Optimal. Leaf size=46 \[ -\frac {b \text {Chi}(a+b x)}{a}-\frac {\text {Chi}(a+b x)}{x}+\frac {b \cosh (a) \text {Chi}(b x)}{a}+\frac {b \sinh (a) \text {Shi}(b x)}{a} \]
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Rubi [A] time = 0.22, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6533, 6742, 3303, 3298, 3301} \[ -\frac {b \text {Chi}(a+b x)}{a}-\frac {\text {Chi}(a+b x)}{x}+\frac {b \cosh (a) \text {Chi}(b x)}{a}+\frac {b \sinh (a) \text {Shi}(b x)}{a} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 6533
Rule 6742
Rubi steps
\begin {align*} \int \frac {\text {Chi}(a+b x)}{x^2} \, dx &=-\frac {\text {Chi}(a+b x)}{x}+b \int \frac {\cosh (a+b x)}{x (a+b x)} \, dx\\ &=-\frac {\text {Chi}(a+b x)}{x}+b \int \left (\frac {\cosh (a+b x)}{a x}-\frac {b \cosh (a+b x)}{a (a+b x)}\right ) \, dx\\ &=-\frac {\text {Chi}(a+b x)}{x}+\frac {b \int \frac {\cosh (a+b x)}{x} \, dx}{a}-\frac {b^2 \int \frac {\cosh (a+b x)}{a+b x} \, dx}{a}\\ &=-\frac {b \text {Chi}(a+b x)}{a}-\frac {\text {Chi}(a+b x)}{x}+\frac {(b \cosh (a)) \int \frac {\cosh (b x)}{x} \, dx}{a}+\frac {(b \sinh (a)) \int \frac {\sinh (b x)}{x} \, dx}{a}\\ &=\frac {b \cosh (a) \text {Chi}(b x)}{a}-\frac {b \text {Chi}(a+b x)}{a}-\frac {\text {Chi}(a+b x)}{x}+\frac {b \sinh (a) \text {Shi}(b x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 39, normalized size = 0.85 \[ \frac {-(a+b x) \text {Chi}(a+b x)+b x \cosh (a) \text {Chi}(b x)+b x \sinh (a) \text {Shi}(b x)}{a x} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Chi}\left (b x + a\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\rm Chi}\left (b x + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\Chi \left (b x +a \right )}{x^{2}}\, dx \]
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 \[ \int \frac {{\rm Chi}\left (b x + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {coshint}\left (a+b\,x\right )}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {Chi}\left (a + b x\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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