Optimal. Leaf size=78 \[ -\frac{\cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}+\frac{\sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}+\frac{\log (c+d x)}{2 d} \]
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Rubi [A] time = 0.168239, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3312, 3303, 3299, 3302} \[ -\frac{\cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}+\frac{\sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}+\frac{\log (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3312
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sin ^2(a+b x)}{c+d x} \, dx &=\int \left (\frac{1}{2 (c+d x)}-\frac{\cos (2 a+2 b x)}{2 (c+d x)}\right ) \, dx\\ &=\frac{\log (c+d x)}{2 d}-\frac{1}{2} \int \frac{\cos (2 a+2 b x)}{c+d x} \, dx\\ &=\frac{\log (c+d x)}{2 d}-\frac{1}{2} \cos \left (2 a-\frac{2 b c}{d}\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx+\frac{1}{2} \sin \left (2 a-\frac{2 b c}{d}\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx\\ &=-\frac{\cos \left (2 a-\frac{2 b c}{d}\right ) \text{Ci}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}+\frac{\log (c+d x)}{2 d}+\frac{\sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.104045, size = 65, normalized size = 0.83 \[ \frac{-\cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b (c+d x)}{d}\right )+\sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )+\log (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 105, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( \left ( bx+a \right ) d-da+cb \right ) }{2\,d}}-{\frac{1}{2\,d}{\it Si} \left ( 2\,bx+2\,a+2\,{\frac{-da+cb}{d}} \right ) \sin \left ( 2\,{\frac{-da+cb}{d}} \right ) }-{\frac{1}{2\,d}{\it Ci} \left ( 2\,bx+2\,a+2\,{\frac{-da+cb}{d}} \right ) \cos \left ( 2\,{\frac{-da+cb}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.23145, size = 216, normalized size = 2.77 \begin{align*} \frac{b{\left (E_{1}\left (\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + E_{1}\left (-\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + b{\left (-i \, E_{1}\left (\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + i \, E_{1}\left (-\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + 2 \, b \log \left (b c +{\left (b x + a\right )} d - a d\right )}{4 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70157, size = 238, normalized size = 3.05 \begin{align*} -\frac{{\left (\operatorname{Ci}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) + \operatorname{Ci}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) - 2 \, \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{Si}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) - 2 \, \log \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{2}{\left (a + b x \right )}}{c + d x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.2331, size = 826, normalized size = 10.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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