Optimal. Leaf size=105 \[ -\frac {\sqrt {2 \pi } \sqrt {b} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d}+\frac {\sqrt {2 \pi } \sqrt {b} \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d}+\frac {(c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d} \]
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Rubi [A] time = 0.07, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3359, 3387, 3354, 3352, 3351} \[ -\frac {\sqrt {2 \pi } \sqrt {b} \cos (a) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{c+d x}\right )}{d}+\frac {\sqrt {2 \pi } \sqrt {b} \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d}+\frac {(c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3354
Rule 3359
Rule 3387
Rubi steps
\begin {align*} \int \sin \left (a+\frac {b}{(c+d x)^2}\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=\frac {(c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d}-\frac {(2 b) \operatorname {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=\frac {(c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d}-\frac {(2 b \cos (a)) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d}+\frac {(2 b \sin (a)) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=-\frac {\sqrt {b} \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d}+\frac {\sqrt {b} \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)}{d}+\frac {(c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 100, normalized size = 0.95 \[ \frac {\sqrt {2 \pi } \left (-\sqrt {b}\right ) \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+\sqrt {2 \pi } \sqrt {b} \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+(c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 137, normalized size = 1.30 \[ -\frac {\sqrt {2} \pi d \sqrt {\frac {b}{\pi d^{2}}} \cos \relax (a) \operatorname {C}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) - \sqrt {2} \pi d \sqrt {\frac {b}{\pi d^{2}}} \operatorname {S}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) \sin \relax (a) - {\left (d x + c\right )} \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{d} \]
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 \[ \int \sin \left (a + \frac {b}{{\left (d x + c\right )}^{2}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 80, normalized size = 0.76 \[ -\frac {-\left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )\right )}{d} \]
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 \[ b d \int \frac {x \cos \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} + b d \int \frac {x \cos \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \cos \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}}\,{d x} + x \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \]
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 \[ \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^2}\right ) \,d x \]
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 \[ \int \sin {\left (a + \frac {b}{\left (c + d x\right )^{2}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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