Integrand size = 18, antiderivative size = 158 \[ \int \sqrt {c+d x} \sin ^2(a+b x) \, dx=\frac {(c+d x)^{3/2}}{3 d}+\frac {\sqrt {d} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}+\frac {\sqrt {d} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{8 b^{3/2}}-\frac {\sqrt {c+d x} \sin (2 a+2 b x)}{4 b} \]
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Time = 0.15 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3393, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int \sqrt {c+d x} \sin ^2(a+b x) \, dx=\frac {\sqrt {\pi } \sqrt {d} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}+\frac {\sqrt {\pi } \sqrt {d} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {\sqrt {c+d x} \sin (2 a+2 b x)}{4 b}+\frac {(c+d x)^{3/2}}{3 d} \]
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Rule 3377
Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3432
Rule 3433
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} \sqrt {c+d x}-\frac {1}{2} \sqrt {c+d x} \cos (2 a+2 b x)\right ) \, dx \\ & = \frac {(c+d x)^{3/2}}{3 d}-\frac {1}{2} \int \sqrt {c+d x} \cos (2 a+2 b x) \, dx \\ & = \frac {(c+d x)^{3/2}}{3 d}-\frac {\sqrt {c+d x} \sin (2 a+2 b x)}{4 b}+\frac {d \int \frac {\sin (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{8 b} \\ & = \frac {(c+d x)^{3/2}}{3 d}-\frac {\sqrt {c+d x} \sin (2 a+2 b x)}{4 b}+\frac {\left (d \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{8 b}+\frac {\left (d \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{8 b} \\ & = \frac {(c+d x)^{3/2}}{3 d}-\frac {\sqrt {c+d x} \sin (2 a+2 b x)}{4 b}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{4 b}+\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{4 b} \\ & = \frac {(c+d x)^{3/2}}{3 d}+\frac {\sqrt {d} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}+\frac {\sqrt {d} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{8 b^{3/2}}-\frac {\sqrt {c+d x} \sin (2 a+2 b x)}{4 b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.82 \[ \int \sqrt {c+d x} \sin ^2(a+b x) \, dx=\frac {(c+d x)^{3/2} \left (16+\frac {3 \sqrt {2} e^{2 i \left (a-\frac {b c}{d}\right )} \Gamma \left (\frac {3}{2},-\frac {2 i b (c+d x)}{d}\right )}{\left (-\frac {i b (c+d x)}{d}\right )^{3/2}}+\frac {3 \sqrt {2} e^{-2 i \left (a-\frac {b c}{d}\right )} \Gamma \left (\frac {3}{2},\frac {2 i b (c+d x)}{d}\right )}{\left (\frac {i b (c+d x)}{d}\right )^{3/2}}\right )}{48 d} \]
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Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 c b}{d}\right ) \operatorname {S}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 d a -2 c b}{d}\right ) \operatorname {C}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}}{d}\) | \(150\) |
default | \(\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 d a -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 c b}{d}\right ) \operatorname {S}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 d a -2 c b}{d}\right ) \operatorname {C}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}}{d}\) | \(150\) |
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Time = 0.33 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.94 \[ \int \sqrt {c+d x} \sin ^2(a+b x) \, dx=\frac {3 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 3 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 4 \, {\left (2 \, b^{2} d x - 3 \, b d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, b^{2} c\right )} \sqrt {d x + c}}{24 \, b^{2} d} \]
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\[ \int \sqrt {c+d x} \sin ^2(a+b x) \, dx=\int \sqrt {c + d x} \sin ^{2}{\left (a + b x \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.45 \[ \int \sqrt {c+d x} \sin ^2(a+b x) \, dx=\frac {\sqrt {2} {\left (\frac {32 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2}}{d} - 24 \, \sqrt {2} \sqrt {d x + c} b \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 3 \, {\left (\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {2 i \, b}{d}}\right ) + 3 \, {\left (-\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {2 i \, b}{d}}\right )\right )}}{192 \, b^{2}} \]
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Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.76 \[ \int \sqrt {c+d x} \sin ^2(a+b x) \, dx=-\frac {12 \, {\left (\frac {i \, \sqrt {\pi } d \operatorname {erf}\left (-\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - \frac {i \, \sqrt {\pi } d \operatorname {erf}\left (\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - 4 \, \sqrt {d x + c}\right )} c - \frac {3 i \, \sqrt {\pi } {\left (4 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {3 i \, \sqrt {\pi } {\left (4 \, b c + i \, d\right )} d \operatorname {erf}\left (\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} - 16 \, {\left (d x + c\right )}^{\frac {3}{2}} + 48 \, \sqrt {d x + c} c + \frac {6 i \, \sqrt {d x + c} d e^{\left (-\frac {2 \, {\left (i \, {\left (d x + c\right )} b - i \, b c + i \, a d\right )}}{d}\right )}}{b} - \frac {6 i \, \sqrt {d x + c} d e^{\left (-\frac {2 \, {\left (-i \, {\left (d x + c\right )} b + i \, b c - i \, a d\right )}}{d}\right )}}{b}}{48 \, d} \]
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Timed out. \[ \int \sqrt {c+d x} \sin ^2(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^2\,\sqrt {c+d\,x} \,d x \]
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