Optimal. Leaf size=113 \[ \frac {b \coth (c+d x)}{a^2 d}+\frac {\left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}+\frac {2 b^3 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d \sqrt {a^2+b^2}}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
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Rubi [A] time = 0.40, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2802, 3055, 3001, 3770, 2660, 618, 204} \[ \frac {2 b^3 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d \sqrt {a^2+b^2}}+\frac {\left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2802
Rule 3001
Rule 3055
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {i \int \frac {\text {csch}^2(c+d x) \left (2 i b+i a \sinh (c+d x)+i b \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{2 a}\\ &=\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\int \frac {\text {csch}(c+d x) \left (a^2-2 b^2+a b \sinh (c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{2 a^2}\\ &=\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b^3 \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac {\left (a^2-2 b^2\right ) \int \text {csch}(c+d x) \, dx}{2 a^3}\\ &=\frac {\left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {\left (2 i b^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 d}\\ &=\frac {\left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\left (4 i b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 d}\\ &=\frac {\left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}+\frac {2 b^3 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 1.81, size = 145, normalized size = 1.28 \[ -\frac {4 \left (a^2-2 b^2\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\frac {16 b^3 \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )-4 a b \tanh \left (\frac {1}{2} (c+d x)\right )-4 a b \coth \left (\frac {1}{2} (c+d x)\right )}{8 a^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 1203, normalized size = 10.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.00, size = 176, normalized size = 1.56 \[ -\frac {\frac {2 \, b^{3} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} - \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{3}} + \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{3}} + \frac {2 \, {\left (a e^{\left (3 \, d x + 3 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a e^{\left (d x + c\right )} + 2 \, b\right )}}{a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 164, normalized size = 1.45 \[ \frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{2 d \,a^{2}}-\frac {2 b^{3} \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,a^{3} \sqrt {a^{2}+b^{2}}}-\frac {1}{8 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{d \,a^{3}}+\frac {b}{2 d \,a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 211, normalized size = 1.87 \[ -\frac {b^{3} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3} d} + \frac {a e^{\left (-d x - c\right )} + 2 \, b e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-3 \, d x - 3 \, c\right )} - 2 \, b}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} + \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{2 \, a^{3} d} - \frac {{\left (a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{2 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 776, normalized size = 6.87 \[ \frac {{\mathrm {e}}^{c+d\,x}}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d-2\,a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+a\,d\,{\mathrm {e}}^{4\,c+4\,d\,x}}-\frac {2\,b}{a^2\,d-a^2\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-24\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+20\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{2\,a\,d}+\frac {\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+24\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-20\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{2\,a\,d}-\frac {b^3\,\ln \left (16\,a^5\,b-48\,a\,b^5-24\,b^5\,\sqrt {a^2+b^2}-32\,a^3\,b^3-40\,a^2\,b^3\,\sqrt {a^2+b^2}-32\,a^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+24\,b^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^4\,b\,\sqrt {a^2+b^2}+112\,a^2\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+56\,a^4\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-32\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}+72\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}+72\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{d\,a^5+d\,a^3\,b^2}+\frac {b^3\,\ln \left (24\,b^5\,\sqrt {a^2+b^2}-48\,a\,b^5+16\,a^5\,b-32\,a^3\,b^3+40\,a^2\,b^3\,\sqrt {a^2+b^2}-32\,a^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+24\,b^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,a^4\,b\,\sqrt {a^2+b^2}+112\,a^2\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+56\,a^4\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}-72\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}-72\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{d\,a^5+d\,a^3\,b^2}+\frac {b^2\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-24\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+20\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^3\,d}-\frac {b^2\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+24\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-20\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^3\,d} \]
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 \[ \int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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