Optimal. Leaf size=159 \[ \frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {2 \sqrt {a^2+b^2} \left (a^2+4 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))} \]
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Rubi [A]
time = 0.48, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2803, 3134,
3080, 3855, 2739, 632, 212} \begin {gather*} -\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {2 \sqrt {a^2+b^2} \left (a^2+4 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5}+\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 2739
Rule 2803
Rule 3080
Rule 3134
Rule 3855
Rubi steps
\begin {align*} \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx &=-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}-\frac {\int \frac {\text {csch}^3(x) \left (6 \left (a^2+2 b^2\right )-a b \sinh (x)+\left (3 a^2+8 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{3 a^2 b}\\ &=\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}-\frac {i \int \frac {\text {csch}^2(x) \left (2 i b \left (7 a^2+12 b^2\right )-4 i a b^2 \sinh (x)+6 i b \left (a^2+2 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{6 a^3 b}\\ &=-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}+\frac {\int \frac {\text {csch}(x) \left (-6 b^2 \left (3 a^2+4 b^2\right )+6 a b \left (a^2+2 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{6 a^4 b}\\ &=-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}+\frac {\left (\left (a^2+b^2\right ) \left (a^2+4 b^2\right )\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{a^5}-\frac {\left (b \left (3 a^2+4 b^2\right )\right ) \int \text {csch}(x) \, dx}{a^5}\\ &=\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}+\frac {\left (2 \left (a^2+b^2\right ) \left (a^2+4 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^5}\\ &=\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}-\frac {\left (4 \left (a^2+b^2\right ) \left (a^2+4 b^2\right )\right ) \text {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 )}{a^5}\\ &=\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {2 \sqrt {a^2+b^2} \left (a^2+4 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 214, normalized size = 1.35 \begin {gather*} \frac {\frac {48 \left (a^4+5 a^2 b^2+4 b^4\right ) \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-4 a \left (4 a^2+9 b^2\right ) \coth \left (\frac {x}{2}\right )+6 a^2 b \text {csch}^2\left (\frac {x}{2}\right )-24 b \left (3 a^2+4 b^2\right ) \log \left (\tanh \left (\frac {x}{2}\right )\right )+6 a^2 b \text {sech}^2\left (\frac {x}{2}\right )+8 a^3 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-\frac {1}{2} a^3 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)-\frac {24 a b \left (a^2+b^2\right ) \cosh (x)}{a+b \sinh (x)}-4 a \left (4 a^2+9 b^2\right ) \tanh \left (\frac {x}{2}\right )}{24 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 221, normalized size = 1.39
method | result | size |
default | \(-\frac {\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) a^{2}}{3}+2 a b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+5 a^{2} \tanh \left (\frac {x}{2}\right )+12 b^{2} \tanh \left (\frac {x}{2}\right )}{8 a^{4}}-\frac {1}{24 a^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {5 a^{2}+12 b^{2}}{8 a^{4} \tanh \left (\frac {x}{2}\right )}+\frac {b}{4 a^{3} \tanh \left (\frac {x}{2}\right )^{2}}-\frac {b \left (3 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{5}}-\frac {2 \left (\frac {-b^{2} \left (a^{2}+b^{2}\right ) \tanh \left (\frac {x}{2}\right )-b a \left (a^{2}+b^{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (a^{4}+5 a^{2} b^{2}+4 b^{4}\right ) \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{a^{5}}\) | \(221\) |
risch | \(\frac {2 a^{3} {\mathrm e}^{7 x}+4 a \,b^{2} {\mathrm e}^{7 x}-2 a^{2} b \,{\mathrm e}^{6 x}-8 b^{3} {\mathrm e}^{6 x}-14 a^{3} {\mathrm e}^{5 x}-20 a \,b^{2} {\mathrm e}^{5 x}+14 a^{2} b \,{\mathrm e}^{4 x}+24 b^{3} {\mathrm e}^{4 x}+14 a^{3} {\mathrm e}^{3 x}+28 a \,b^{2} {\mathrm e}^{3 x}-\frac {50 a^{2} b \,{\mathrm e}^{2 x}}{3}-24 b^{3} {\mathrm e}^{2 x}-\frac {22 a^{3} {\mathrm e}^{x}}{3}-12 a \,b^{2} {\mathrm e}^{x}+\frac {14 a^{2} b}{3}+8 b^{3}}{a^{4} \left ({\mathrm e}^{2 x}-1\right )^{3} \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}+\frac {3 b \ln \left ({\mathrm e}^{x}+1\right )}{a^{3}}+\frac {4 b^{3} \ln \left ({\mathrm e}^{x}+1\right )}{a^{5}}-\frac {3 b \ln \left ({\mathrm e}^{x}-1\right )}{a^{3}}-\frac {4 b^{3} \ln \left ({\mathrm e}^{x}-1\right )}{a^{5}}+\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{a^{3}}+\frac {4 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right ) b^{2}}{a^{5}}-\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{a^{3}}-\frac {4 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right ) b^{2}}{a^{5}}\) | \(364\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 339 vs.
