Optimal. Leaf size=106 \[ -\frac {1}{6} a d^3 \cos (c) \text {Ci}(d x)+\frac {1}{6} a d^3 \sin (c) \text {Si}(d x)+\frac {a d^2 \sin (c+d x)}{6 x}-\frac {a \sin (c+d x)}{3 x^3}-\frac {a d \cos (c+d x)}{6 x^2}+b d \cos (c) \text {Ci}(d x)-b d \sin (c) \text {Si}(d x)-\frac {b \sin (c+d x)}{x} \]
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Rubi [A] time = 0.21, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3339, 3297, 3303, 3299, 3302} \[ -\frac {1}{6} a d^3 \cos (c) \text {CosIntegral}(d x)+\frac {1}{6} a d^3 \sin (c) \text {Si}(d x)+\frac {a d^2 \sin (c+d x)}{6 x}-\frac {a \sin (c+d x)}{3 x^3}-\frac {a d \cos (c+d x)}{6 x^2}+b d \cos (c) \text {CosIntegral}(d x)-b d \sin (c) \text {Si}(d x)-\frac {b \sin (c+d x)}{x} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3339
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^4} \, dx &=\int \left (\frac {a \sin (c+d x)}{x^4}+\frac {b \sin (c+d x)}{x^2}\right ) \, dx\\ &=a \int \frac {\sin (c+d x)}{x^4} \, dx+b \int \frac {\sin (c+d x)}{x^2} \, dx\\ &=-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{x}+\frac {1}{3} (a d) \int \frac {\cos (c+d x)}{x^3} \, dx+(b d) \int \frac {\cos (c+d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{6 x^2}-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{x}-\frac {1}{6} \left (a d^2\right ) \int \frac {\sin (c+d x)}{x^2} \, dx+(b d \cos (c)) \int \frac {\cos (d x)}{x} \, dx-(b d \sin (c)) \int \frac {\sin (d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{6 x^2}+b d \cos (c) \text {Ci}(d x)-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{x}+\frac {a d^2 \sin (c+d x)}{6 x}-b d \sin (c) \text {Si}(d x)-\frac {1}{6} \left (a d^3\right ) \int \frac {\cos (c+d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{6 x^2}+b d \cos (c) \text {Ci}(d x)-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{x}+\frac {a d^2 \sin (c+d x)}{6 x}-b d \sin (c) \text {Si}(d x)-\frac {1}{6} \left (a d^3 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx+\frac {1}{6} \left (a d^3 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{6 x^2}+b d \cos (c) \text {Ci}(d x)-\frac {1}{6} a d^3 \cos (c) \text {Ci}(d x)-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{x}+\frac {a d^2 \sin (c+d x)}{6 x}-b d \sin (c) \text {Si}(d x)+\frac {1}{6} a d^3 \sin (c) \text {Si}(d x)\\ \end {align*}
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Mathematica [A] time = 0.20, size = 95, normalized size = 0.90 \[ \frac {d x^3 \cos (c) \left (6 b-a d^2\right ) \text {Ci}(d x)+d x^3 \sin (c) \left (a d^2-6 b\right ) \text {Si}(d x)+a d^2 x^2 \sin (c+d x)-2 a \sin (c+d x)-a d x \cos (c+d x)-6 b x^2 \sin (c+d x)}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 105, normalized size = 0.99 \[ \frac {2 \, {\left (a d^{3} - 6 \, b d\right )} x^{3} \sin \relax (c) \operatorname {Si}\left (d x\right ) - 2 \, a d x \cos \left (d x + c\right ) - {\left ({\left (a d^{3} - 6 \, b d\right )} x^{3} \operatorname {Ci}\left (d x\right ) + {\left (a d^{3} - 6 \, b d\right )} x^{3} \operatorname {Ci}\left (-d x\right )\right )} \cos \relax (c) + 2 \, {\left ({\left (a d^{2} - 6 \, b\right )} x^{2} - 2 \, a\right )} \sin \left (d x + c\right )}{12 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.42, size = 834, normalized size = 7.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 102, normalized size = 0.96 \[ d^{3} \left (\frac {b \left (-\frac {\sin \left (d x +c \right )}{x d}-\Si \left (d x \right ) \sin \relax (c )+\Ci \left (d x \right ) \cos \relax (c )\right )}{d^{2}}+a \left (-\frac {\sin \left (d x +c \right )}{3 x^{3} d^{3}}-\frac {\cos \left (d x +c \right )}{6 x^{2} d^{2}}+\frac {\sin \left (d x +c \right )}{6 x d}+\frac {\Si \left (d x \right ) \sin \relax (c )}{6}-\frac {\Ci \left (d x \right ) \cos \relax (c )}{6}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.13, size = 123, normalized size = 1.16 \[ -\frac {{\left ({\left (a {\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \cos \relax (c) + a {\left (-i \, \Gamma \left (-3, i \, d x\right ) + i \, \Gamma \left (-3, -i \, d x\right )\right )} \sin \relax (c)\right )} d^{5} - {\left (6 \, b {\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \cos \relax (c) - b {\left (6 i \, \Gamma \left (-3, i \, d x\right ) - 6 i \, \Gamma \left (-3, -i \, d x\right )\right )} \sin \relax (c)\right )} d^{3}\right )} x^{3} + 2 \, b d x \cos \left (d x + c\right ) + 4 \, b \sin \left (d x + c\right )}{2 \, d^{2} x^{3}} \]
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 \[ \int \frac {\sin \left (c+d\,x\right )\,\left (b\,x^2+a\right )}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b x^{2}\right ) \sin {\left (c + d x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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