Optimal. Leaf size=28 \[ \frac {B \tan ^3(c+d x)}{3 d}+\frac {B \tan (c+d x)}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {21, 3767} \[ \frac {B \tan ^3(c+d x)}{3 d}+\frac {B \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 3767
Rubi steps
\begin {align*} \int \frac {(a B+b B \cos (c+d x)) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx &=B \int \sec ^4(c+d x) \, dx\\ &=-\frac {B \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {B \tan (c+d x)}{d}+\frac {B \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 24, normalized size = 0.86 \[ \frac {B \left (\frac {1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 32, normalized size = 1.14 \[ \frac {{\left (2 \, B \cos \left (d x + c\right )^{2} + B\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 25, normalized size = 0.89 \[ \frac {B \tan \left (d x + c\right )^{3} + 3 \, B \tan \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 25, normalized size = 0.89 \[ -\frac {B \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d} \]
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 = 0.52, size = 39, normalized size = 1.39 \[ \frac {2\,B\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^2+B\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.65, size = 42, normalized size = 1.50 \[ \begin {cases} \frac {B \left (\frac {\tan ^{3}{\left (c + d x \right )}}{3} + \tan {\left (c + d x \right )}\right )}{d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \cos {\relax (c )}\right ) \sec ^{4}{\relax (c )}}{a + b \cos {\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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