Optimal. Leaf size=182 \[ -\frac {b d \sinh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}+\frac {b d \sinh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \cosh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \cosh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \]
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Rubi [A] time = 0.38, 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 = {5365, 5289, 5280, 3303, 3298, 3301} \[ -\frac {b d \sinh \left (a-b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}+\frac {b d \sinh \left (a+b \sqrt {c}\right ) \text {Chi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \cosh \left (a+b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \cosh \left (a-b \sqrt {c}\right ) \text {Shi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5280
Rule 5289
Rule 5365
Rubi steps
\begin {align*} \int \frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx &=d \operatorname {Subst}\left (\int \frac {\cosh \left (a+b \sqrt {x}\right )}{(-c+x)^2} \, dx,x,c+d x\right )\\ &=(2 d) \operatorname {Subst}\left (\int \frac {x \cosh (a+b x)}{\left (c-x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x}-(b d) \operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{c-x^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x}-(b d) \operatorname {Subst}\left (\int \left (\frac {\sinh (a+b x)}{2 \sqrt {c} \left (\sqrt {c}-x\right )}+\frac {\sinh (a+b x)}{2 \sqrt {c} \left (\sqrt {c}+x\right )}\right ) \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x}-\frac {(b d) \operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}-\frac {(b d) \operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}\\ &=-\frac {\cosh \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 {\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 \cosh \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 {\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 {\cosh \left (b \sqrt {c}-b x\right )}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}\\ &=-\frac {\cosh \left (a+b \sqrt {c+d x}\right )}{x}-\frac {b d \text {Chi}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \sinh \left (a-b \sqrt {c}\right )}{2 \sqrt {c}}+\frac {b d \text {Chi}\left (b \sqrt {c}-b \sqrt {c+d x}\right ) \sinh \left (a+b \sqrt {c}\right )}{2 \sqrt {c}}-\frac {b d \cosh \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 \cosh \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 = 1.88, size = 199, normalized size = 1.09 \[ \frac {e^{-a} \left (-b d x e^{-b \sqrt {c}} \text {Ei}\left (b \left (\sqrt {c}-\sqrt {c+d x}\right )\right )+b d x e^{b \sqrt {c}} \text {Ei}\left (-b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )-2 \sqrt {c} e^{-b \sqrt {c+d x}}\right )+e^a \left (b d x e^{b \sqrt {c}} \text {Ei}\left (b \left (\sqrt {c+d x}-\sqrt {c}\right )\right )-b d x e^{-b \sqrt {c}} \text {Ei}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )-2 \sqrt {c} e^{b \sqrt {c+d x}}\right )}{4 \sqrt {c} x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 317, normalized size = 1.74 \[ -\frac {4 \, c \cosh \left (\sqrt {d x + c} b + a\right ) - {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) - \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \cosh \left (a + \sqrt {b^{2} c}\right ) + {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) - \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \cosh \left (-a + \sqrt {b^{2} c}\right ) - {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b - \sqrt {b^{2} c}\right ) + \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b + \sqrt {b^{2} c}\right )\right )} \sinh \left (a + \sqrt {b^{2} c}\right ) - {\left (\sqrt {b^{2} c} d x {\rm Ei}\left (\sqrt {d x + c} b + \sqrt {b^{2} c}\right ) + \sqrt {b^{2} c} d x {\rm Ei}\left (-\sqrt {d x + c} b - \sqrt {b^{2} c}\right )\right )} \sinh \left (-a + \sqrt {b^{2} c}\right )}{4 \, c x} \]
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 \[ \int \frac {\cosh \left (\sqrt {d x + c} b + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (a +b \sqrt {d x +c}\right )}{x^{2}}\, dx \]
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 \[ \int \frac {\cosh \left (\sqrt {d x + c} b + a\right )}{x^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {cosh}\left (a+b\,\sqrt {c+d\,x}\right )}{x^2} \,d x \]
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 \[ \int \frac {\cosh {\left (a + b \sqrt {c + d x} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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