Optimal. Leaf size=132 \[ -\frac {a d \cos (c+d x)}{6 x^2}-\frac {b d \cos (c+d x)}{2 x}-\frac {1}{6} a d^3 \cos (c) \text {Ci}(d x)-\frac {1}{2} b d^2 \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{2 x^2}+\frac {a d^2 \sin (c+d x)}{6 x}-\frac {1}{2} b d^2 \cos (c) \text {Si}(d x)+\frac {1}{6} a d^3 \sin (c) \text {Si}(d x) \]
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Rubi [A]
time = 0.22, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6874, 3378,
3384, 3380, 3383} \begin {gather*} -\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}-\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 6874
Rubi steps
\begin {align*} \int \frac {(a+b x) \sin (c+d x)}{x^4} \, dx &=\int \left (\frac {a \sin (c+d x)}{x^4}+\frac {b \sin (c+d x)}{x^3}\right ) \, dx\\ &=a \int \frac {\sin (c+d x)}{x^4} \, dx+b \int \frac {\sin (c+d x)}{x^3} \, dx\\ &=-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{2 x^2}+\frac {1}{3} (a d) \int \frac {\cos (c+d x)}{x^3} \, dx+\frac {1}{2} (b d) \int \frac {\cos (c+d x)}{x^2} \, dx\\ &=-\frac {a d \cos (c+d x)}{6 x^2}-\frac {b d \cos (c+d x)}{2 x}-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{2 x^2}-\frac {1}{6} \left (a d^2\right ) \int \frac {\sin (c+d x)}{x^2} \, dx-\frac {1}{2} \left (b d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{6 x^2}-\frac {b d \cos (c+d x)}{2 x}-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{2 x^2}+\frac {a d^2 \sin (c+d x)}{6 x}-\frac {1}{6} \left (a d^3\right ) \int \frac {\cos (c+d x)}{x} \, 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)}{6 x^2}-\frac {b d \cos (c+d x)}{2 x}-\frac {1}{2} b d^2 \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{2 x^2}+\frac {a d^2 \sin (c+d x)}{6 x}-\frac {1}{2} b d^2 \cos (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}-\frac {b d \cos (c+d x)}{2 x}-\frac {1}{6} a d^3 \cos (c) \text {Ci}(d x)-\frac {1}{2} b d^2 \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{3 x^3}-\frac {b \sin (c+d x)}{2 x^2}+\frac {a d^2 \sin (c+d x)}{6 x}-\frac {1}{2} b d^2 \cos (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 = 0.23, size = 110, normalized size = 0.83 \begin {gather*} -\frac {a d x \cos (c+d x)+3 b d x^2 \cos (c+d x)+d^2 x^3 \text {Ci}(d x) (a d \cos (c)+3 b \sin (c))+2 a \sin (c+d x)+3 b x \sin (c+d x)-a d^2 x^2 \sin (c+d x)+d^2 x^3 (3 b \cos (c)-a d \sin (c)) \text {Si}(d x)}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 117, normalized size = 0.89
method | result | size |
derivativedivides | \(d^{3} \left (a \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\sinIntegral \left (d x \right ) \sin \left (c \right )}{6}-\frac {\cosineIntegral \left (d x \right ) \cos \left (c \right )}{6}\right )+\frac {b \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )}{2}-\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{2}\right )}{d}\right )\) | \(117\) |
default | \(d^{3} \left (a \left (-\frac {\sin \left (d x +c \right )}{3 d^{3} x^{3}}-\frac {\cos \left (d x +c \right )}{6 d^{2} x^{2}}+\frac {\sin \left (d x +c \right )}{6 d x}+\frac {\sinIntegral \left (d x \right ) \sin \left (c \right )}{6}-\frac {\cosineIntegral \left (d x \right ) \cos \left (c \right )}{6}\right )+\frac {b \left (-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )}{2}-\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{2}\right )}{d}\right )\) | \(117\) |
risch | \(\frac {i \expIntegral \left (1, i d x \right ) \cos \left (c \right ) b \,d^{2}}{4}+\frac {\expIntegral \left (1, i d x \right ) \cos \left (c \right ) a \,d^{3}}{12}-\frac {i \expIntegral \left (1, -i d x \right ) \cos \left (c \right ) b \,d^{2}}{4}+\frac {\expIntegral \left (1, -i d x \right ) \cos \left (c \right ) a \,d^{3}}{12}+\frac {\expIntegral \left (1, i d x \right ) \sin \left (c \right ) b \,d^{2}}{4}-\frac {i \expIntegral \left (1, i d x \right ) \sin \left (c \right ) a \,d^{3}}{12}+\frac {\expIntegral \left (1, -i d x \right ) \sin \left (c \right ) b \,d^{2}}{4}+\frac {i \expIntegral \left (1, -i d x \right ) \sin \left (c \right ) a \,d^{3}}{12}-\frac {\left (6 b \,d^{5} x^{5}+2 a \,d^{5} x^{4}\right ) \cos \left (d x +c \right )}{12 d^{4} x^{6}}+\frac {i \left (-2 i a \,d^{6} x^{5}+6 i b \,d^{4} x^{4}+4 i a \,d^{4} x^{3}\right ) \sin \left (d x +c \right )}{12 d^{4} x^{6}}\) | \(205\) |
