Optimal. Leaf size=15 \[ A x+\frac {C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3767, 8} \[ A x+\frac {C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rubi steps
\begin {align*} \int \left (A+C \sec ^2(c+d x)\right ) \, dx &=A x+C \int \sec ^2(c+d x) \, dx\\ &=A x-\frac {C \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=A x+\frac {C \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ A x+\frac {C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 31, normalized size = 2.07 \[ \frac {A d x \cos \left (d x + c\right ) + C \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 15, normalized size = 1.00 \[ A x + \frac {C \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.98, size = 16, normalized size = 1.07 \[ A x +\frac {C \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 15, normalized size = 1.00 \[ A x + \frac {C \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.34, size = 17, normalized size = 1.13 \[ \frac {C\,\mathrm {tan}\left (c+d\,x\right )+A\,d\,x}{d} \]
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 \[ \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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