Optimal. Leaf size=198 \[ -\frac {b f \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {\sqrt {b} (d e-c f) \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^2}+\frac {\sqrt {b} (d e-c f) \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)}{d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{2 d^2}+\frac {b f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2} \]
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Rubi [A]
time = 0.19, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {3514, 3440,
3468, 3435, 3433, 3432, 3460, 3378, 3384, 3380, 3383} \begin {gather*} -\frac {b f \cos (a) \text {CosIntegral}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {\sqrt {2 \pi } \sqrt {b} \cos (a) (d e-c f) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{c+d x}\right )}{d^2}+\frac {\sqrt {2 \pi } \sqrt {b} \sin (a) (d e-c f) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^2}+\frac {(c+d x) (d e-c f) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {b f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3432
Rule 3433
Rule 3435
Rule 3440
Rule 3460
Rule 3468
Rule 3514
Rubi steps
\begin {align*} \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^2}\right ) \, dx &=\frac {\text {Subst}\left (\int \left (d e \left (1-\frac {c f}{d e}\right ) \sin \left (a+\frac {b}{x^2}\right )+f x \sin \left (a+\frac {b}{x^2}\right )\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {f \text {Subst}\left (\int x \sin \left (a+\frac {b}{x^2}\right ) \, dx,x,c+d x\right )}{d^2}+\frac {(d e-c f) \text {Subst}\left (\int \sin \left (a+\frac {b}{x^2}\right ) \, dx,x,c+d x\right )}{d^2}\\ &=-\frac {f \text {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^2}\right )}{2 d^2}-\frac {(d e-c f) \text {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {(b f) \text {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,\frac {1}{(c+d x)^2}\right )}{2 d^2}-\frac {(2 b (d e-c f)) \text {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {(b f \cos (a)) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{(c+d x)^2}\right )}{2 d^2}-\frac {(2 b (d e-c f) \cos (a)) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^2}+\frac {(b f \sin (a)) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{(c+d x)^2}\right )}{2 d^2}+\frac {(2 b (d e-c f) \sin (a)) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=-\frac {b f \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2}-\frac {\sqrt {b} (d e-c f) \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )}{d^2}+\frac {\sqrt {b} (d e-c f) \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)}{d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{(c+d x)^2}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )}{2 d^2}+\frac {b f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 242, normalized size = 1.22 \begin {gather*} \frac {-b f \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^2}\right )-2 \sqrt {b} (d e-c f) \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right )+2 \sqrt {b} d e \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)-2 \sqrt {b} c f \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{c+d x}\right ) \sin (a)+2 c d e \sin \left (a+\frac {b}{(c+d x)^2}\right )-c^2 f \sin \left (a+\frac {b}{(c+d x)^2}\right )+2 d^2 e x \sin \left (a+\frac {b}{(c+d x)^2}\right )+d^2 f x^2 \sin \left (a+\frac {b}{(c+d x)^2}\right )+b f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^2}\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 150, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {-\left (c f -d e \right ) \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )+\left (c f -d e \right ) \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )\right )+\frac {f \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-f b \left (\frac {\cos \left (a \right ) \cosineIntegral \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-\frac {\sin \left (a \right ) \sinIntegral \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d^{2}}\) | \(150\) |
default | \(\frac {-\left (c f -d e \right ) \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )+\left (c f -d e \right ) \sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )}\right )\right )+\frac {f \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-f b \left (\frac {\cos \left (a \right ) \cosineIntegral \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}-\frac {\sin \left (a \right ) \sinIntegral \left (\frac {b}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d^{2}}\) | \(150\) |
risch | \(-\frac {e \,{\mathrm e}^{i a} b \sqrt {\pi }\, \erf \left (\frac {\sqrt {-i b}}{d x +c}\right )}{2 d \sqrt {-i b}}+\frac {f \,{\mathrm e}^{i a} b \expIntegral \left (1, -\frac {i b}{\left (d x +c \right )^{2}}\right )}{4 d^{2}}+\frac {c f \,{\mathrm e}^{i a} b \sqrt {\pi }\, \erf \left (\frac {\sqrt {-i b}}{d x +c}\right )}{2 d^{2} \sqrt {-i b}}-\frac {e \,{\mathrm e}^{-i a} b \sqrt {\pi }\, \erf \left (\frac {\sqrt {i b}}{d x +c}\right )}{2 d \sqrt {i b}}+\frac {f \,{\mathrm e}^{-i a} b \expIntegral \left (1, \frac {i b}{\left (d x +c \right )^{2}}\right )}{4 d^{2}}+\frac {c f \,{\mathrm e}^{-i a} b \sqrt {\pi }\, \erf \left (\frac {\sqrt {i b}}{d x +c}\right )}{2 d^{2} \sqrt {i b}}-\frac {\left (e \left (-d x -c \right )+\frac {f \left (-\frac {1}{2} d^{2} x^{2}-c d x -\frac {1}{2} c^{2}\right )}{d}-\frac {c f \left (-d x -c \right )}{d}\right ) \sin \left (\frac {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}{\left (d x +c \right )^{2}}\right )}{d}\) | \(283\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 266, normalized size = 1.34 \begin {gather*} \frac {4 \, \sqrt {2} {\left (\pi c d f - \pi d^{2} e\right )} \sqrt {\frac {b}{\pi d^{2}}} \cos \left (a\right ) \operatorname {C}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) - 4 \, \sqrt {2} {\left (\pi c d f - \pi d^{2} e\right )} \sqrt {\frac {b}{\pi d^{2}}} \operatorname {S}\left (\frac {\sqrt {2} d \sqrt {\frac {b}{\pi d^{2}}}}{d x + c}\right ) \sin \left (a\right ) + 2 \, b f \sin \left (a\right ) \operatorname {Si}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - {\left (b f \operatorname {Ci}\left (\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + b f \operatorname {Ci}\left (-\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )\right )} \cos \left (a\right ) + 2 \, {\left (d^{2} f x^{2} - c^{2} f + 2 \, {\left (d^{2} x + c d\right )} e\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 )}{4 \, d^{2}} \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 \left (e + f x\right ) \sin {\left (a + \frac {b}{c^{2} + 2 c d x + d^{2} x^{2}} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^2}\right )\,\left (e+f\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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