Optimal. Leaf size=125 \[ -\frac {\left (4 a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (2 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}-\frac {2 a b \tan (c+d x)}{3 d}+\frac {a b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac {\tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b)^2}{4 d} \]
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Rubi [A] time = 0.44, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {4397, 2889, 3048, 3031, 3021, 2748, 3767, 8, 3770} \[ -\frac {\left (4 a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (2 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}-\frac {2 a b \tan (c+d x)}{3 d}+\frac {a b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac {\tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b)^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2889
Rule 3021
Rule 3031
Rule 3048
Rule 3767
Rule 3770
Rule 4397
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx &=\int (b+a \cos (c+d x))^2 \sec ^3(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 ^5(c+d x) \, dx\\ &=\frac {(b+a \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (b+a \cos (c+d x)) \left (2 a-b \cos (c+d x)-3 a \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {a b \sec ^2(c+d x) \tan (c+d x)}{6 d}+\frac {(b+a \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{12} \int \left (-3 \left (2 a^2-b^2\right )+8 a b \cos (c+d x)+9 a^2 \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {\left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a b \sec ^2(c+d x) \tan (c+d x)}{6 d}+\frac {(b+a \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (16 a b+3 \left (4 a^2+b^2\right ) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {\left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a b \sec ^2(c+d x) \tan (c+d x)}{6 d}+\frac {(b+a \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{3} (2 a b) \int \sec ^2(c+d x) \, dx-\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 {\left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a b \sec ^2(c+d x) \tan (c+d x)}{6 d}+\frac {(b+a \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {(2 a b) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=-\frac {\left (4 a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {2 a b \tan (c+d x)}{3 d}+\frac {\left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a b \sec ^2(c+d x) \tan (c+d x)}{6 d}+\frac {(b+a \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end {align*}
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Mathematica [B] time = 0.61, size = 336, normalized size = 2.69 \[ \frac {\sec ^4(c+d x) \left (12 \left (4 a^2+b^2\right ) \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+3 \left (4 a^2+b^2\right ) \cos (4 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+24 a^2 \sin (c+d x)+24 a^2 \sin (3 (c+d x))+36 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-36 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+32 a b \sin (2 (c+d x))-16 a b \sin (4 (c+d x))+42 b^2 \sin (c+d x)-6 b^2 \sin (3 (c+d x))+9 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-9 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 129, normalized size = 1.03 \[ -\frac {3 \, {\left (4 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, a b \cos \left (d x + c\right )^{3} - 16 \, a b \cos \left (d x + c\right ) - 3 \, {\left (4 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 6 \, b^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.89, size = 226, normalized size = 1.81 \[ -\frac {3 \, {\left (4 \, a^{2} + b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (4 \, a^{2} + b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 64 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 64 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 169, normalized size = 1.35 \[ \frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {a^{2} \sin \left (d x +c \right )}{2 d}-\frac {a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 a b \left (\sin ^{3}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {b^{2} \sin \left (d x +c \right )}{8 d}-\frac {b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 129, normalized size = 1.03 \[ \frac {32 \, a b \tan \left (d x + c\right )^{3} + 3 \, b^{2} {\left (\frac {2 \, {\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.27, size = 177, normalized size = 1.42 \[ \frac {\left (a^2+\frac {b^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (-a^2-\frac {16\,a\,b}{3}+\frac {7\,b^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-a^2+\frac {16\,a\,b}{3}+\frac {7\,b^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (a^2+\frac {b^2}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2+\frac {b^2}{4}\right )}{d} \]
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 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{2} \sec ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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