Optimal. Leaf size=107 \[ -\frac{\sin \left (\frac{2 b}{d}\right ) (b c-a d) \text{CosIntegral}\left (\frac{2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac{\cos \left (\frac{2 b}{d}\right ) (b c-a d) \text{Si}\left (\frac{2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \cos ^2\left (\frac{a+b x}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.160648, antiderivative size = 107, normalized size of antiderivative = 1., 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 = {4564, 3313, 12, 3303, 3299, 3302} \[ -\frac{\sin \left (\frac{2 b}{d}\right ) (b c-a d) \text{CosIntegral}\left (\frac{2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac{\cos \left (\frac{2 b}{d}\right ) (b c-a d) \text{Si}\left (\frac{2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \cos ^2\left (\frac{a+b x}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4564
Rule 3313
Rule 12
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \cos ^2\left (\frac{a+b x}{c+d x}\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^2\left (\frac{b}{d}-\frac{(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=\frac{(c+d x) \cos ^2\left (\frac{a+b x}{c+d x}\right )}{d}+\frac{(2 (b c-a d)) \operatorname{Subst}\left (\int -\frac{\sin \left (\frac{2 b}{d}-\frac{2 (b c-a d) x}{d}\right )}{2 x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \cos ^2\left (\frac{a+b x}{c+d x}\right )}{d}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 b}{d}-\frac{2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \cos ^2\left (\frac{a+b x}{c+d x}\right )}{d}+\frac{\left ((b c-a d) \cos \left (\frac{2 b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}-\frac{\left ((b c-a d) \sin \left (\frac{2 b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \cos ^2\left (\frac{a+b x}{c+d x}\right )}{d}-\frac{(b c-a d) \text{Ci}\left (\frac{2 (b c-a d)}{d (c+d x)}\right ) \sin \left (\frac{2 b}{d}\right )}{d^2}+\frac{(b c-a d) \cos \left (\frac{2 b}{d}\right ) \text{Si}\left (\frac{2 (b c-a d)}{d (c+d x)}\right )}{d^2}\\ \end{align*}
Mathematica [C] time = 6.11297, size = 400, normalized size = 3.74 \[ \frac{\left (a c d-b c^2\right ) \left (\frac{\left (-1+e^{\frac{4 i b}{d}}\right ) \left (e^{\frac{4 i b c}{d (c+d x)}}-e^{\frac{4 i a}{c+d x}}\right ) \exp \left (-\frac{2 i (a d+2 b c+b d x)}{d (c+d x)}\right )}{8 (b c-a d)}-\frac{\left (1+e^{\frac{4 i b}{d}}\right ) \left (e^{\frac{4 i a}{c+d x}}+e^{\frac{4 i b c}{d (c+d x)}}\right ) \exp \left (-\frac{2 i (a d+2 b c+b d x)}{d (c+d x)}\right )}{8 (b c-a d)}\right )}{d}+\frac{2 a d \sin \left (\frac{2 b}{d}\right ) \text{CosIntegral}\left (\frac{2 (a d-b c)}{d (c+d x)}\right )-2 b c \sin \left (\frac{2 b}{d}\right ) \text{CosIntegral}\left (\frac{2 (a d-b c)}{d (c+d x)}\right )+2 a d \cos \left (\frac{2 b}{d}\right ) \text{Si}\left (\frac{2 (a d-b c)}{d (c+d x)}\right )-2 b c \cos \left (\frac{2 b}{d}\right ) \text{Si}\left (\frac{2 (a d-b c)}{d (c+d x)}\right )+d^2 x}{2 d^2}-\frac{1}{2} x \sin \left (\frac{2 b}{d}\right ) \sin \left (\frac{2 (a d-b c)}{d (c+d x)}\right )+\frac{1}{2} x \cos \left (\frac{2 b}{d}\right ) \cos \left (\frac{2 (a d-b c)}{d (c+d x)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.02, size = 195, normalized size = 1.8 \begin{align*} -{\frac{ad-cb}{{d}^{2}} \left ({\frac{{d}^{2}}{4} \left ( -2\,{\frac{1}{d}\cos \left ( 2\,{\frac{ad-cb}{d \left ( dx+c \right ) }}+2\,{\frac{b}{d}} \right ) \left ( d \left ({\frac{b}{d}}+{\frac{ad-cb}{d \left ( dx+c \right ) }} \right ) -b \right ) ^{-1}}-2\,{\frac{1}{d} \left ( 2\,{\frac{1}{d}{\it Si} \left ( 2\,{\frac{ad-cb}{d \left ( dx+c \right ) }} \right ) \cos \left ( 2\,{\frac{b}{d}} \right ) }+2\,{\frac{1}{d}{\it Ci} \left ( 2\,{\frac{ad-cb}{d \left ( dx+c \right ) }} \right ) \sin \left ( 2\,{\frac{b}{d}} \right ) } \right ) } \right ) }-{\frac{d}{2} \left ( d \left ({\frac{b}{d}}+{\frac{ad-cb}{d \left ( dx+c \right ) }} \right ) -b \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, x + \frac{1}{2} \, \int \cos \left (\frac{2 \,{\left (b x + a\right )}}{d x + c}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.583, size = 338, normalized size = 3.16 \begin{align*} \frac{2 \,{\left (d^{2} x + c d\right )} \cos \left (\frac{b x + a}{d x + c}\right )^{2} - 2 \,{\left (b c - a d\right )} \cos \left (\frac{2 \, b}{d}\right ) \operatorname{Si}\left (-\frac{2 \,{\left (b c - a d\right )}}{d^{2} x + c d}\right ) -{\left ({\left (b c - a d\right )} \operatorname{Ci}\left (\frac{2 \,{\left (b c - a d\right )}}{d^{2} x + c d}\right ) +{\left (b c - a d\right )} \operatorname{Ci}\left (-\frac{2 \,{\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \sin \left (\frac{2 \, b}{d}\right )}{2 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (\frac{b x + a}{d x + c}\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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