Optimal. Leaf size=124 \[ \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \sinh \left (a-b \sqrt {c}\right )+\text {Chi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right ) \sinh \left (a+b \sqrt {c}\right )-\cosh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+\cosh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5472, 5400,
3384, 3379, 3382} \begin {gather*} \sinh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )+\sinh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )-\cosh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+\cosh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5400
Rule 5472
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+b \sqrt {c+d x}\right )}{x} \, dx &=\text {Subst}\left (\int \frac {\sinh \left (a+b \sqrt {x}\right )}{-c+x} \, dx,x,c+d x\right )\\ &=2 \text {Subst}\left (\int \frac {x \sinh (a+b x)}{-c+x^2} \, dx,x,\sqrt {c+d x}\right )\\ &=2 \text {Subst}\left (\int \left (-\frac {\sinh (a+b x)}{2 \left (\sqrt {c}-x\right )}+\frac {\sinh (a+b x)}{2 \left (\sqrt {c}+x\right )}\right ) \, dx,x,\sqrt {c+d x}\right )\\ &=-\text {Subst}\left (\int \frac {\sinh (a+b x)}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )+\text {Subst}\left (\int \frac {\sinh (a+b x)}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )\\ &=\cosh \left (a-b \sqrt {c}\right ) \text {Subst}\left (\int \frac {\sinh \left (b \sqrt {c}+b x\right )}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )+\cosh \left (a+b \sqrt {c}\right ) \text {Subst}\left (\int \frac {\sinh \left (b \sqrt {c}-b x\right )}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )+\sinh \left (a-b \sqrt {c}\right ) \text {Subst}\left (\int \frac {\cosh \left (b \sqrt {c}+b x\right )}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )-\sinh \left (a+b \sqrt {c}\right ) \text {Subst}\left (\int \frac {\cosh \left (b \sqrt {c}-b x\right )}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )\\ &=\text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \sinh \left (a-b \sqrt {c}\right )+\text {Chi}\left (b \sqrt {c}-b \sqrt {c+d x}\right ) \sinh \left (a+b \sqrt {c}\right )+\cosh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )-\cosh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \sqrt {c}-b \sqrt {c+d x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 130, normalized size = 1.05 \begin {gather*} \frac {1}{2} e^{-a-b \sqrt {c}} \left (-\text {Ei}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+e^{2 \left (a+b \sqrt {c}\right )} \text {Ei}\left (b \left (-\sqrt {c}+\sqrt {c+d x}\right )\right )-e^{2 b \sqrt {c}} \text {Ei}\left (-b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )+e^{2 a} \text {Ei}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\sinh \left (a +b \sqrt {d x +c}\right )}{x}\, dx\]
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. 217 vs.
\(2 (102) = 204\).
time = 0.38, size = 217, normalized size = 1.75 \begin {gather*} \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) - {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \cosh \left (a + \sqrt {b^{2} c}\right ) + \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) - {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \cosh \left (-a + \sqrt {b^{2} c}\right ) + \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) + {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \sinh \left (a + \sqrt {b^{2} c}\right ) - \frac {1}{2} \, {\left ({\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) + {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \sinh \left (-a + \sqrt {b^{2} c}\right ) \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 {\sinh {\left (a + b \sqrt {c + d x} \right )}}{x}\, 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 {\mathrm {sinh}\left (a+b\,\sqrt {c+d\,x}\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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