Optimal. Leaf size=182 \[ \frac {b d \cosh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \cosh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {\sinh \left (a+b \sqrt {c+d x}\right )}{x}-\frac {b d \sinh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \sinh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}} \]
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Rubi [A]
time = 0.26, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5472, 5396,
5389, 3384, 3379, 3382} \begin {gather*} \frac {b d \cosh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \cosh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \sinh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \sinh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {\sinh \left (a+b \sqrt {c+d x}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5389
Rule 5396
Rule 5472
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx &=d \text {Subst}\left (\int \frac {\sinh \left (a+b \sqrt {x}\right )}{(-c+x)^2} \, dx,x,c+d x\right )\\ &=(2 d) \text {Subst}\left (\int \frac {x \sinh (a+b x)}{\left (c-x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\sinh \left (a+b \sqrt {c+d x}\right )}{x}-(b d) \text {Subst}\left (\int \frac {\cosh (a+b x)}{c-x^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\sinh \left (a+b \sqrt {c+d x}\right )}{x}-(b d) \text {Subst}\left (\int \left (\frac {\cosh (a+b x)}{2 \sqrt {c} \left (\sqrt {c}-x\right )}+\frac {\cosh (a+b x)}{2 \sqrt {c} \left (\sqrt {c}+x\right )}\right ) \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\sinh \left (a+b \sqrt {c+d x}\right )}{x}-\frac {(b d) \text {Subst}\left (\int \frac {\cosh (a+b x)}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}-\frac {(b d) \text {Subst}\left (\int \frac {\cosh (a+b x)}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}\\ &=-\frac {\sinh \left (a+b \sqrt {c+d x}\right )}{x}-\frac {\left (b d \cosh \left (a-b \sqrt {c}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (b \sqrt {c}+b x\right )}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}-\frac {\left (b d \cosh \left (a+b \sqrt {c}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (b \sqrt {c}-b x\right )}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}-\frac {\left (b d \sinh \left (a-b \sqrt {c}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (b \sqrt {c}+b x\right )}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}+\frac {\left (b d \sinh \left (a+b \sqrt {c}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (b \sqrt {c}-b x\right )}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}\\ &=-\frac {b d \cosh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}+\frac {b d \cosh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \sqrt {c}-b \sqrt {c+d x}\right )}{2 \sqrt {c}}-\frac {\sinh \left (a+b \sqrt {c+d x}\right )}{x}-\frac {b d \sinh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \sinh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \sqrt {c}-b \sqrt {c+d x}\right )}{2 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 2.20, size = 199, normalized size = 1.09 \begin {gather*} \frac {e^{-a} \left (2 \sqrt {c} e^{-b \sqrt {c+d x}}+b d e^{-b \sqrt {c}} x \text {Ei}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )-b d e^{b \sqrt {c}} x \text {Ei}\left (-b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right )+e^a \left (-2 \sqrt {c} e^{b \sqrt {c+d x}}+b d e^{b \sqrt {c}} x \text {Ei}\left (b \left (-\sqrt {c}+\sqrt {c+d x}\right )\right )-b d e^{-b \sqrt {c}} x \text {Ei}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right )}{4 \sqrt {c} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\sinh \left (a +b \sqrt {d x +c}\right )}{x^{2}}\, dx\]
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. 315 vs.
\(2 (142) = 284\).
time = 0.37, size = 315, normalized size = 1.73 \begin {gather*} \frac {{\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) + \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \cosh \left (a + \sqrt {b^{2} c}\right ) - {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) + \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \cosh \left (-a + \sqrt {b^{2} c}\right ) - 4 \, c \sinh \left (\sqrt {d x + c} b + a\right ) + {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) - \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \sinh \left (a + \sqrt {b^{2} c}\right ) + {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) - \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \sinh \left (-a + \sqrt {b^{2} c}\right )}{4 \, c x} \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^{2}}\, 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^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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