Optimal. Leaf size=120 \[ \frac {b \left (3 a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )^2}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )}-\frac {a^3 \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {a^3 \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.20, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2721, 1647, 801, 635, 203, 260} \[ -\frac {a^3 \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {b \left (3 a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )^2}+\frac {a^3 \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 801
Rule 1647
Rule 2721
Rubi steps
\begin {align*} \int \frac {\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{(a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {\operatorname {Subst}\left (\int \frac {\frac {a b^4}{a^2+b^2}+\frac {b^2 \left (2 a^2+b^2\right ) x}{a^2+b^2}}{(a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{2 b^2 d}\\ &=\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {\operatorname {Subst}\left (\int \left (\frac {2 a^3 b^2}{\left (a^2+b^2\right )^2 (a+x)}-\frac {b^2 \left (3 a^2 b^2+b^4+2 a^3 x\right )}{\left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{2 b^2 d}\\ &=-\frac {a^3 \log (a+b \sinh (c+d x))}{\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 {\operatorname {Subst}\left (\int \frac {3 a^2 b^2+b^4+2 a^3 x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {a^3 \log (a+b \sinh (c+d x))}{\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 {a^3 \operatorname {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 (b^2 \left (3 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=\frac {b \left (3 a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d}+\frac {a^3 \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a^3 \log (a+b \sinh (c+d x))}{\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] time = 0.39, size = 152, normalized size = 1.27 \[ -\frac {2 a^3 \log (a+b \sinh (c+d x))-a \left (a^2+b^2\right ) \text {sech}^2(c+d x)+b \left (a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))+b \left (a^2+b^2\right ) \tanh (c+d x) \text {sech}(c+d x)-\left (a^3-i \left (2 a^2 b+b^3\right )\right ) \log (-\sinh (c+d x)+i)-\left (a^3+i \left (2 a^2 b+b^3\right )\right ) \log (\sinh (c+d x)+i)}{2 d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 896, normalized size = 7.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.78, size = 223, normalized size = 1.86 \[ \frac {\frac {a^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {a^{3} \log \left ({\left | -b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a e^{\left (d x + c\right )} + b \right |}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (3 \, a^{2} b e^{c} + b^{3} e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (a^{2} b e^{\left (3 \, c\right )} + b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - 2 \, {\left (a^{3} e^{\left (2 \, c\right )} + a b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - {\left (a^{2} b e^{c} + b^{3} e^{c}\right )} e^{\left (d x\right )}}{{\left (a^{2} + b^{2}\right )}^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 472, normalized size = 3.93 \[ -\frac {8 a^{3} \ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right )}{d \left (8 a^{4}+16 a^{2} b^{2}+8 b^{4}\right )}+\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {a^{3} \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 217, normalized size = 1.81 \[ -\frac {a^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {a^{3} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {{\left (3 \, a^{2} b + b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {b e^{\left (-d x - c\right )} - 2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.37, size = 381, normalized size = 3.18 \[ \frac {\frac {2\,\left (a^3+a\,b^2\right )}{d\,{\left (a^2+b^2\right )}^2}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^2\,b+b^3\right )}{d\,{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {2\,a}{d\,\left (a^2+b^2\right )}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2+b^2\right )}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+\frac {\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )\,\left (2\,a+b\,1{}\mathrm {i}\right )}{2\,\left (d\,a^2+2{}\mathrm {i}\,d\,a\,b-d\,b^2\right )}+\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\left (b+a\,2{}\mathrm {i}\right )}{2\,\left (1{}\mathrm {i}\,d\,a^2+2\,d\,a\,b-1{}\mathrm {i}\,d\,b^2\right )}-\frac {a^3\,\ln \left (32\,a^7\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b^7-6\,a^2\,b^5-9\,a^4\,b^3-16\,a^6\,b+b^7\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+16\,a^6\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+12\,a^3\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+18\,a^5\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+6\,a^2\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+9\,a^4\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,a\,b^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{d\,a^4+2\,d\,a^2\,b^2+d\,b^4} \]
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 \[ \int \frac {\tanh ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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