tanh \relax (x)^{6}}{{\left (b \operatorname {sech}\relax (x)^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {tanh}\relax (x)}^6}{{\left (a+\frac {b}{{\mathrm {cosh}\relax (x)}^2}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\tanh ^{6}{\relax (x )}}{\left (a + b \operatorname {sech}^{2}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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