Optimal. Leaf size=224 \[ -\frac {\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}+\frac {2 a b \text {sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {2 a^5 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}-\frac {\left (2 a^4-3 a^2 b^2-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}+\frac {8 a^3 b^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {4 a^3 b \text {sech}(x)}{\left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.43, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {2731, 2664, 12, 2660, 618, 206, 2669, 3767, 8} \[ -\frac {2 a^5 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac {8 a^3 b^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {\left (-3 a^2 b^2+2 a^4-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}+\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}+\frac {2 a b \text {sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {4 a^3 b \text {sech}(x)}{\left (a^2+b^2\right )^3}-\frac {a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 206
Rule 618
Rule 2660
Rule 2664
Rule 2669
Rule 2731
Rule 3767
Rubi steps
\begin {align*} \int \frac {\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx &=\int \left (\frac {a^4}{\left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac {4 a^3 b^2}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac {\text {sech}^4(x) \left (a^2 \left (1-\frac {b^2}{a^2}\right )-2 a b \sinh (x)\right )}{\left (a^2+b^2\right )^2}+\frac {\text {sech}^2(x) \left (-2 a^4 \left (1-\frac {3 a^2 b^2+b^4}{2 a^4}\right )+4 a^3 b \sinh (x)\right )}{\left (a^2+b^2\right )^3}\right ) \, dx\\ &=\frac {\int \text {sech}^2(x) \left (-2 a^4 \left (1-\frac {3 a^2 b^2+b^4}{2 a^4}\right )+4 a^3 b \sinh (x)\right ) \, dx}{\left (a^2+b^2\right )^3}-\frac {\left (4 a^3 b^2\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^3}+\frac {\int \text {sech}^4(x) \left (a^2 \left (1-\frac {b^2}{a^2}\right )-2 a b \sinh (x)\right ) \, dx}{\left (a^2+b^2\right )^2}+\frac {a^4 \int \frac {1}{(a+b \sinh (x))^2} \, dx}{\left (a^2+b^2\right )^2}\\ &=-\frac {4 a^3 b \text {sech}(x)}{\left (a^2+b^2\right )^3}+\frac {2 a b \text {sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac {a^4 \int \frac {a}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^3}-\frac {\left (8 a^3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^3}+\frac {\left (a^2-b^2\right ) \int \text {sech}^4(x) \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (2 a^4-3 a^2 b^2-b^4\right ) \int \text {sech}^2(x) \, dx}{\left (a^2+b^2\right )^3}\\ &=-\frac {4 a^3 b \text {sech}(x)}{\left (a^2+b^2\right )^3}+\frac {2 a b \text {sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac {a^5 \int \frac {1}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (16 a^3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^3}+\frac {\left (i \left (a^2-b^2\right )\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (x)\right )}{\left (a^2+b^2\right )^2}-\frac {\left (i \left (2 a^4-3 a^2 b^2-b^4\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (x))}{\left (a^2+b^2\right )^3}\\ &=\frac {8 a^3 b^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {4 a^3 b \text {sech}(x)}{\left (a^2+b^2\right )^3}+\frac {2 a b \text {sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}-\frac {\left (2 a^4-3 a^2 b^2-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}-\frac {\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac {\left (2 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^3}\\ &=\frac {8 a^3 b^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {4 a^3 b \text {sech}(x)}{\left (a^2+b^2\right )^3}+\frac {2 a b \text {sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}-\frac {\left (2 a^4-3 a^2 b^2-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}-\frac {\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {\left (4 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^3}\\ &=-\frac {2 a^5 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac {8 a^3 b^2 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {4 a^3 b \text {sech}(x)}{\left (a^2+b^2\right )^3}+\frac {2 a b \text {sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}-\frac {\left (2 a^4-3 a^2 b^2-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}-\frac {\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 144, normalized size = 0.64 \[ \frac {-\frac {3 a^4 b \cosh (x)}{a+b \sinh (x)}-12 a^3 b \text {sech}(x)+\left (a^2+b^2\right ) \text {sech}^3(x) \left (\left (a^2-b^2\right ) \sinh (x)+2 a b\right )+\left (-4 a^4+9 a^2 b^2+b^4\right ) \tanh (x)+\frac {6 a^3 \left (a^2-4 b^2\right ) \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}}{3 \left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 3534, normalized size = 15.