Optimal. Leaf size=80 \[ \frac {\left (2 a^2+3 b^2\right ) \tan (c+d x)}{3 d}+\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac {a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a b \tan (c+d x) \sec (c+d x)}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2789, 3768, 3770, 3012, 3767, 8} \[ \frac {\left (2 a^2+3 b^2\right ) \tan (c+d x)}{3 d}+\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac {a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a b \tan (c+d x) \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2789
Rule 3012
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 \sec ^4(c+d x) \, dx &=(2 a b) \int \sec ^3(c+d x) \, dx+\int \left (a^2+b^2 \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {a b \sec (c+d x) \tan (c+d x)}{d}+\frac {a^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}+(a b) \int \sec (c+d x) \, dx+\frac {1}{3} \left (2 a^2+3 b^2\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a b \sec (c+d x) \tan (c+d x)}{d}+\frac {a^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {\left (2 a^2+3 b^2\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac {a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {\left (2 a^2+3 b^2\right ) \tan (c+d x)}{3 d}+\frac {a b \sec (c+d x) \tan (c+d x)}{d}+\frac {a^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 71, normalized size = 0.89 \[ \frac {a^2 \left (\frac {1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac {a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a b \tan (c+d x) \sec (c+d x)}{d}+\frac {b^2 \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 100, normalized size = 1.25 \[ \frac {3 \, a b \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a b \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.56, size = 178, normalized size = 2.22 \[ \frac {3 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 89, normalized size = 1.11 \[ \frac {2 a^{2} \tan \left (d x +c \right )}{3 d}+\frac {a^{2} \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{3 d}+\frac {a b \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{2} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 84, normalized size = 1.05 \[ \frac {2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} - 3 \, a b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, b^{2} \tan \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 141, normalized size = 1.76 \[ \frac {2\,a\,b\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {\left (2\,a^2-2\,a\,b+2\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {4\,a^2}{3}-4\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^2+2\,a\,b+2\,b^2\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \cos {\left (c + d x \right )}\right )^{2} \sec ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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