Optimal. Leaf size=111 \[ \frac {b^2 \text {Ci}(a+b x)}{2 a^2}-\frac {b^2 \cos (a) \text {Ci}(b x)}{2 a^2}+\frac {b^2 \sin (a) \text {Si}(b x)}{2 a^2}-\frac {b^2 \sin (a) \text {Ci}(b x)}{2 a}-\frac {b^2 \cos (a) \text {Si}(b x)}{2 a}-\frac {\text {Ci}(a+b x)}{2 x^2}-\frac {b \cos (a+b x)}{2 a x} \]
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Rubi [A] time = 0.33, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6504, 6742, 3297, 3303, 3299, 3302} \[ \frac {b^2 \text {CosIntegral}(a+b x)}{2 a^2}-\frac {b^2 \cos (a) \text {CosIntegral}(b x)}{2 a^2}+\frac {b^2 \sin (a) \text {Si}(b x)}{2 a^2}-\frac {b^2 \sin (a) \text {CosIntegral}(b x)}{2 a}-\frac {b^2 \cos (a) \text {Si}(b x)}{2 a}-\frac {\text {CosIntegral}(a+b x)}{2 x^2}-\frac {b \cos (a+b x)}{2 a x} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 6504
Rule 6742
Rubi steps
\begin {align*} \int \frac {\text {Ci}(a+b x)}{x^3} \, dx &=-\frac {\text {Ci}(a+b x)}{2 x^2}+\frac {1}{2} b \int \frac {\cos (a+b x)}{x^2 (a+b x)} \, dx\\ &=-\frac {\text {Ci}(a+b x)}{2 x^2}+\frac {1}{2} b \int \left (\frac {\cos (a+b x)}{a x^2}-\frac {b \cos (a+b x)}{a^2 x}+\frac {b^2 \cos (a+b x)}{a^2 (a+b x)}\right ) \, dx\\ &=-\frac {\text {Ci}(a+b x)}{2 x^2}+\frac {b \int \frac {\cos (a+b x)}{x^2} \, dx}{2 a}-\frac {b^2 \int \frac {\cos (a+b x)}{x} \, dx}{2 a^2}+\frac {b^3 \int \frac {\cos (a+b x)}{a+b x} \, dx}{2 a^2}\\ &=-\frac {b \cos (a+b x)}{2 a x}+\frac {b^2 \text {Ci}(a+b x)}{2 a^2}-\frac {\text {Ci}(a+b x)}{2 x^2}-\frac {b^2 \int \frac {\sin (a+b x)}{x} \, dx}{2 a}-\frac {\left (b^2 \cos (a)\right ) \int \frac {\cos (b x)}{x} \, dx}{2 a^2}+\frac {\left (b^2 \sin (a)\right ) \int \frac {\sin (b x)}{x} \, dx}{2 a^2}\\ &=-\frac {b \cos (a+b x)}{2 a x}-\frac {b^2 \cos (a) \text {Ci}(b x)}{2 a^2}+\frac {b^2 \text {Ci}(a+b x)}{2 a^2}-\frac {\text {Ci}(a+b x)}{2 x^2}+\frac {b^2 \sin (a) \text {Si}(b x)}{2 a^2}-\frac {\left (b^2 \cos (a)\right ) \int \frac {\sin (b x)}{x} \, dx}{2 a}-\frac {\left (b^2 \sin (a)\right ) \int \frac {\cos (b x)}{x} \, dx}{2 a}\\ &=-\frac {b \cos (a+b x)}{2 a x}-\frac {b^2 \cos (a) \text {Ci}(b x)}{2 a^2}+\frac {b^2 \text {Ci}(a+b x)}{2 a^2}-\frac {\text {Ci}(a+b x)}{2 x^2}-\frac {b^2 \text {Ci}(b x) \sin (a)}{2 a}-\frac {b^2 \cos (a) \text {Si}(b x)}{2 a}+\frac {b^2 \sin (a) \text {Si}(b x)}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 76, normalized size = 0.68 \[ -\frac {\left (a^2-b^2 x^2\right ) \text {Ci}(a+b x)+b^2 x^2 (a \sin (a)+\cos (a)) \text {Ci}(b x)+b x (b x (a \cos (a)-\sin (a)) \text {Si}(b x)+a \cos (a+b x))}{2 a^2 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Ci}\left (b x + a\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.55, size = 748, normalized size = 6.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 88, normalized size = 0.79 \[ b^{2} \left (-\frac {\Ci \left (b x +a \right )}{2 b^{2} x^{2}}-\frac {-\Si \left (b x \right ) \sin \relax (a )+\Ci \left (b x \right ) \cos \relax (a )}{2 a^{2}}+\frac {-\frac {\cos \left (b x +a \right )}{b x}-\Si \left (b x \right ) \cos \relax (a )-\Ci \left (b x \right ) \sin \relax (a )}{2 a}+\frac {\Ci \left (b x +a \right )}{2 a^{2}}\right ) \]
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 \[ \int \frac {{\rm Ci}\left (b x + a\right )}{x^{3}}\,{d x} \]
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 \[ \int \frac {\mathrm {cosint}\left (a+b\,x\right )}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {Ci}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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