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 292, normalized size = 1.30 \[ \frac {{\left (a^{5} - 4 \, a^{3} b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a^{5} e^{x} - a^{4} b\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} - \frac {2 \, {\left (12 \, a^{3} b e^{\left (5 \, x\right )} - 6 \, a^{4} e^{\left (4 \, x\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, x\right )} + 3 \, b^{4} e^{\left (4 \, x\right )} + 16 \, a^{3} b e^{\left (3 \, x\right )} - 8 \, a b^{3} e^{\left (3 \, x\right )} - 6 \, a^{4} e^{\left (2 \, x\right )} + 18 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 12 \, a^{3} b e^{x} - 4 \, a^{4} + 9 \, a^{2} b^{2} + b^{4}\right )}}{3 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 262, normalized size = 1.17 \[ -\frac {2 a^{3} \left (\frac {-b^{2} \tanh \left (\frac {x}{2}\right )-a b}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a}-\frac {\left (a^{2}-4 b^{2}\right ) \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {2 \left (-a^{4}+3 a^{2} b^{2}\right ) \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )+2 \left (-2 a^{3} b +2 a \,b^{3}\right ) \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )+2 \left (-\frac {10}{3} a^{4}+6 a^{2} b^{2}+\frac {4}{3} b^{4}\right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )-16 a^{3} b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \left (-a^{4}+3 a^{2} b^{2}\right ) \tanh \left (\frac {x}{2}\right )-\frac {20 a^{3} b}{3}+\frac {4 a \,b^{3}}{3}}{\left (a^{2}+b^{2}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 523, normalized size = 2.33 \[ \frac {{\left (a^{2} - 4 \, b^{2}\right )} a^{3} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (7 \, a^{4} b - 9 \, a^{2} b^{3} - b^{5} + {\left (11 \, a^{5} - 6 \, a^{3} b^{2} - 2 \, a b^{4}\right )} e^{\left (-x\right )} + {\left (35 \, a^{4} b - 9 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-2 \, x\right )} + {\left (21 \, a^{5} - 32 \, a^{3} b^{2} - 8 \, a b^{4}\right )} e^{\left (-3 \, x\right )} + {\left (41 \, a^{4} b - 7 \, a^{2} b^{3} - 3 \, b^{5}\right )} e^{\left (-4 \, x\right )} + {\left (21 \, a^{5} - 22 \, a^{3} b^{2} + 2 \, a b^{4}\right )} e^{\left (-5 \, x\right )} + 3 \, {\left (7 \, a^{4} b + 3 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-6 \, x\right )} + 3 \, {\left (a^{5} - 4 \, a^{3} b^{2}\right )} e^{\left (-7 \, x\right )}\right )}}{3 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-x\right )} + 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-3 \, x\right )} + 6 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-5 \, x\right )} - 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-6 \, x\right )} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-7 \, x\right )} - {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-8 \, x\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 543, normalized size = 2.42 \[ \frac {\frac {8\,\left (a^2-b^2\right )}{3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {16\,a\,b\,{\mathrm {e}}^x}{3\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {4\,\left (a^6+a^4\,b^2-a^2\,b^4-b^6\right )}{{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}-\frac {16\,{\mathrm {e}}^x\,\left (a^5\,b+2\,a^3\,b^3+a\,b^5\right )}{3\,{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {\frac {2\,\left (a^6\,b^5+a^4\,b^7\right )}{b^3\,\left (a^2\,b+b^3\right )\,{\left (a^2+b^2\right )}^3}-\frac {2\,{\mathrm {e}}^x\,\left (a^7\,b^5+a^5\,b^7\right )}{b^4\,\left (a^2\,b+b^3\right )\,{\left (a^2+b^2\right )}^3}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {\frac {2\,\left (-2\,a^6+a^4\,b^2+4\,a^2\,b^4+b^6\right )}{{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}+\frac {8\,{\mathrm {e}}^x\,\left (a^5\,b+a^3\,b^3\right )}{{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x\,\left (a^5-4\,a^3\,b^2\right )}{b\,{\left (a^2+b^2\right )}^3}-\frac {2\,\left (a^5-4\,a^3\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b\,{\left (a^2+b^2\right )}^{7/2}}\right )\,\left (a^5-4\,a^3\,b^2\right )}{{\left (a^2+b^2\right )}^{7/2}}+\frac {\ln \left (\frac {2\,\left (a^5-4\,a^3\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b\,{\left (a^2+b^2\right )}^{7/2}}-\frac {2\,{\mathrm {e}}^x\,\left (a^5-4\,a^3\,b^2\right )}{b\,{\left (a^2+b^2\right )}^3}\right )\,\left (a^5-4\,a^3\,b^2\right )}{{\left (a^2+b^2\right )}^{7/2}} \]
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 \[ \int \frac {\tanh ^{4}{\relax (x )}}{\left (a + b \sinh {\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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