Optimal. Leaf size=74 \[ -\frac {a}{4 x^4}-\frac {b d \cos \left (c+d x^2\right )}{4 x^2}-\frac {1}{4} b d^2 \text {Ci}\left (d x^2\right ) \sin (c)-\frac {b \sin \left (c+d x^2\right )}{4 x^4}-\frac {1}{4} b d^2 \cos (c) \text {Si}\left (d x^2\right ) \]
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Rubi [A]
time = 0.08, 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, 3460, 3378,
3384, 3380, 3383} \begin {gather*} -\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 d \cos \left (c+d x^2\right )}{4 x^2}-\frac {b \sin \left (c+d x^2\right )}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3460
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 \text {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) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {\sin (d x)}{x} \, dx,x,x^2\right )-\frac {1}{4} \left (b d^2 \sin (c)\right ) \text {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 = 0.06, size = 86, normalized size = 1.16 \begin {gather*} -\frac {a}{4 x^4}-\frac {b \cos \left (d x^2\right ) \left (d x^2 \cos (c)+\sin (c)\right )}{4 x^4}+\frac {b \left (-\cos (c)+d x^2 \sin (c)\right ) \sin \left (d x^2\right )}{4 x^4}-\frac {1}{4} b d^2 \left (\text {Ci}\left (d x^2\right ) \sin (c)+\cos (c) \text {Si}\left (d x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 65, normalized size = 0.88
method | result | size |
default | \(-\frac {a}{4 x^{4}}+b \left (-\frac {\sin \left (d \,x^{2}+c \right )}{4 x^{4}}+\frac {d \left (-\frac {\cos \left (d \,x^{2}+c \right )}{2 x^{2}}-d \left (\frac {\cos \left (c \right ) \sinIntegral \left (d \,x^{2}\right )}{2}+\frac {\sin \left (c \right ) \cosineIntegral \left (d \,x^{2}\right )}{2}\right )\right )}{2}\right )\) | \(65\) |
risch | \(-\frac {a}{4 x^{4}}+\frac {\pi \,\mathrm {csgn}\left (d \,x^{2}\right ) {\mathrm e}^{-i c} b \,d^{2}}{8}-\frac {\sinIntegral \left (d \,x^{2}\right ) {\mathrm e}^{-i c} b \,d^{2}}{4}+\frac {i \expIntegral \left (1, -i d \,x^{2}\right ) {\mathrm e}^{-i c} b \,d^{2}}{8}-\frac {i b \,d^{2} \expIntegral \left (1, -i d \,x^{2}\right ) {\mathrm e}^{i c}}{8}-\frac {b d \cos \left (d \,x^{2}+c \right )}{4 x^{2}}-\frac {b \sin \left (d \,x^{2}+c \right )}{4 x^{4}}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.36, size = 58, normalized size = 0.78 \begin {gather*} \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 \left (c\right ) + {\left (\Gamma \left (-2, i \, d x^{2}\right ) + \Gamma \left (-2, -i \, d x^{2}\right )\right )} \sin \left (c\right )\right )} b d^{2} - \frac {a}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 85, normalized size = 1.15 \begin {gather*} -\frac {2 \, b d^{2} x^{4} \cos \left (c\right ) \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 \left (c\right ) + 2 \, a}{8 \, x^{4}} \end {gather*}
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 \begin {gather*} \int \frac {a + b \sin {\left (c + d x^{2} \right )}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 204 vs.
\(2 (64) = 128\).
time = 3.70, size = 204, normalized size = 2.76 \begin {gather*} -\frac {{\left (d x^{2} + c\right )}^{2} b d^{3} \operatorname {Ci}\left (d x^{2}\right ) \sin \left (c\right ) - 2 \, {\left (d x^{2} + c\right )} b c d^{3} \operatorname {Ci}\left (d x^{2}\right ) \sin \left (c\right ) + b c^{2} d^{3} \operatorname {Ci}\left (d x^{2}\right ) \sin \left (c\right ) + {\left (d x^{2} + c\right )}^{2} b d^{3} \cos \left (c\right ) \operatorname {Si}\left (d x^{2}\right ) - 2 \, {\left (d x^{2} + c\right )} b c d^{3} \cos \left (c\right ) \operatorname {Si}\left (d x^{2}\right ) + b c^{2} d^{3} \cos \left (c\right ) \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} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\sin \left (d\,x^2+c\right )}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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