Optimal. Leaf size=33 \[ \frac {(a-b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} x (a+b) \]
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Rubi [A] time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3675, 385, 203} \[ \frac {(a-b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} x (a+b) \]
Antiderivative was successfully verified.
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Rule 203
Rule 385
Rule 3675
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {(a-b) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {1}{2} (a+b) x+\frac {(a-b) \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 32, normalized size = 0.97 \[ \frac {2 (a+b) (c+d x)+(a-b) \sin (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 30, normalized size = 0.91 \[ \frac {{\left (a + b\right )} d x + {\left (a - b\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.50, size = 169, normalized size = 5.12 \[ \frac {a d x \tan \left (d x\right )^{2} \tan \relax (c)^{2} + b d x \tan \left (d x\right )^{2} \tan \relax (c)^{2} + a d x \tan \left (d x\right )^{2} + b d x \tan \left (d x\right )^{2} + a d x \tan \relax (c)^{2} + b d x \tan \relax (c)^{2} - a \tan \left (d x\right )^{2} \tan \relax (c) + b \tan \left (d x\right )^{2} \tan \relax (c) - a \tan \left (d x\right ) \tan \relax (c)^{2} + b \tan \left (d x\right ) \tan \relax (c)^{2} + a d x + b d x + a \tan \left (d x\right ) - b \tan \left (d x\right ) + a \tan \relax (c) - b \tan \relax (c)}{2 \, {\left (d \tan \left (d x\right )^{2} \tan \relax (c)^{2} + d \tan \left (d x\right )^{2} + d \tan \relax (c)^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 54, normalized size = 1.64 \[ \frac {b \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 39, normalized size = 1.18 \[ \frac {{\left (d x + c\right )} {\left (a + b\right )} + \frac {{\left (a - b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.94, size = 32, normalized size = 0.97 \[ \frac {\sin \left (2\,c+2\,d\,x\right )\,\left (\frac {a}{4}-\frac {b}{4}\right )+d\,x\,\left (\frac {a}{2}+\frac {b}{2}\right )}{d} \]
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 \[ \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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