Optimal. Leaf size=99 \[ \frac {\left (4 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (4 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {5 a b \sec ^3(c+d x)}{12 d}+\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))}{4 d} \]
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Rubi [A] time = 0.10, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3508, 3486, 3768, 3770} \[ \frac {\left (4 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (4 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {5 a b \sec ^3(c+d x)}{12 d}+\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3508
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))}{4 d}+\frac {1}{4} \int \sec ^3(c+d x) \left (4 a^2-b^2+5 a b \tan (c+d x)\right ) \, dx\\ &=\frac {5 a b \sec ^3(c+d x)}{12 d}+\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))}{4 d}+\frac {1}{4} \left (4 a^2-b^2\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac {5 a b \sec ^3(c+d x)}{12 d}+\frac {\left (4 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))}{4 d}+\frac {1}{8} \left (4 a^2-b^2\right ) \int \sec (c+d x) \, dx\\ &=\frac {\left (4 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {5 a b \sec ^3(c+d x)}{12 d}+\frac {\left (4 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) (a+b \tan (c+d x))}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 120, normalized size = 1.21 \[ \frac {a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 a b \sec ^3(c+d x)}{3 d}-\frac {b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac {b^2 \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 120, normalized size = 1.21 \[ \frac {3 \, {\left (4 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 32 \, a b \cos \left (d x + c\right ) + 6 \, {\left ({\left (4 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, b^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.57, size = 249, normalized size = 2.52 \[ \frac {3 \, {\left (4 \, a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (4 \, a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16 \, a b\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 143, normalized size = 1.44 \[ \frac {a^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 a b}{3 d \cos \left (d x +c \right )^{3}}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {b^{2} \sin \left (d x +c \right )}{8 d}-\frac {b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 129, normalized size = 1.30 \[ \frac {3 \, b^{2} {\left (\frac {2 \, {\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {32 \, a b}{\cos \left (d x + c\right )^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.24, size = 216, normalized size = 2.18 \[ \frac {\left (a^2+\frac {b^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {7\,b^2}{4}-a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {7\,b^2}{4}-a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\left (a^2+\frac {b^2}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {4\,a\,b}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2-\frac {b^2}{4}\right )}{d} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{2} \sec ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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