Optimal. Leaf size=99 \[ \frac {\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+a^2 (-x)-\frac {a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a b \tan (c+d x) \sec (c+d x)}{3 d}+\frac {\tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b)^2}{3 d} \]
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Rubi [A] time = 0.47, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4397, 2889, 3048, 3031, 3021, 2735, 3770} \[ \frac {\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+a^2 (-x)-\frac {a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a b \tan (c+d x) \sec (c+d x)}{3 d}+\frac {\tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2889
Rule 3021
Rule 3031
Rule 3048
Rule 3770
Rule 4397
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx &=\int (b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan ^2(c+d x) \, dx\\ &=\int (b+a \cos (c+d x))^2 \left (1-\cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {(b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{3} \int (b+a \cos (c+d x)) \left (2 a-b \cos (c+d x)-3 a \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {a b \sec (c+d x) \tan (c+d x)}{3 d}+\frac {(b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{6} \int \left (-2 \left (2 a^2-b^2\right )+6 a b \cos (c+d x)+6 a^2 \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+\frac {a b \sec (c+d x) \tan (c+d x)}{3 d}+\frac {(b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{6} \int \left (6 a b+6 a^2 \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=-a^2 x+\frac {\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+\frac {a b \sec (c+d x) \tan (c+d x)}{3 d}+\frac {(b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}-(a b) \int \sec (c+d x) \, dx\\ &=-a^2 x-\frac {a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+\frac {a b \sec (c+d x) \tan (c+d x)}{3 d}+\frac {(b+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 1.12, size = 201, normalized size = 2.03 \[ \frac {\sec ^3(c+d x) \left (2 \sin (c+d x) \left (\left (3 a^2-b^2\right ) \cos (2 (c+d x))+3 a^2+6 a b \cos (c+d x)+b^2\right )-9 a \cos (c+d x) \left (a (c+d x)-b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-3 a \cos (3 (c+d x)) \left (a (c+d x)-b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 115, normalized size = 1.16 \[ -\frac {6 \, a^{2} d x \cos \left (d x + c\right )^{3} + 3 \, a b \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a b \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (3 \, a b \cos \left (d x + c\right ) + {\left (3 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.09, size = 158, normalized size = 1.60 \[ -\frac {3 \, {\left (d x + c\right )} a^{2} + 3 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 109, normalized size = 1.10 \[ -a^{2} x +\frac {a^{2} \tan \left (d x +c \right )}{d}-\frac {a^{2} c}{d}+\frac {a b \left (\sin ^{3}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {a b \sin \left (d x +c \right )}{d}-\frac {a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 82, normalized size = 0.83 \[ \frac {2 \, b^{2} \tan \left (d x + c\right )^{3} - 6 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} - 3 \, a b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 227, normalized size = 2.29 \[ -\frac {\frac {b^2\,\sin \left (3\,c+3\,d\,x\right )}{12}-\frac {b^2\,\sin \left (c+d\,x\right )}{4}-\frac {a^2\,\sin \left (3\,c+3\,d\,x\right )}{4}-\frac {a^2\,\sin \left (c+d\,x\right )}{4}+\frac {3\,a^2\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\frac {a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{2}-\frac {a\,b\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {3\,a\,b\,\cos \left (c+d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\frac {a\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{2}}{d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \]
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 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{2} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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