meijerg | \(\frac {d^{2} b \sqrt {\pi }\, \sin \left (c \right ) \left (-\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+\ln \left (d^{2}\right )\right )}{\sqrt {\pi }}+\frac {-6 d^{2} x^{2}+4}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \gamma }{\sqrt {\pi }}+\frac {4 \ln \left (2\right )}{\sqrt {\pi }}+\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \cos \left (d x \right )}{\sqrt {\pi }\, d^{2} x^{2}}+\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, d x}-\frac {4 \cosineIntegral \left (d x \right )}{\sqrt {\pi }}\right )}{8}+\frac {d^{2} b \sqrt {\pi }\, \cos \left (c \right ) \left (-\frac {4 \cos \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{d^{2} x^{2} \sqrt {\pi }}-\frac {4 \sinIntegral \left (d x \right )}{\sqrt {\pi }}\right )}{8}+\frac {a \sqrt {\pi }\, \sin \left (c \right ) d^{4} \left (-\frac {8 \left (-d^{2} x^{2}+2\right ) d^{2} \cos \left (x \sqrt {d^{2}}\right )}{3 x^{3} \left (d^{2}\right )^{\frac {5}{2}} \sqrt {\pi }}+\frac {8 \sin \left (x \sqrt {d^{2}}\right )}{3 d^{2} x^{2} \sqrt {\pi }}+\frac {8 \sinIntegral \left (x \sqrt {d^{2}}\right )}{3 \sqrt {\pi }}\right )}{16 \sqrt {d^{2}}}+\frac {a \sqrt {\pi }\, \cos \left (c \right ) d^{3} \left (-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (d \right )\right )}{3 \sqrt {\pi }}+\frac {-\frac {44 d^{2} x^{2}}{9}+8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \gamma }{3 \sqrt {\pi }}+\frac {8 \ln \left (2\right )}{3 \sqrt {\pi }}+\frac {8 \ln \left (\frac {d x}{2}\right )}{3 \sqrt {\pi }}-\frac {8 \cos \left (d x \right )}{3 \sqrt {\pi }\, d^{2} x^{2}}-\frac {16 \left (-\frac {5 d^{2} x^{2}}{2}+5\right ) \sin \left (d x \right )}{15 \sqrt {\pi }\, d^{3} x^{3}}-\frac {8 \cosineIntegral \left (d x \right )}{3 \sqrt {\pi }}\right )}{16}\) | \(394\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.65, size = 111, normalized size = 0.84 \begin {gather*} -\frac {{\left ({\left (a {\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \cos \left (c\right ) + a {\left (-i \, \Gamma \left (-3, i \, d x\right ) + i \, \Gamma \left (-3, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4} - 3 \, {\left (b {\left (-i \, \Gamma \left (-3, i \, d x\right ) + i \, \Gamma \left (-3, -i \, d x\right )\right )} \cos \left (c\right ) - b {\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{3}\right )} x^{3} + 2 \, b \cos \left (d x + c\right )}{2 \, d x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 137, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (3 \, b d x^{2} + a d x\right )} \cos \left (d x + c\right ) + {\left (a d^{3} x^{3} \operatorname {Ci}\left (d x\right ) + a d^{3} x^{3} \operatorname {Ci}\left (-d x\right ) + 6 \, b d^{2} x^{3} \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) - 2 \, {\left (a d^{2} x^{2} - 3 \, b x - 2 \, a\right )} \sin \left (d x + c\right ) - {\left (2 \, a d^{3} x^{3} \operatorname {Si}\left (d x\right ) - 3 \, b d^{2} x^{3} \operatorname {Ci}\left (d x\right ) - 3 \, b d^{2} x^{3} \operatorname {Ci}\left (-d x\right )\right )} \sin \left (c\right )}{12 \, x^{3}} \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\right ) \sin {\left (c + d x \right )}}{x^{4}}\, 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 = 3.25, size = 961, normalized size = 7.28 \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 (a+b\,x\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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