Optimal. Leaf size=78 \[ -\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\log (c+d x)}{2 d} \]
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Rubi [A] time = 0.17, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 3299
Rule 3302
Rule 3303
Rule 3312
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*}
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Mathematica [A] time = 0.11, size = 65, normalized size = 0.83 \[ \frac {-\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b (c+d x)}{d}\right )+\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )+\log (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 88, normalized size = 1.13 \[ -\frac {{\left (\operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + \operatorname {Ci}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - 2 \, \log \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.66, size = 612, normalized size = 7.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 105, normalized size = 1.35 \[ \frac {\ln \left (\left (b x +a \right ) d -d a +c b \right )}{2 d}-\frac {\Si \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \sin \left (\frac {-2 d a +2 c b}{d}\right )}{2 d}-\frac {\Ci \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \cos \left (\frac {-2 d a +2 c b}{d}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.42, size = 160, normalized size = 2.05 \[ \frac {b {\left (E_{1}\left (\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + E_{1}\left (-\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b {\left (-i \, E_{1}\left (\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + i \, E_{1}\left (-\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, b \log \left (b c + {\left (b x + a\right )} d - a d\right )}{4 \, b d} \]
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 \[ \int \frac {{\sin \left (a+b\,x\right )}^2}{c+d\,x} \,d x \]
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 \[ \int \frac {\sin ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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