Optimal. Leaf size=149 \[ -\frac {a d \cos (c+d x)}{12 x^3}-\frac {b d \cos (c+d x)}{2 x}+\frac {a d^3 \cos (c+d x)}{24 x}-\frac {1}{2} b d^2 \text {Ci}(d x) \sin (c)+\frac {1}{24} a d^4 \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{4 x^4}-\frac {b \sin (c+d x)}{2 x^2}+\frac {a d^2 \sin (c+d x)}{24 x^2}-\frac {1}{2} b d^2 \cos (c) \text {Si}(d x)+\frac {1}{24} a d^4 \cos (c) \text {Si}(d x) \]
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Rubi [A]
time = 0.19, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3420, 3378,
3384, 3380, 3383} \begin {gather*} \frac {1}{24} a d^4 \sin (c) \text {CosIntegral}(d x)+\frac {1}{24} a d^4 \cos (c) \text {Si}(d x)+\frac {a d^3 \cos (c+d x)}{24 x}+\frac {a d^2 \sin (c+d x)}{24 x^2}-\frac {a \sin (c+d x)}{4 x^4}-\frac {a d \cos (c+d x)}{12 x^3}-\frac {1}{2} b d^2 \sin (c) \text {CosIntegral}(d x)-\frac {1}{2} b d^2 \cos (c) \text {Si}(d x)-\frac {b \sin (c+d x)}{2 x^2}-\frac {b d \cos (c+d x)}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3420
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^5} \, dx &=\int \left (\frac {a \sin (c+d x)}{x^5}+\frac {b \sin (c+d x)}{x^3}\right ) \, dx\\ &=a \int \frac {\sin (c+d x)}{x^5} \, dx+b \int \frac {\sin (c+d x)}{x^3} \, dx\\ &=-\frac {a \sin (c+d x)}{4 x^4}-\frac {b \sin (c+d x)}{2 x^2}+\frac {1}{4} (a d) \int \frac {\cos (c+d x)}{x^4} \, dx+\frac {1}{2} (b d) \int \frac {\cos (c+d x)}{x^2} \, dx\\ &=-\frac {a d \cos (c+d x)}{12 x^3}-\frac {b d \cos (c+d x)}{2 x}-\frac {a \sin (c+d x)}{4 x^4}-\frac {b \sin (c+d x)}{2 x^2}-\frac {1}{12} \left (a d^2\right ) \int \frac {\sin (c+d x)}{x^3} \, dx-\frac {1}{2} \left (b d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{12 x^3}-\frac {b d \cos (c+d x)}{2 x}-\frac {a \sin (c+d x)}{4 x^4}-\frac {b \sin (c+d x)}{2 x^2}+\frac {a d^2 \sin (c+d x)}{24 x^2}-\frac {1}{24} \left (a d^3\right ) \int \frac {\cos (c+d x)}{x^2} \, dx-\frac {1}{2} \left (b d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx-\frac {1}{2} \left (b d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{12 x^3}-\frac {b d \cos (c+d x)}{2 x}+\frac {a d^3 \cos (c+d x)}{24 x}-\frac {1}{2} b d^2 \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{4 x^4}-\frac {b \sin (c+d x)}{2 x^2}+\frac {a d^2 \sin (c+d x)}{24 x^2}-\frac {1}{2} b d^2 \cos (c) \text {Si}(d x)+\frac {1}{24} \left (a d^4\right ) \int \frac {\sin (c+d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{12 x^3}-\frac {b d \cos (c+d x)}{2 x}+\frac {a d^3 \cos (c+d x)}{24 x}-\frac {1}{2} b d^2 \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{4 x^4}-\frac {b \sin (c+d x)}{2 x^2}+\frac {a d^2 \sin (c+d x)}{24 x^2}-\frac {1}{2} b d^2 \cos (c) \text {Si}(d x)+\frac {1}{24} \left (a d^4 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx+\frac {1}{24} \left (a d^4 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{12 x^3}-\frac {b d \cos (c+d x)}{2 x}+\frac {a d^3 \cos (c+d x)}{24 x}-\frac {1}{2} b d^2 \text {Ci}(d x) \sin (c)+\frac {1}{24} a d^4 \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{4 x^4}-\frac {b \sin (c+d x)}{2 x^2}+\frac {a d^2 \sin (c+d x)}{24 x^2}-\frac {1}{2} b d^2 \cos (c) \text {Si}(d x)+\frac {1}{24} a d^4 \cos (c) \text {Si}(d x)\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 125, normalized size = 0.84 \begin {gather*} \frac {-2 a d x \cos (c+d x)-12 b d x^3 \cos (c+d x)+a d^3 x^3 \cos (c+d x)+d^2 \left (-12 b+a d^2\right ) x^4 \text {Ci}(d x) \sin (c)-6 a \sin (c+d x)-12 b x^2 \sin (c+d x)+a d^2 x^2 \sin (c+d x)+d^2 \left (-12 b+a d^2\right ) x^4 \cos (c) \text {Si}(d x)}{24 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 131, normalized size = 0.88
method | result | size |
derivativedivides | \(d^{4} \left (a \left (-\frac {\sin \left (d x +c \right )}{4 d^{4} x^{4}}-\frac {\cos \left (d x +c \right )}{12 d^{3} x^{3}}+\frac {\sin \left (d x +c \right )}{24 d^{2} x^{2}}+\frac {\cos \left (d x +c \right )}{24 d x}+\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )}{24}+\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{24}\right )+\frac {b \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )}{2}-\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{2}\right )}{d^{2}}\right )\) | \(131\) |
default | \(d^{4} \left (a \left (-\frac {\sin \left (d x +c \right )}{4 d^{4} x^{4}}-\frac {\cos \left (d x +c \right )}{12 d^{3} x^{3}}+\frac {\sin \left (d x +c \right )}{24 d^{2} x^{2}}+\frac {\cos \left (d x +c \right )}{24 d x}+\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )}{24}+\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{24}\right )+\frac {b \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )}{2}-\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{2}\right )}{d^{2}}\right )\) | \(131\) |
