Optimal. Leaf size=47 \[ -\frac {b \text {Ci}(a+b x)}{a}-\frac {\text {Ci}(a+b x)}{x}+\frac {b \cos (a) \text {Ci}(b x)}{a}-\frac {b \sin (a) \text {Si}(b x)}{a} \]
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Rubi [A] time = 0.22, antiderivative size = 47, 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 = {6504, 6742, 3303, 3299, 3302} \[ -\frac {b \text {CosIntegral}(a+b x)}{a}-\frac {\text {CosIntegral}(a+b x)}{x}+\frac {b \cos (a) \text {CosIntegral}(b x)}{a}-\frac {b \sin (a) \text {Si}(b x)}{a} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 6504
Rule 6742
Rubi steps
\begin {align*} \int \frac {\text {Ci}(a+b x)}{x^2} \, dx &=-\frac {\text {Ci}(a+b x)}{x}+b \int \frac {\cos (a+b x)}{x (a+b x)} \, dx\\ &=-\frac {\text {Ci}(a+b x)}{x}+b \int \left (\frac {\cos (a+b x)}{a x}-\frac {b \cos (a+b x)}{a (a+b x)}\right ) \, dx\\ &=-\frac {\text {Ci}(a+b x)}{x}+\frac {b \int \frac {\cos (a+b x)}{x} \, dx}{a}-\frac {b^2 \int \frac {\cos (a+b x)}{a+b x} \, dx}{a}\\ &=-\frac {b \text {Ci}(a+b x)}{a}-\frac {\text {Ci}(a+b x)}{x}+\frac {(b \cos (a)) \int \frac {\cos (b x)}{x} \, dx}{a}-\frac {(b \sin (a)) \int \frac {\sin (b x)}{x} \, dx}{a}\\ &=\frac {b \cos (a) \text {Ci}(b x)}{a}-\frac {b \text {Ci}(a+b x)}{a}-\frac {\text {Ci}(a+b x)}{x}-\frac {b \sin (a) \text {Si}(b x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 40, normalized size = 0.85 \[ \frac {-(a+b x) \text {Ci}(a+b x)+b x \cos (a) \text {Ci}(b x)-b x \sin (a) \text {Si}(b x)}{a x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Ci}\left (b x + a\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.17, size = 149, normalized size = 3.17 \[ -\frac {{\left (\Re \left (\operatorname {Ci}\left (b x + a\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + \Re \left (\operatorname {Ci}\left (b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + \Re \left (\operatorname {Ci}\left (-b x - a\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + \Re \left (\operatorname {Ci}\left (-b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, \Im \left (\operatorname {Ci}\left (b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) - 2 \, \Im \left (\operatorname {Ci}\left (-b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) + 4 \, \operatorname {Si}\left (b x\right ) \tan \left (\frac {1}{2} \, a\right ) + \Re \left (\operatorname {Ci}\left (b x + a\right ) \right ) - \Re \left (\operatorname {Ci}\left (b x\right ) \right ) + \Re \left (\operatorname {Ci}\left (-b x - a\right ) \right ) - \Re \left (\operatorname {Ci}\left (-b x\right ) \right )\right )} b}{2 \, {\left (a \tan \left (\frac {1}{2} \, a\right )^{2} + a\right )}} - \frac {\operatorname {Ci}\left (b x + a\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 49, normalized size = 1.04 \[ b \left (-\frac {\Ci \left (b x +a \right )}{b x}+\frac {-\Si \left (b x \right ) \sin \relax (a )+\Ci \left (b x \right ) \cos \relax (a )}{a}-\frac {\Ci \left (b x +a \right )}{a}\right ) \]
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 \[ \int \frac {{\rm Ci}\left (b x + a\right )}{x^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {cosint}\left (a+b\,x\right )}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {Ci}{\left (a + b x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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