Optimal. Leaf size=74 \[ -\frac {a}{4 x^4}-\frac {1}{4} b d^2 \sin (c) \text {Ci}\left (d x^2\right )-\frac {1}{4} b d^2 \cos (c) \text {Si}\left (d x^2\right )-\frac {b d \cos \left (c+d x^2\right )}{4 x^2}-\frac {b \sin \left (c+d x^2\right )}{4 x^4} \]
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Rubi [A] time = 0.13, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {14, 3379, 3297, 3303, 3299, 3302} \[ -\frac {a}{4 x^4}-\frac {1}{4} b d^2 \sin (c) \text {CosIntegral}\left (d x^2\right )-\frac {1}{4} b d^2 \cos (c) \text {Si}\left (d x^2\right )-\frac {b \sin \left (c+d x^2\right )}{4 x^4}-\frac {b d \cos \left (c+d x^2\right )}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3379
Rubi steps
\begin {align*} \int \frac {a+b \sin \left (c+d x^2\right )}{x^5} \, dx &=\int \left (\frac {a}{x^5}+\frac {b \sin \left (c+d x^2\right )}{x^5}\right ) \, dx\\ &=-\frac {a}{4 x^4}+b \int \frac {\sin \left (c+d x^2\right )}{x^5} \, dx\\ &=-\frac {a}{4 x^4}+\frac {1}{2} b \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{x^3} \, dx,x,x^2\right )\\ &=-\frac {a}{4 x^4}-\frac {b \sin \left (c+d x^2\right )}{4 x^4}+\frac {1}{4} (b d) \operatorname {Subst}\left (\int \frac {\cos (c+d x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {a}{4 x^4}-\frac {b d \cos \left (c+d x^2\right )}{4 x^2}-\frac {b \sin \left (c+d x^2\right )}{4 x^4}-\frac {1}{4} \left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{x} \, dx,x,x^2\right )\\ &=-\frac {a}{4 x^4}-\frac {b d \cos \left (c+d x^2\right )}{4 x^2}-\frac {b \sin \left (c+d x^2\right )}{4 x^4}-\frac {1}{4} \left (b d^2 \cos (c)\right ) \operatorname {Subst}\left (\int \frac {\sin (d x)}{x} \, dx,x,x^2\right )-\frac {1}{4} \left (b d^2 \sin (c)\right ) \operatorname {Subst}\left (\int \frac {\cos (d x)}{x} \, dx,x,x^2\right )\\ &=-\frac {a}{4 x^4}-\frac {b d \cos \left (c+d x^2\right )}{4 x^2}-\frac {1}{4} b d^2 \text {Ci}\left (d x^2\right ) \sin (c)-\frac {b \sin \left (c+d x^2\right )}{4 x^4}-\frac {1}{4} b d^2 \cos (c) \text {Si}\left (d x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 86, normalized size = 1.16 \[ -\frac {a}{4 x^4}-\frac {1}{4} b d^2 \left (\sin (c) \text {Ci}\left (d x^2\right )+\cos (c) \text {Si}\left (d x^2\right )\right )-\frac {b \cos \left (d x^2\right ) \left (d x^2 \cos (c)+\sin (c)\right )}{4 x^4}+\frac {b \sin \left (d x^2\right ) \left (d x^2 \sin (c)-\cos (c)\right )}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 85, normalized size = 1.15 \[ -\frac {2 \, b d^{2} x^{4} \cos \relax (c) \operatorname {Si}\left (d x^{2}\right ) + 2 \, b d x^{2} \cos \left (d x^{2} + c\right ) + 2 \, b \sin \left (d x^{2} + c\right ) + {\left (b d^{2} x^{4} \operatorname {Ci}\left (d x^{2}\right ) + b d^{2} x^{4} \operatorname {Ci}\left (-d x^{2}\right )\right )} \sin \relax (c) + 2 \, a}{8 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 204, normalized size = 2.76 \[ -\frac {{\left (d x^{2} + c\right )}^{2} b d^{3} \operatorname {Ci}\left (d x^{2}\right ) \sin \relax (c) - 2 \, {\left (d x^{2} + c\right )} b c d^{3} \operatorname {Ci}\left (d x^{2}\right ) \sin \relax (c) + b c^{2} d^{3} \operatorname {Ci}\left (d x^{2}\right ) \sin \relax (c) + {\left (d x^{2} + c\right )}^{2} b d^{3} \cos \relax (c) \operatorname {Si}\left (d x^{2}\right ) - 2 \, {\left (d x^{2} + c\right )} b c d^{3} \cos \relax (c) \operatorname {Si}\left (d x^{2}\right ) + b c^{2} d^{3} \cos \relax (c) \operatorname {Si}\left (d x^{2}\right ) + {\left (d x^{2} + c\right )} b d^{3} \cos \left (d x^{2} + c\right ) - b c d^{3} \cos \left (d x^{2} + c\right ) + b d^{3} \sin \left (d x^{2} + c\right ) + a d^{3}}{4 \, {\left ({\left (d x^{2} + c\right )}^{2} - 2 \, {\left (d x^{2} + c\right )} c + c^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 65, normalized size = 0.88 \[ -\frac {a}{4 x^{4}}+b \left (-\frac {\sin \left (d \,x^{2}+c \right )}{4 x^{4}}+\frac {d \left (-\frac {\cos \left (d \,x^{2}+c \right )}{2 x^{2}}-d \left (\frac {\cos \relax (c ) \Si \left (d \,x^{2}\right )}{2}+\frac {\sin \relax (c ) \Ci \left (d \,x^{2}\right )}{2}\right )\right )}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.00, size = 58, normalized size = 0.78 \[ \frac {1}{4} \, {\left ({\left (i \, \Gamma \left (-2, i \, d x^{2}\right ) - i \, \Gamma \left (-2, -i \, d x^{2}\right )\right )} \cos \relax (c) + {\left (\Gamma \left (-2, i \, d x^{2}\right ) + \Gamma \left (-2, -i \, d x^{2}\right )\right )} \sin \relax (c)\right )} b d^{2} - \frac {a}{4 \, x^{4}} \]
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 \[ \int \frac {a+b\,\sin \left (d\,x^2+c\right )}{x^5} \,d x \]
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 \[ \int \frac {a + b \sin {\left (c + d x^{2} \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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