Optimal. Leaf size=180 \[ -\frac {a \text {ArcTan}(\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {a \left (a^2+2 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\text {csch}(c+d x)}{a d}+\frac {b \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \]
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Rubi [A]
time = 0.18, antiderivative size = 180, 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 {a \left (a^2+2 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {a \text {ArcTan}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {\text {sech}^2(c+d x) (a \sinh (c+d x)+b)}{2 d \left (a^2+b^2\right )}+\frac {b^5 \log (a+b \sinh (c+d x))}{a^2 d \left (a^2+b^2\right )^2}-\frac {b \log (\sinh (c+d x))}{a^2 d}-\frac {\text {csch}(c+d x)}{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}^2(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^2}{x^2 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b^5 \text {Subst}\left (\int \frac {1}{x^2 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b^5 \text {Subst}\left (\int \left (\frac {1}{a b^4 x^2}-\frac {1}{a^2 b^4 x}+\frac {1}{a^2 \left (a^2+b^2\right )^2 (a+x)}+\frac {-a+x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )^2}-\frac {\left (a^2+2 b^2\right ) (a-x)}{b^4 \left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac {\text {csch}(c+d x)}{a d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b^3 \text {Subst}\left (\int \frac {-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 {\left (b \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {a-x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {\text {csch}(c+d x)}{a d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {(a b) \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 \left (a^2+2 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 {\left (a b \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 {a \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {a \left (a^2+2 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\text {csch}(c+d x)}{a d}+\frac {b \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (b+a \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 = 0.82, size = 227, normalized size = 1.26 \begin {gather*} -\frac {\text {csch}(c+d x) (a+b \sinh (c+d x)) \left (\frac {a \text {ArcTan}(\sinh (c+d x))}{a^2+b^2}+\frac {2 \text {csch}(c+d x)}{a}-\frac {(i a+b) \left (a^2+2 b^2\right ) \log (i-\sinh (c+d x))}{\left (a^2+b^2\right )^2}+\frac {2 b \log (\sinh (c+d x))}{a^2}+\frac {(i a-b) \left (a^2+2 b^2\right ) \log (i+\sinh (c+d x))}{\left (a^2+b^2\right )^2}-\frac {2 b^5 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )^2}+\frac {b \text {sech}^2(c+d x)}{a^2+b^2}+\frac {a \text {sech}(c+d x) \tanh (c+d x)}{a^2+b^2}\right )}{2 d (b+a \text {csch}(c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.68, size = 249, normalized size = 1.38
method | result | size |
derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{3}-\frac {1}{2} a \,b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{2} b -b^{3}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{3}+\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (-2 a^{2} b -4 b^{3}\right ) \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4}+\frac {\left (3 a^{3}+5 a \,b^{2}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {b^{5} \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right )^{2} a^{2}}}{d}\) | \(249\) |
default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{3}-\frac {1}{2} a \,b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{2} b -b^{3}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{3}+\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (-2 a^{2} b -4 b^{3}\right ) \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4}+\frac {\left (3 a^{3}+5 a \,b^{2}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {b^{5} \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right )^{2} a^{2}}}{d}\) | \(249\) |
risch | \(-\frac {2 a^{2} b \,d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {2 a^{2} b d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {4 b^{3} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {4 b^{3} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {2 b^{5} x}{a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 b^{5} c}{a^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b x}{a^{2}}+\frac {2 b c}{a^{2} d}-\frac {{\mathrm e}^{d x +c} \left (3 a^{2} {\mathrm e}^{4 d x +4 c}+2 b^{2} {\mathrm e}^{4 d x +4 c}+2 a b \,{\mathrm e}^{3 d x +3 c}+2 a^{2} {\mathrm e}^{2 d x +2 c}+4 b^{2} {\mathrm e}^{2 d x +2 c}-2 a b \,{\mathrm e}^{d x +c}+3 a^{2}+2 b^{2}\right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} a \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {b^{5} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{2} d}\) | \(717\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 350, normalized size = 1.94 \begin {gather*} \frac {b^{5} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d} + \frac {{\left (3 \, a^{3} + 5 \, a b^{2}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {{\left (a^{2} b + 2 \, b^{3}\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 {2 \, a b e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, a b e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{2} + 2 \, b^{2}\right )} e^{\left (-d x - c\right )} + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )} + {\left (3 \, a^{2} + 2 \, b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{{\left (a^{3} + a b^{2} + {\left (a^{3} + a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{3} + a b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (a^{3} + a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} - \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} 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. 2568 vs.
\(2 (176) = 352\).
time = 0.67, size = 2568, normalized size = 14.27 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 458 vs.
\(2 (176) = 352\).
time = 0.49, size = 458, normalized size = 2.54 \begin {gather*} \frac {\frac {12 \, b^{6} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}} - \frac {3 \, {\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^{3} + 5 \, a b^{2}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {6 \, {\left (a^{2} b + 2 \, b^{3}\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 {12 \, b \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{2}} + \frac {4 \, {\left (b^{5} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 9 \, a^{5} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 15 \, a^{3} b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 6 \, a b^{4} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 6 \, a^{4} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 6 \, a^{2} b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 4 \, b^{5} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 24 \, a^{5} - 48 \, a^{3} b^{2} - 24 \, a b^{4}\right )}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 4 \, e^{\left (d x + c\right )} - 4 \, e^{\left (-d x - c\right )}\right )}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.50, size = 398, normalized size = 2.21 \begin {gather*} \frac {2\,b}{d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}^2\,\left (a^2+b^2\right )}-\frac {b\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{a^2\,d}+\frac {2\,b\,\ln \left (1+{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,1{}\mathrm {i}\right )}{d\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}-\frac {2\,b^3}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {2\,b\,\ln \left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+1{}\mathrm {i}\right )}{d\,{\left (b+a\,1{}\mathrm {i}\right )}^2}-\frac {2\,a^2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}+\frac {b^5\,\ln \left (2\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{a^2\,d\,{\left (a^2+b^2\right )}^2}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}^2\,\left (a^2+b^2\right )}-\frac {a\,b^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,{\left (a^2+b^2\right )}^2}-\frac {a\,\ln \left (1+{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,d\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}+\frac {a\,\ln \left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,d\,{\left (b+a\,1{}\mathrm {i}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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