\(2 (149) = 298\).
time = 0.49, size = 339, normalized size = 2.13 \begin {gather*} -\frac {2 \, {\left (7 \, a^{2} b + 12 \, b^{3} + {\left (11 \, a^{3} + 18 \, a b^{2}\right )} e^{\left (-x\right )} - {\left (25 \, a^{2} b + 36 \, b^{3}\right )} e^{\left (-2 \, x\right )} - 21 \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (7 \, a^{2} b + 12 \, b^{3}\right )} e^{\left (-4 \, x\right )} + 3 \, {\left (7 \, a^{3} + 10 \, a b^{2}\right )} e^{\left (-5 \, x\right )} - 3 \, {\left (a^{2} b + 4 \, b^{3}\right )} e^{\left (-6 \, x\right )} - 3 \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (-7 \, x\right )}\right )}}{3 \, {\left (2 \, a^{5} e^{\left (-x\right )} - 4 \, a^{4} b e^{\left (-2 \, x\right )} - 6 \, a^{5} e^{\left (-3 \, x\right )} + 6 \, a^{4} b e^{\left (-4 \, x\right )} + 6 \, a^{5} e^{\left (-5 \, x\right )} - 4 \, a^{4} b e^{\left (-6 \, x\right )} - 2 \, a^{5} e^{\left (-7 \, x\right )} + a^{4} b e^{\left (-8 \, x\right )} + a^{4} b\right )}} + \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{5}} - \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{5}} + \frac {{\left (a^{4} + 5 \, a^{2} b^{2} + 4 \, b^{4}\right )} \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 )}{\sqrt {a^{2} + b^{2}} a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3648 vs.
\(2 (149) = 298\).
time = 0.53, size = 3648, normalized size = 22.94 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {\coth ^{4}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 242, normalized size = 1.52 \begin {gather*} \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{a^{5}} - \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{5}} + \frac {{\left (a^{4} + 5 \, a^{2} b^{2} + 4 \, b^{4}\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 )}{\sqrt {a^{2} + b^{2}} a^{5}} + \frac {2 \, {\left (a^{3} e^{x} + a b^{2} e^{x} - a^{2} b - b^{3}\right )}}{{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )} a^{4}} + \frac {2 \, {\left (3 \, a b e^{\left (5 \, x\right )} - 6 \, a^{2} e^{\left (4 \, x\right )} - 9 \, b^{2} e^{\left (4 \, x\right )} + 6 \, a^{2} e^{\left (2 \, x\right )} + 18 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} - 4 \, a^{2} - 9 \, b^{2}\right )}}{3 \, a^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.53, size = 1450, normalized size = 9.12 \begin {gather*} \frac {3\,b\,\ln \left (96\,a^4+128\,b^4+224\,a^2\,b^2+96\,a^4\,{\mathrm {e}}^x+128\,b^4\,{\mathrm {e}}^x+224\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^3}-\frac {4}{a^2\,{\mathrm {e}}^{2\,x}-a^2}-\frac {6\,b^2}{a^4\,{\mathrm {e}}^{2\,x}-a^4}-\frac {8}{3\,\left (3\,a^2\,{\mathrm {e}}^{2\,x}-3\,a^2\,{\mathrm {e}}^{4\,x}+a^2\,{\mathrm {e}}^{6\,x}-a^2\right )}-\frac {4\,a^3\,b^7}{a^5\,b^7\,{\mathrm {e}}^{2\,x}-a^7\,b^5-a^5\,b^7+a^7\,b^5\,{\mathrm {e}}^{2\,x}+2\,a^6\,b^6\,{\mathrm {e}}^x+2\,a^8\,b^4\,{\mathrm {e}}^x}-\frac {2\,a^5\,b^5}{a^5\,b^7\,{\mathrm {e}}^{2\,x}-a^7\,b^5-a^5\,b^7+a^7\,b^5\,{\mathrm {e}}^{2\,x}+2\,a^6\,b^6\,{\mathrm {e}}^x+2\,a^8\,b^4\,{\mathrm {e}}^x}-\frac {3\,b\,\ln \left (96\,a^4+128\,b^4+224\,a^2\,b^2-96\,a^4\,{\mathrm {e}}^x-128\,b^4\,{\mathrm {e}}^x-224\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^3}-\frac {4}{a^2\,{\mathrm {e}}^{4\,x}-2\,a^2\,{\mathrm {e}}^{2\,x}+a^2}-\frac {4\,b^3\,\ln \left (96\,a^4+128\,b^4+224\,a^2\,b^2-96\,a^4\,{\mathrm {e}}^x-128\,b^4\,{\mathrm {e}}^x-224\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^5}+\frac {4\,b^3\,\ln \left (96\,a^4+128\,b^4+224\,a^2\,b^2+96\,a^4\,{\mathrm {e}}^x+128\,b^4\,{\mathrm {e}}^x+224\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^5}+\frac {\ln \left (128\,a^6\,{\mathrm {e}}^x-256\,a\,b^5-64\,a^5\,b-320\,a^3\,b^3-128\,b^5\,\sqrt {a^2+b^2}+128\,b^6\,{\mathrm {e}}^x-288\,a^2\,b^3\,\sqrt {a^2+b^2}+128\,a^5\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+672\,a^2\,b^4\,{\mathrm {e}}^x+672\,a^4\,b^2\,{\mathrm {e}}^x-64\,a^4\,b\,\sqrt {a^2+b^2}+384\,a\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+608\,a^3\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^3}-\frac {\ln \left (128\,b^5\,\sqrt {a^2+b^2}-256\,a\,b^5-64\,a^5\,b-320\,a^3\,b^3+128\,a^6\,{\mathrm {e}}^x+128\,b^6\,{\mathrm {e}}^x+288\,a^2\,b^3\,\sqrt {a^2+b^2}-128\,a^5\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+672\,a^2\,b^4\,{\mathrm {e}}^x+672\,a^4\,b^2\,{\mathrm {e}}^x+64\,a^4\,b\,\sqrt {a^2+b^2}-384\,a\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}-608\,a^3\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^3}-\frac {2\,a\,b^9}{a^5\,b^7\,{\mathrm {e}}^{2\,x}-a^7\,b^5-a^5\,b^7+a^7\,b^5\,{\mathrm {e}}^{2\,x}+2\,a^6\,b^6\,{\mathrm {e}}^x+2\,a^8\,b^4\,{\mathrm {e}}^x}+\frac {4\,b\,{\mathrm {e}}^x}{a^3\,{\mathrm {e}}^{4\,x}-2\,a^3\,{\mathrm {e}}^{2\,x}+a^3}+\frac {2\,b\,{\mathrm {e}}^x}{a^3\,{\mathrm {e}}^{2\,x}-a^3}+\frac {4\,b^2\,\ln \left (128\,a^6\,{\mathrm {e}}^x-256\,a\,b^5-64\,a^5\,b-320\,a^3\,b^3-128\,b^5\,\sqrt {a^2+b^2}+128\,b^6\,{\mathrm {e}}^x-288\,a^2\,b^3\,\sqrt {a^2+b^2}+128\,a^5\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+672\,a^2\,b^4\,{\mathrm {e}}^x+672\,a^4\,b^2\,{\mathrm {e}}^x-64\,a^4\,b\,\sqrt {a^2+b^2}+384\,a\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+608\,a^3\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^5}-\frac {4\,b^2\,\ln \left (128\,b^5\,\sqrt {a^2+b^2}-256\,a\,b^5-64\,a^5\,b-320\,a^3\,b^3+128\,a^6\,{\mathrm {e}}^x+128\,b^6\,{\mathrm {e}}^x+288\,a^2\,b^3\,\sqrt {a^2+b^2}-128\,a^5\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+672\,a^2\,b^4\,{\mathrm {e}}^x+672\,a^4\,b^2\,{\mathrm {e}}^x+64\,a^4\,b\,\sqrt {a^2+b^2}-384\,a\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}-608\,a^3\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^5}+\frac {2\,a^2\,b^9\,{\mathrm {e}}^x}{a^5\,b^8\,{\mathrm {e}}^{2\,x}-a^7\,b^6-a^5\,b^8+a^7\,b^6\,{\mathrm {e}}^{2\,x}+2\,a^6\,b^7\,{\mathrm {e}}^x+2\,a^8\,b^5\,{\mathrm {e}}^x}+\frac {4\,a^4\,b^7\,{\mathrm {e}}^x}{a^5\,b^8\,{\mathrm {e}}^{2\,x}-a^7\,b^6-a^5\,b^8+a^7\,b^6\,{\mathrm {e}}^{2\,x}+2\,a^6\,b^7\,{\mathrm {e}}^x+2\,a^8\,b^5\,{\mathrm {e}}^x}+\frac {2\,a^6\,b^5\,{\mathrm {e}}^x}{a^5\,b^8\,{\mathrm {e}}^{2\,x}-a^7\,b^6-a^5\,b^8+a^7\,b^6\,{\mathrm {e}}^{2\,x}+2\,a^6\,b^7\,{\mathrm {e}}^x+2\,a^8\,b^5\,{\mathrm {e}}^x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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