Optimal. Leaf size=182 \[ \frac{b d \cosh \left (a+b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{b d \cosh \left (a-b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{b d \sinh \left (a+b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{b d \sinh \left (a-b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{\sinh \left (a+b \sqrt{c+d x}\right )}{x} \]
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Rubi [A] time = 0.360614, antiderivative size = 182, normalized size of antiderivative = 1., 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 = {5364, 5288, 5281, 3303, 3298, 3301} \[ \frac{b d \cosh \left (a+b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{b d \cosh \left (a-b \sqrt{c}\right ) \text{Chi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{b d \sinh \left (a+b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{b d \sinh \left (a-b \sqrt{c}\right ) \text{Shi}\left (b \left (\sqrt{c}+\sqrt{c+d x}\right )\right )}{2 \sqrt{c}}-\frac{\sinh \left (a+b \sqrt{c+d x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 5364
Rule 5288
Rule 5281
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\sinh \left (a+b \sqrt{c+d x}\right )}{x^2} \, dx &=d \operatorname{Subst}\left (\int \frac{\sinh \left (a+b \sqrt{x}\right )}{(-c+x)^2} \, dx,x,c+d x\right )\\ &=(2 d) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{\sqrt{c}-x} \, dx,x,\sqrt{c+d x}\right )}{2 \sqrt{c}}-\frac{(b d) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 3.04243, size = 199, normalized size = 1.09 \[ \frac{e^{-a} \left (b d x e^{-b \sqrt{c}} \text{ExpIntegralEi}\left (b \left (\sqrt{c}-\sqrt{c+d x}\right )\right )-b d x e^{b \sqrt{c}} \text{ExpIntegralEi}\left (-b \left (\sqrt{c+d x}+\sqrt{c}\right )\right )+2 \sqrt{c} e^{-b \sqrt{c+d x}}\right )+e^a \left (b d x e^{b \sqrt{c}} \text{ExpIntegralEi}\left (b \left (\sqrt{c+d x}-\sqrt{c}\right )\right )-b d x e^{-b \sqrt{c}} \text{ExpIntegralEi}\left (b \left (\sqrt{c+d x}+\sqrt{c}\right )\right )-2 \sqrt{c} e^{b \sqrt{c+d x}}\right )}{4 \sqrt{c} x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sinh \left ( a+b\sqrt{dx+c} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14669, size = 757, normalized size = 4.16 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (a + b \sqrt{c + d x} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (\sqrt{d x + c} b + a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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