Optimal. Leaf size=11 \[ \frac {B \tan (c+d x)}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {21, 3767, 8} \[ \frac {B \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 21
Rule 3767
Rubi steps
\begin {align*} \int \frac {(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx &=B \int \sec ^2(c+d x) \, dx\\ &=-\frac {B \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac {B \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 11, normalized size = 1.00 \[ \frac {B \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 19, normalized size = 1.73 \[ \frac {B \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 11, normalized size = 1.00 \[ \frac {B \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 12, normalized size = 1.09 \[ \frac {B \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 30, normalized size = 2.73 \[ -\frac {2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.10, size = 32, normalized size = 2.91 \[ \begin {cases} \frac {B \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \cos {\relax (c )}\right ) \sec ^{2}{\relax (c )}}{a + b \cos {\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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