Optimal. Leaf size=213 \[ -\frac {\sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Ci}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Ci}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Rubi [A] time = 0.54, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3333, 3303, 3299, 3302} \[ -\frac {\sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 3333
Rubi steps
\begin {align*} \int \frac {\sin (a+b x)}{c+d x^2} \, dx &=\int \left (\frac {\sqrt {-c} \sin (a+b x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \sin (a+b x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\sin (a+b x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\int \frac {\sin (a+b x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}}\\ &=-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\sin \left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}}+\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\sin \left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\cos \left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\cos \left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}\\ &=-\frac {\text {Ci}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right ) \sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Ci}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right ) \sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{2 \sqrt {-c} \sqrt {d}}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 172, normalized size = 0.81 \[ \frac {i \left (\sin \left (a-\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Ci}\left (b \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )-\sin \left (a+\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Ci}\left (b \left (x-\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )+\cos \left (a-\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Si}\left (b \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )+\cos \left (a+\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {i b \sqrt {c}}{\sqrt {d}}-b x\right )\right )}{2 \sqrt {c} \sqrt {d}} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.56, size = 187, normalized size = 0.88 \[ \frac {\sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (i \, b x - \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (i \, a + \sqrt {\frac {b^{2} c}{d}}\right )} - \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (i \, b x + \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (i \, a - \sqrt {\frac {b^{2} c}{d}}\right )} + \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (-i \, b x - \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (-i \, a + \sqrt {\frac {b^{2} c}{d}}\right )} - \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (-i \, b x + \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (-i \, a - \sqrt {\frac {b^{2} c}{d}}\right )}}{4 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x + a\right )}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 229, normalized size = 1.08 \[ b \left (\frac {\Si \left (b x +a -\frac {b \sqrt {-c d}+d a}{d}\right ) \cos \left (\frac {b \sqrt {-c d}+d a}{d}\right )+\Ci \left (b x +a -\frac {b \sqrt {-c d}+d a}{d}\right ) \sin \left (\frac {b \sqrt {-c d}+d a}{d}\right )}{2 \left (\frac {b \sqrt {-c d}+d a}{d}-a \right ) d}+\frac {\Si \left (b x +a +\frac {b \sqrt {-c d}-d a}{d}\right ) \cos \left (\frac {b \sqrt {-c d}-d a}{d}\right )-\Ci \left (b x +a +\frac {b \sqrt {-c d}-d a}{d}\right ) \sin \left (\frac {b \sqrt {-c d}-d a}{d}\right )}{2 \left (-\frac {b \sqrt {-c d}-d a}{d}-a \right ) d}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x + a\right )}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (a+b\,x\right )}{d\,x^2+c} \,d x \]
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 \[ \int \frac {\sin {\left (a + b x \right )}}{c + d x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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