Optimal. Leaf size=44 \[ -\frac {b \cos (c+d x)}{d}+a d \cos (c) \text {Ci}(d x)-\frac {a \sin (c+d x)}{x}-a d \sin (c) \text {Si}(d x) \]
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Rubi [A]
time = 0.07, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3420, 2718,
3378, 3384, 3380, 3383} \begin {gather*} a d \cos (c) \text {CosIntegral}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}-\frac {b \cos (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
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^2} \, dx &=\int \left (b \sin (c+d x)+\frac {a \sin (c+d x)}{x^2}\right ) \, dx\\ &=a \int \frac {\sin (c+d x)}{x^2} \, dx+b \int \sin (c+d x) \, dx\\ &=-\frac {b \cos (c+d x)}{d}-\frac {a \sin (c+d x)}{x}+(a d) \int \frac {\cos (c+d x)}{x} \, dx\\ &=-\frac {b \cos (c+d x)}{d}-\frac {a \sin (c+d x)}{x}+(a d \cos (c)) \int \frac {\cos (d x)}{x} \, dx-(a d \sin (c)) \int \frac {\sin (d x)}{x} \, dx\\ &=-\frac {b \cos (c+d x)}{d}+a d \cos (c) \text {Ci}(d x)-\frac {a \sin (c+d x)}{x}-a d \sin (c) \text {Si}(d x)\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 44, normalized size = 1.00 \begin {gather*} -\frac {b \cos (c+d x)}{d}+a d \cos (c) \text {Ci}(d x)-\frac {a \sin (c+d x)}{x}-a d \sin (c) \text {Si}(d x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 48, normalized size = 1.09
method | result | size |
derivativedivides | \(d \left (a \left (-\frac {\sin \left (d x +c \right )}{d x}-\sinIntegral \left (d x \right ) \sin \left (c \right )+\cosineIntegral \left (d x \right ) \cos \left (c \right )\right )-\frac {b \cos \left (d x +c \right )}{d^{2}}\right )\) | \(48\) |
default | \(d \left (a \left (-\frac {\sin \left (d x +c \right )}{d x}-\sinIntegral \left (d x \right ) \sin \left (c \right )+\cosineIntegral \left (d x \right ) \cos \left (c \right )\right )-\frac {b \cos \left (d x +c \right )}{d^{2}}\right )\) | \(48\) |
risch | \(-\frac {d \cos \left (c \right ) a \expIntegral \left (1, i d x \right )}{2}-\frac {d \cos \left (c \right ) a \expIntegral \left (1, -i d x \right )}{2}+\frac {i d \sin \left (c \right ) a \expIntegral \left (1, i d x \right )}{2}-\frac {i d \sin \left (c \right ) a \expIntegral \left (1, -i d x \right )}{2}-\frac {b \cos \left (d x +c \right )}{d}-\frac {a \sin \left (d x +c \right )}{x}\) | \(80\) |
meijerg | \(\frac {b \sin \left (c \right ) \sin \left (d x \right )}{d}+\frac {b \sqrt {\pi }\, \cos \left (c \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (d x \right )}{\sqrt {\pi }}\right )}{d}+\frac {a \sqrt {\pi }\, \sin \left (c \right ) d^{2} \left (-\frac {4 d^{2} \cos \left (x \sqrt {d^{2}}\right )}{x \left (d^{2}\right )^{\frac {3}{2}} \sqrt {\pi }}-\frac {4 \sinIntegral \left (x \sqrt {d^{2}}\right )}{\sqrt {\pi }}\right )}{4 \sqrt {d^{2}}}+\frac {a \sqrt {\pi }\, \cos \left (c \right ) d \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \gamma }{\sqrt {\pi }}-\frac {4 \ln \left (2\right )}{\sqrt {\pi }}-\frac {4 \ln \left (\frac {d x}{2}\right )}{\sqrt {\pi }}-\frac {4 \sin \left (d x \right )}{\sqrt {\pi }\, d x}+\frac {4 \cosineIntegral \left (d x \right )}{\sqrt {\pi }}\right )}{4}\) | \(170\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.46, size = 937, normalized size = 21.30 \begin {gather*} -\frac {1}{4} \, {\left (\frac {{\left ({\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )} \cos \left (c\right )^{3} + {\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )} \cos \left (c\right ) \sin \left (c\right )^{2} + {\left (E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \sin \left (c\right )^{3} + {\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )} \cos \left (c\right ) + {\left ({\left (E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \cos \left (c\right )^{2} + E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \sin \left (c\right )\right )} b c^{2}}{{\left (d x + c\right )} {\left (\cos \left (c\right )^{2} + \sin \left (c\right )^{2}\right )} d^{2} - {\left (c \cos \left (c\right )^{2} + c \sin \left (c\right )^{2}\right )} d^{2}} - \frac {{\left ({\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )} \cos \left (c\right )^{3} + {\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )} \cos \left (c\right ) \sin \left (c\right )^{2} + {\left (E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \sin \left (c\right )^{3} + {\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )} \cos \left (c\right ) + {\left ({\left (E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \cos \left (c\right )^{2} + E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \sin \left (c\right )\right )} a}{c \cos \left (c\right )^{2} + c \sin \left (c\right )^{2} - {\left (d x + c\right )} {\left (\cos \left (c\right )^{2} + \sin \left (c\right )^{2}\right )}} + \frac {2 \, {\left ({\left ({\left (b \cos \left (c\right )^{2} + b \sin \left (c\right )^{2}\right )} {\left (d x + c\right )}^{2} - 2 \, {\left (b c \cos \left (c\right )^{2} + b c \sin \left (c\right )^{2}\right )} {\left (d x + c\right )}\right )} \cos \left (d x + c\right )^{3} + {\left (b c^{2} {\left (E_{3}\left (i \, d x\right ) + E_{3}\left (-i \, d x\right )\right )} \cos \left (c\right )^{3} + b c^{2} {\left (E_{3}\left (i \, d x\right ) + E_{3}\left (-i \, d x\right )\right )} \cos \left (c\right ) \sin \left (c\right )^{2} + b c^{2} {\left (-i \, E_{3}\left (i \, d x\right ) + i \, E_{3}\left (-i \, d x\right )\right )} \sin \left (c\right )^{3} + b c^{2} {\left (E_{3}\left (i \, d x\right ) + E_{3}\left (-i \, d x\right )\right )} \cos \left (c\right ) + {\left (b c^{2} {\left (-i \, E_{3}\left (i \, d x\right ) + i \, E_{3}\left (-i \, d x\right )\right )} \cos \left (c\right )^{2} + b c^{2} {\left (-i \, E_{3}\left (i \, d x\right ) + i \, E_{3}\left (-i \, d x\right )\right )}\right )} \sin \left (c\right )\right )} \cos \left (d x + c\right )^{2} + {\left (b c^{2} {\left (E_{3}\left (i \, d x\right ) + E_{3}\left (-i \, d x\right )\right )} \cos \left (c\right )^{3} + b c^{2} {\left (E_{3}\left (i \, d x\right ) + E_{3}\left (-i \, d x\right )\right )} \cos \left (c\right ) \sin \left (c\right )^{2} + b c^{2} {\left (-i \, E_{3}\left (i \, d x\right ) + i \, E_{3}\left (-i \, d x\right )\right )} \sin \left (c\right )^{3} + b c^{2} {\left (E_{3}\left (i \, d x\right ) + E_{3}\left (-i \, d x\right )\right )} \cos \left (c\right ) + {\left ({\left (b \cos \left (c\right )^{2} + b \sin \left (c\right )^{2}\right )} {\left (d x + c\right )}^{2} - 2 \, {\left (b c \cos \left (c\right )^{2} + b c \sin \left (c\right )^{2}\right )} {\left (d x + c\right )}\right )} \cos \left (d x + c\right ) + {\left (b c^{2} {\left (-i \, E_{3}\left (i \, d x\right ) + i \, E_{3}\left (-i \, d x\right )\right )} \cos \left (c\right )^{2} + b c^{2} {\left (-i \, E_{3}\left (i \, d x\right ) + i \, E_{3}\left (-i \, d x\right )\right )}\right )} \sin \left (c\right )\right )} \sin \left (d x + c\right )^{2} + {\left ({\left (b \cos \left (c\right )^{2} + b \sin \left (c\right )^{2}\right )} {\left (d x + c\right )}^{2} - 2 \, {\left (b c \cos \left (c\right )^{2} + b c \sin \left (c\right )^{2}\right )} {\left (d x + c\right )}\right )} \cos \left (d x + c\right )\right )}}{{\left ({\left (d x + c\right )}^{2} {\left (\cos \left (c\right )^{2} + \sin \left (c\right )^{2}\right )} d^{2} - 2 \, {\left (c \cos \left (c\right )^{2} + c \sin \left (c\right )^{2}\right )} {\left (d x + c\right )} d^{2} + {\left (c^{2} \cos \left (c\right )^{2} + c^{2} \sin \left (c\right )^{2}\right )} d^{2}\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (d x + c\right )}^{2} {\left (\cos \left (c\right )^{2} + \sin \left (c\right )^{2}\right )} d^{2} - 2 \, {\left (c \cos \left (c\right )^{2} + c \sin \left (c\right )^{2}\right )} {\left (d x + c\right )} d^{2} + {\left (c^{2} \cos \left (c\right )^{2} + c^{2} \sin \left (c\right )^{2}\right )} d^{2}\right )} \sin \left (d x + c\right )^{2}}\right )} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 68, normalized size = 1.55 \begin {gather*} -\frac {2 \, a d^{2} x \sin \left (c\right ) \operatorname {Si}\left (d x\right ) + 2 \, b x \cos \left (d x + c\right ) + 2 \, a d \sin \left (d x + c\right ) - {\left (a d^{2} x \operatorname {Ci}\left (d x\right ) + a d^{2} x \operatorname {Ci}\left (-d x\right )\right )} \cos \left (c\right )}{2 \, d x} \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^{2}}\, 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 = 4.03, size = 411, normalized size = 9.34 \begin {gather*} -\frac {a d^{2} x \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + a d^{2} x \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{2} x \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{2} x \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, a d^{2} x \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - a d^{2} x \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} - a d^{2} x \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{2} x \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + a d^{2} x \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{2} x \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{2} x \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, a d^{2} x \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right ) + 2 \, b x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{2} x \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) - a d^{2} x \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) - 4 \, a d \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 4 \, a d \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, b x \tan \left (\frac {1}{2} \, d x\right )^{2} - 8 \, b x \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, b x \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, a d \tan \left (\frac {1}{2} \, d x\right ) + 4 \, a d \tan \left (\frac {1}{2} \, c\right ) + 2 \, b x}{2 \, {\left (d x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d x \tan \left (\frac {1}{2} \, d x\right )^{2} + d x \tan \left (\frac {1}{2} \, c\right )^{2} + d x\right )}} \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 = -0.02 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )\,\left (b\,x^2+a\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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