Optimal. Leaf size=238 \[ \frac {\text {Ci}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b \sqrt {c+d x}\right ) \sin \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )}{f}+\frac {\text {Ci}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b \sqrt {c+d x}\right ) \sin \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )}{f}-\frac {\cos \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b \sqrt {c+d x}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b \sqrt {c+d x}\right )}{f} \]
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Rubi [A]
time = 0.52, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3512, 3384,
3380, 3383} \begin {gather*} \frac {\sin \left (a-\frac {b \sqrt {c f-d e}}{\sqrt {f}}\right ) \text {CosIntegral}\left (\frac {b \sqrt {c f-d e}}{\sqrt {f}}+b \sqrt {c+d x}\right )}{f}+\frac {\sin \left (a+\frac {b \sqrt {c f-d e}}{\sqrt {f}}\right ) \text {CosIntegral}\left (\frac {b \sqrt {c f-d e}}{\sqrt {f}}-b \sqrt {c+d x}\right )}{f}-\frac {\cos \left (a+\frac {b \sqrt {c f-d e}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {c f-d e}}{\sqrt {f}}-b \sqrt {c+d x}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt {c f-d e}}{\sqrt {f}}\right ) \text {Si}\left (\frac {\sqrt {c f-d e} b}{\sqrt {f}}+\sqrt {c+d x} b\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 3512
Rubi steps
\begin {align*} \int \frac {\sin \left (a+b \sqrt {c+d x}\right )}{e+f x} \, dx &=\frac {2 \text {Subst}\left (\int \left (-\frac {d \sin (a+b x)}{2 \sqrt {f} \left (\sqrt {-d e+c f}-\sqrt {f} x\right )}+\frac {d \sin (a+b x)}{2 \sqrt {f} \left (\sqrt {-d e+c f}+\sqrt {f} x\right )}\right ) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt {-d e+c f}-\sqrt {f} x} \, dx,x,\sqrt {c+d x}\right )}{\sqrt {f}}+\frac {\text {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt {-d e+c f}+\sqrt {f} x} \, dx,x,\sqrt {c+d x}\right )}{\sqrt {f}}\\ &=\frac {\cos \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b x\right )}{\sqrt {-d e+c f}+\sqrt {f} x} \, dx,x,\sqrt {c+d x}\right )}{\sqrt {f}}+\frac {\cos \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b x\right )}{\sqrt {-d e+c f}-\sqrt {f} x} \, dx,x,\sqrt {c+d x}\right )}{\sqrt {f}}+\frac {\sin \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b x\right )}{\sqrt {-d e+c f}+\sqrt {f} x} \, dx,x,\sqrt {c+d x}\right )}{\sqrt {f}}-\frac {\sin \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b x\right )}{\sqrt {-d e+c f}-\sqrt {f} x} \, dx,x,\sqrt {c+d x}\right )}{\sqrt {f}}\\ &=\frac {\text {Ci}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b \sqrt {c+d x}\right ) \sin \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )}{f}+\frac {\text {Ci}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b \sqrt {c+d x}\right ) \sin \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )}{f}-\frac {\cos \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}-b \sqrt {c+d x}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right ) \text {Si}\left (\frac {b \sqrt {-d e+c f}}{\sqrt {f}}+b \sqrt {c+d x}\right )}{f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.64, size = 238, normalized size = 1.00 \begin {gather*} \frac {i e^{-i \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )} \left (\text {Ei}\left (-i b \left (-\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )-e^{2 i \left (a+\frac {b \sqrt {-d e+c f}}{\sqrt {f}}\right )} \text {Ei}\left (i b \left (-\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )+e^{\frac {2 i b \sqrt {-d e+c f}}{\sqrt {f}}} \text {Ei}\left (-i b \left (\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )-e^{2 i a} \text {Ei}\left (i b \left (\frac {\sqrt {-d e+c f}}{\sqrt {f}}+\sqrt {c+d x}\right )\right )\right )}{2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(792\) vs.
\(2(197)=394\).
time = 0.06, size = 793, normalized size = 3.33
method | result | size |
derivativedivides | \(\frac {-\frac {b^{2} \left (a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\sinIntegral \left (-b \sqrt {d x +c}-a +\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\cosineIntegral \left (b \sqrt {d x +c}+a -\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (-\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}+\frac {b^{2} \left (-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\sinIntegral \left (-b \sqrt {d x +c}-a -\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\cosineIntegral \left (b \sqrt {d x +c}+a +\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-2 a \,b^{2} \left (-\frac {-\sinIntegral \left (-b \sqrt {d x +c}-a +\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\cosineIntegral \left (b \sqrt {d x +c}+a -\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (-\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-\frac {-\sinIntegral \left (-b \sqrt {d x +c}-a -\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\cosineIntegral \left (b \sqrt {d x +c}+a +\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}\right )}{b^{2}}\) | \(793\) |
default | \(\frac {-\frac {b^{2} \left (a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\sinIntegral \left (-b \sqrt {d x +c}-a +\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\cosineIntegral \left (b \sqrt {d x +c}+a -\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (-\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}+\frac {b^{2} \left (-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}\right ) \left (-\sinIntegral \left (-b \sqrt {d x +c}-a -\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\cosineIntegral \left (b \sqrt {d x +c}+a +\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )\right )}{f^{2} \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-2 a \,b^{2} \left (-\frac {-\sinIntegral \left (-b \sqrt {d x +c}-a +\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )+\cosineIntegral \left (b \sqrt {d x +c}+a -\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (-\frac {a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}-\frac {-\sinIntegral \left (-b \sqrt {d x +c}-a -\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \cos \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )-\cosineIntegral \left (b \sqrt {d x +c}+a +\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right ) \sin \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}\right )}{2 f \left (\frac {-a f +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{f}+a \right )}\right )}{b^{2}}\) | \(793\) |
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 [C] Result contains complex when optimal does not.
time = 0.50, size = 266, normalized size = 1.12 \begin {gather*} \frac {-i \, {\rm Ei}\left (i \, \sqrt {d x + c} b - \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right ) e^{\left (i \, a + \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right )} - i \, {\rm Ei}\left (i \, \sqrt {d x + c} b + \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right ) e^{\left (i \, a - \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right )} + i \, {\rm Ei}\left (-i \, \sqrt {d x + c} b - \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right ) e^{\left (-i \, a + \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right )} + i \, {\rm Ei}\left (-i \, \sqrt {d x + c} b + \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right ) e^{\left (-i \, a - \sqrt {-\frac {b^{2} c f - b^{2} d e}{f}}\right )}}{2 \, f} \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 {\sin {\left (a + b \sqrt {c + d x} \right )}}{e + f x}\, 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 = -0.00 \begin {gather*} \int \frac {\sin \left (a+b\,\sqrt {c+d\,x}\right )}{e+f\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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