Optimal. Leaf size=211 \[ \frac {b \text {ArcTan}(\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac {b \left (a^2+2 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {a \left (2 a^2+3 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \]
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Rubi [A]
time = 0.26, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2916, 12, 908,
653, 209, 649, 266} \begin {gather*} \frac {b \left (a^2+2 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {b \text {ArcTan}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac {a \left (2 a^2+3 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 d \left (a^2+b^2\right )^2}-\frac {\text {csch}^2(c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 266
Rule 649
Rule 653
Rule 908
Rule 2916
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {b^3 \text {Subst}\left (\int \frac {b^3}{x^3 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b^6 \text {Subst}\left (\int \frac {1}{x^3 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b^6 \text {Subst}\left (\int \left (\frac {1}{a b^4 x^3}-\frac {1}{a^2 b^4 x^2}+\frac {-2 a^2+b^2}{a^3 b^6 x}-\frac {1}{a^3 \left (a^2+b^2\right )^2 (a+x)}+\frac {b^2+a x}{b^4 \left (a^2+b^2\right ) \left (b^2+x^2\right )^2}+\frac {b^2 \left (a^2+2 b^2\right )+a \left (2 a^2+3 b^2\right ) x}{b^6 \left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}+\frac {\text {Subst}\left (\int \frac {b^2 \left (a^2+2 b^2\right )+a \left (2 a^2+3 b^2\right ) x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^2 \text {Subst}\left (\int \frac {b^2+a x}{\left (b^2+x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac {\left (b^2 \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (a \left (2 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}\\ &=\frac {b \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac {b \left (a^2+2 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {a \left (2 a^2+3 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.62, size = 237, normalized size = 1.12 \begin {gather*} \frac {\frac {b \text {ArcTan}(\sinh (c+d x))}{a^2+b^2}+\frac {2 b \text {csch}(c+d x)}{a^2}-\frac {\text {csch}^2(c+d x)}{a}+\frac {(a-i b) \left (2 a^2+i a b+2 b^2\right ) \log (i-\sinh (c+d x))}{\left (a^2+b^2\right )^2}-\frac {2 \left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3}+\frac {(a+i b) \left (2 a^2-i a b+2 b^2\right ) \log (i+\sinh (c+d x))}{\left (a^2+b^2\right )^2}-\frac {2 b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2}-\frac {a \text {sech}^2(c+d x)}{a^2+b^2}+\frac {b \text {sech}(c+d x) \tanh (c+d x)}{a^2+b^2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.02, size = 292, normalized size = 1.38
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {b^{6} \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{3}+a \,b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (4 a^{3}+6 a \,b^{2}\right ) \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}+\left (3 a^{2} b +5 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-8 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(292\) |
default | \(\frac {-\frac {\frac {a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {b^{6} \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{3}+a \,b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (4 a^{3}+6 a \,b^{2}\right ) \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}+\left (3 a^{2} b +5 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-8 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(292\) |
risch | \(-\frac {4 a^{3} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {4 a^{3} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {6 a \,b^{2} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {6 a \,b^{2} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {4 x}{a}+\frac {4 c}{a d}-\frac {2 b^{2} x}{a^{3}}-\frac {2 b^{2} c}{a^{3} d}+\frac {2 b^{6} x}{a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{6} c}{a^{3} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {{\mathrm e}^{d x +c} \left (-3 a^{2} b \,{\mathrm e}^{6 d x +6 c}-2 b^{3} {\mathrm e}^{6 d x +6 c}+4 a^{3} {\mathrm e}^{5 d x +5 c}+2 a \,b^{2} {\mathrm e}^{5 d x +5 c}+a^{2} b \,{\mathrm e}^{4 d x +4 c}-2 b^{3} {\mathrm e}^{4 d x +4 c}+4 a \,b^{2} {\mathrm e}^{3 d x +3 c}-a^{2} b \,{\mathrm e}^{2 d x +2 c}+2 b^{3} {\mathrm e}^{2 d x +2 c}+4 a^{3} {\mathrm e}^{d x +c}+2 a \,b^{2} {\mathrm e}^{d x +c}+3 a^{2} b +2 b^{3}\right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} a^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {5 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a d}+\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{3} d}-\frac {b^{6} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a^{3} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(835\) |
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. 418 vs.
\(2 (206) = 412\).
time = 0.50, size = 418, normalized size = 1.98 \begin {gather*} -\frac {b^{6} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d} - \frac {{\left (3 \, a^{2} b + 5 \, b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {4 \, a b^{2} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (3 \, a^{2} b + 2 \, b^{3}\right )} e^{\left (-d x - c\right )} + 2 \, {\left (2 \, a^{3} + a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )} - {\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )} + 2 \, {\left (2 \, a^{3} + a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{{\left (a^{4} + a^{2} b^{2} - 2 \, {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} - \frac {{\left (2 \, a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} - \frac {{\left (2 \, a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \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. 3148 vs.
\(2 (206) = 412\).
time = 0.72, size = 3148, normalized size = 14.92 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 464 vs.
\(2 (206) = 412\).
time = 0.45, size = 464, normalized size = 2.20 \begin {gather*} -\frac {\frac {4 \, b^{7} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}} - \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (3 \, a^{2} b + 5 \, b^{3}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (2 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 3 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 2 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 12 \, a^{3} + 16 \, a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}} + \frac {4 \, {\left (2 \, a^{2} - b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{3}} - \frac {2 \, {\left (6 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 3 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4 \, a^{2}\right )}}{a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.91, size = 554, normalized size = 2.63 \begin {gather*} -\frac {\frac {4\,b^5}{a\,d\,\left (a^2\,b^3+b^5\right )}-\frac {4\,b^4\,{\mathrm {e}}^{3\,c+3\,d\,x}}{d\,\left (a^2\,b^3+b^5\right )}+\frac {4\,b^4\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2\,b^3+b^5\right )}+\frac {4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^2+b^2\right )}{a\,d\,\left (a^2\,b^3+b^5\right )}}{{\mathrm {e}}^{8\,c+8\,d\,x}-2\,{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {4\,\left (a^2\,b^5+b^7\right )}{a\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^4\,b^3+3\,a^2\,b^5+b^7\right )}{a\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (3\,a^4\,b^4+5\,a^2\,b^6+2\,b^8\right )}{a^2\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}-\frac {b^4\,{\mathrm {e}}^{c+d\,x}\,\left (-a^4+a^2\,b^2+2\,b^4\right )}{a^2\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}+\frac {\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )\,\left (4\,a+b\,5{}\mathrm {i}\right )}{2\,\left (d\,a^2+2{}\mathrm {i}\,d\,a\,b-d\,b^2\right )}-\frac {b^6\,\ln \left (2\,a\,{\mathrm {e}}^{c+d\,x}-b+b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{d\,a^7+2\,d\,a^5\,b^2+d\,a^3\,b^4}+\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\left (5\,b+a\,4{}\mathrm {i}\right )}{2\,\left (1{}\mathrm {i}\,d\,a^2+2\,d\,a\,b-1{}\mathrm {i}\,d\,b^2\right )}-\frac {\ln \left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )\,\left (2\,a^2-b^2\right )}{a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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