Optimal. Leaf size=142 \[ -\frac {\sqrt {c+d x} \cos (a+b x)}{b}+\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{b^{3/2}}-\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{b^{3/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3377, 3387,
3386, 3432, 3385, 3433} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \cos \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \sin \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{b^{3/2}}-\frac {\sqrt {c+d x} \cos (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3377
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rubi steps
\begin {align*} \int \sqrt {c+d x} \sin (a+b x) \, dx &=-\frac {\sqrt {c+d x} \cos (a+b x)}{b}+\frac {d \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{2 b}\\ &=-\frac {\sqrt {c+d x} \cos (a+b x)}{b}+\frac {\left (d \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{2 b}-\frac {\left (d \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{2 b}\\ &=-\frac {\sqrt {c+d x} \cos (a+b x)}{b}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{b}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{b}\\ &=-\frac {\sqrt {c+d x} \cos (a+b x)}{b}+\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{b^{3/2}}-\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.06, size = 123, normalized size = 0.87 \begin {gather*} \frac {e^{-\frac {i (b c+a d)}{d}} \sqrt {c+d x} \left (-\frac {e^{2 i a} \Gamma \left (\frac {3}{2},-\frac {i b (c+d x)}{d}\right )}{\sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{\frac {2 i b c}{d}} \Gamma \left (\frac {3}{2},\frac {i b (c+d x)}{d}\right )}{\sqrt {\frac {i b (c+d x)}{d}}}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 145, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{2 b \sqrt {\frac {b}{d}}}}{d}\) | \(145\) |
default | \(\frac {-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{2 b \sqrt {\frac {b}{d}}}}{d}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.33, size = 196, normalized size = 1.38 \begin {gather*} -\frac {\sqrt {2} {\left (4 \, \sqrt {2} \sqrt {d x + c} b \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) + {\left (\left (i - 1\right ) \, \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) + \left (i + 1\right ) \, \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) + {\left (-\left (i + 1\right ) \, \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) - \left (i - 1\right ) \, \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right )\right )}}{8 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 127, normalized size = 0.89 \begin {gather*} \frac {\sqrt {2} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - \sqrt {2} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 2 \, \sqrt {d x + c} b \cos \left (b x + a\right )}{2 \, b^{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 \sqrt {c + d x} \sin {\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 5.07, size = 426, normalized size = 3.00 \begin {gather*} -\frac {-\frac {i \, \sqrt {2} \sqrt {\pi } {\left (2 \, b c + i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {i \, \sqrt {2} \sqrt {\pi } {\left (2 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + 2 \, {\left (\frac {i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - \frac {i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}\right )} c + \frac {2 \, \sqrt {d x + c} d e^{\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{d}\right )}}{b} + \frac {2 \, \sqrt {d x + c} d e^{\left (\frac {-i \, {\left (d x + c\right )} b + i \, b c - i \, a d}{d}\right )}}{b}}{4 \, d} \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+b\,x\right )\,\sqrt {c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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