Optimal. Leaf size=187 \[ \frac {(c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {d \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {d \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2} \]
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Rubi [A]
time = 0.25, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3403, 2296,
2221, 2317, 2438} \begin {gather*} \frac {(c+d x) \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{f \sqrt {a^2+b^2}}-\frac {(c+d x) \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{f \sqrt {a^2+b^2}}+\frac {d \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^2 \sqrt {a^2+b^2}}-\frac {d \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^2 \sqrt {a^2+b^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3403
Rubi steps
\begin {align*} \int \frac {c+d x}{a+b \sinh (e+f x)} \, dx &=2 \int \frac {e^{e+f x} (c+d x)}{-b+2 a e^{e+f x}+b e^{2 (e+f x)}} \, dx\\ &=\frac {(2 b) \int \frac {e^{e+f x} (c+d x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{e+f x}} \, dx}{\sqrt {a^2+b^2}}-\frac {(2 b) \int \frac {e^{e+f x} (c+d x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{e+f x}} \, dx}{\sqrt {a^2+b^2}}\\ &=\frac {(c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {d \int \log \left (1+\frac {2 b e^{e+f x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f}+\frac {d \int \log \left (1+\frac {2 b e^{e+f x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f}\\ &=\frac {(c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {d \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\sqrt {a^2+b^2} f^2}+\frac {d \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\sqrt {a^2+b^2} f^2}\\ &=\frac {(c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {d \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {d \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 142, normalized size = 0.76 \begin {gather*} \frac {f (c+d x) \left (\log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )-\log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )+d \text {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )-d \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(392\) vs.
\(2(167)=334\).
time = 1.03, size = 393, normalized size = 2.10
method | result | size |
risch | \(-\frac {2 c \arctanh \left (\frac {2 b \,{\mathrm e}^{f x +e}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{f \sqrt {a^{2}+b^{2}}}+\frac {d \ln \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{f \sqrt {a^{2}+b^{2}}}+\frac {d \ln \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) e}{f^{2} \sqrt {a^{2}+b^{2}}}-\frac {d \ln \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{f \sqrt {a^{2}+b^{2}}}-\frac {d \ln \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) e}{f^{2} \sqrt {a^{2}+b^{2}}}+\frac {d \dilog \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{f^{2} \sqrt {a^{2}+b^{2}}}-\frac {d \dilog \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{f^{2} \sqrt {a^{2}+b^{2}}}+\frac {2 d e \arctanh \left (\frac {2 b \,{\mathrm e}^{f x +e}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{f^{2} \sqrt {a^{2}+b^{2}}}\) | \(393\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 541 vs.
\(2 (169) = 338\).
time = 0.35, size = 541, normalized size = 2.89 \begin {gather*} \frac {b d \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) - b d \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - {\left (b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) - {\left (b c f - b d \cosh \left (1\right ) - b d \sinh \left (1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) + {\left (b c f - b d \cosh \left (1\right ) - b d \sinh \left (1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) + {\left (b d f x + b d \cosh \left (1\right ) + b d \sinh \left (1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) - {\left (b d f x + b d \cosh \left (1\right ) + b d \sinh \left (1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - {\left (b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right )}{{\left (a^{2} + b^{2}\right )} f^{2}} \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 {c + d x}{a + b \sinh {\left (e + f x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {c+d\,x}{a+b\,\mathrm {sinh}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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