Optimal. Leaf size=49 \[ \frac {1}{2} \left (a^2+b^2\right ) x-\frac {\cos ^2(c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))}{2 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 49, 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 = {3587, 737, 209}
\begin {gather*} \frac {1}{2} x \left (a^2+b^2\right )-\frac {\cos ^2(c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 737
Rule 3587
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac {\text {Subst}\left (\int \frac {(a+x)^2}{\left (1+\frac {x^2}{b^2}\right )^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac {\cos ^2(c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))}{2 d}+\frac {\left (a^2+b^2\right ) \text {Subst}\left (\int \frac {1}{1+\frac {x^2}{b^2}} \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=\frac {1}{2} \left (a^2+b^2\right ) x-\frac {\cos ^2(c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 52, normalized size = 1.06 \begin {gather*} \frac {2 \left (a^2+b^2\right ) (c+d x)-2 a b \cos (2 (c+d x))+\left (a^2-b^2\right ) \sin (2 (c+d x))}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 70, normalized size = 1.43
method | result | size |
risch | \(\frac {a^{2} x}{2}+\frac {b^{2} x}{2}-\frac {a b \cos \left (2 d x +2 c \right )}{2 d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{4 d}-\frac {\sin \left (2 d x +2 c \right ) b^{2}}{4 d}\) | \(64\) |
derivativedivides | \(\frac {b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-a b \left (\cos ^{2}\left (d x +c \right )\right )+a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(70\) |
default | \(\frac {b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-a b \left (\cos ^{2}\left (d x +c \right )\right )+a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 55, normalized size = 1.12 \begin {gather*} \frac {{\left (a^{2} + b^{2}\right )} {\left (d x + c\right )} - \frac {2 \, a b - {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 52, normalized size = 1.06 \begin {gather*} -\frac {2 \, a b \cos \left (d x + c\right )^{2} - {\left (a^{2} + b^{2}\right )} d x - {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d} \end {gather*}
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 \begin {gather*} \int \left (a + b \tan {\left (c + d x \right )}\right )^{2} \cos ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 245 vs.
\(2 (46) = 92\).
time = 0.61, size = 245, normalized size = 5.00 \begin {gather*} \frac {a^{2} d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + b^{2} d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + a^{2} d x \tan \left (d x\right )^{2} + b^{2} d x \tan \left (d x\right )^{2} + a^{2} d x \tan \left (c\right )^{2} + b^{2} d x \tan \left (c\right )^{2} - a b \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - a^{2} \tan \left (d x\right )^{2} \tan \left (c\right ) + b^{2} \tan \left (d x\right )^{2} \tan \left (c\right ) - a^{2} \tan \left (d x\right ) \tan \left (c\right )^{2} + b^{2} \tan \left (d x\right ) \tan \left (c\right )^{2} + a^{2} d x + b^{2} d x + a b \tan \left (d x\right )^{2} + 4 \, a b \tan \left (d x\right ) \tan \left (c\right ) + a b \tan \left (c\right )^{2} + a^{2} \tan \left (d x\right ) - b^{2} \tan \left (d x\right ) + a^{2} \tan \left (c\right ) - b^{2} \tan \left (c\right ) - a b}{2 \, {\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + d \tan \left (d x\right )^{2} + d \tan \left (c\right )^{2} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.58, size = 50, normalized size = 1.02 \begin {gather*} x\,\left (\frac {a^2}{2}+\frac {b^2}{2}\right )-\frac {{\cos \left (c+d\,x\right )}^2\,\left (a\,b-\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {a^2}{2}-\frac {b^2}{2}\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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