risch | \(-\frac {i \cos \left (c \right ) \expIntegral \left (1, i d x \right ) a \,d^{4}}{48}+\frac {i \cos \left (c \right ) \expIntegral \left (1, -i d x \right ) a \,d^{4}}{48}+\frac {i \expIntegral \left (1, i d x \right ) \cos \left (c \right ) b \,d^{2}}{4}-\frac {i \expIntegral \left (1, -i d x \right ) \cos \left (c \right ) b \,d^{2}}{4}-\frac {\sin \left (c \right ) \expIntegral \left (1, i d x \right ) a \,d^{4}}{48}-\frac {\sin \left (c \right ) \expIntegral \left (1, -i d x \right ) a \,d^{4}}{48}+\frac {\expIntegral \left (1, i d x \right ) \sin \left (c \right ) b \,d^{2}}{4}+\frac {\expIntegral \left (1, -i d x \right ) \sin \left (c \right ) b \,d^{2}}{4}-\frac {i \left (2 i a \,d^{9} x^{7}-24 i b \,d^{7} x^{7}-4 i a \,d^{7} x^{5}\right ) \cos \left (d x +c \right )}{48 d^{6} x^{8}}-\frac {\left (-2 a \,d^{8} x^{6}+24 b \,d^{6} x^{6}+12 a \,d^{6} x^{4}\right ) \sin \left (d x +c \right )}{48 d^{6} x^{8}}\) | \(214\) |
meijerg | \(\frac {d^{2} b \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+\ln \left (d^{2}\right )\right )}{\sqrt {\pi }}+\frac {-6 d^{2} x^{2}+4}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \gamma }{\sqrt {\pi }}+\frac {4 \ln \left (2\right )}{\sqrt {\pi }}+\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \cos \left (d x \right )}{\sqrt {\pi }\, d^{2} x^{2}}+\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, d x}-\frac {4 \cosineIntegral \left (d x \right )}{\sqrt {\pi }}\right )}{8}+\frac {d^{2} b \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {4 \cos \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{d^{2} x^{2} \sqrt {\pi }}-\frac {4 \sinIntegral \left (d x \right )}{\sqrt {\pi }}\right )}{8}+\frac {a \sqrt {\pi }\, \sin \left (c \right ) d^{4} \left (-\frac {8}{\sqrt {\pi }\, x^{4} d^{4}}+\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {\frac {4 \gamma }{3}-\frac {25}{9}+\frac {4 \ln \left (x \right )}{3}+\frac {2 \ln \left (d^{2}\right )}{3}}{\sqrt {\pi }}+\frac {\frac {25}{9} d^{4} x^{4}-8 d^{2} x^{2}+8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {4 \gamma }{3 \sqrt {\pi }}-\frac {4 \ln \left (2\right )}{3 \sqrt {\pi }}-\frac {4 \ln \left (\frac {d x}{2}\right )}{3 \sqrt {\pi }}-\frac {8 \left (-\frac {15 d^{2} x^{2}}{2}+45\right ) \cos \left (d x \right )}{45 \sqrt {\pi }\, d^{4} x^{4}}+\frac {8 \left (-\frac {15 d^{2} x^{2}}{2}+15\right ) \sin \left (d x \right )}{45 \sqrt {\pi }\, d^{3} x^{3}}+\frac {4 \cosineIntegral \left (d x \right )}{3 \sqrt {\pi }}\right )}{32}+\frac {a \sqrt {\pi }\, \cos \left (c \right ) d^{4} \left (-\frac {8 \left (-\frac {d^{2} x^{2}}{2}+1\right ) \cos \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 \left (-\frac {d^{2} x^{2}}{2}+3\right ) \sin \left (d x \right )}{3 d^{4} x^{4} \sqrt {\pi }}+\frac {4 \sinIntegral \left (d x \right )}{3 \sqrt {\pi }}\right )}{32}\) | \(411\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.89, size = 121, normalized size = 0.81 \begin {gather*} -\frac {{\left ({\left (a {\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{6} - 12 \, {\left (b {\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + b {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4}\right )} x^{4} + 2 \, b d x \cos \left (d x + c\right ) + 6 \, b \sin \left (d x + c\right )}{2 \, d^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 127, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (a d^{4} - 12 \, b d^{2}\right )} x^{4} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) + 2 \, {\left ({\left (a d^{3} - 12 \, b d\right )} x^{3} - 2 \, a d x\right )} \cos \left (d x + c\right ) + 2 \, {\left ({\left (a d^{2} - 12 \, b\right )} x^{2} - 6 \, a\right )} \sin \left (d x + c\right ) + {\left ({\left (a d^{4} - 12 \, b d^{2}\right )} x^{4} \operatorname {Ci}\left (d x\right ) + {\left (a d^{4} - 12 \, b d^{2}\right )} x^{4} \operatorname {Ci}\left (-d x\right )\right )} \sin \left (c\right )}{48 \, 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 {\left (a + b x^{2}\right ) \sin {\left (c + d x \right )}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 5.90, size = 1086, normalized size = 7.29 \begin {gather*} \text {Too large to display} \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 {\sin \left (c+d\,x\right )\,\left (b\,x^2+a\right )}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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