Optimal. Leaf size=96 \[ \frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {3 (2 a-b) b \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) \left (a-(a-b) \sin ^2(c+d x)\right ) \tan (c+d x)}{4 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 96, 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 = {3757, 424, 393,
212} \begin {gather*} \frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {3 b (2 a-b) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b \tan (c+d x) \sec ^3(c+d x) \left (a-(a-b) \sin ^2(c+d x)\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 393
Rule 424
Rule 3757
Rubi steps
\begin {align*} \int \sec (c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a-(a-b) x^2\right )^2}{\left (1-x^2\right )^3} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {b \sec ^3(c+d x) \left (a-(a-b) \sin ^2(c+d x)\right ) \tan (c+d x)}{4 d}-\frac {\text {Subst}\left (\int \frac {-a (4 a-b)+(4 a-3 b) (a-b) x^2}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{4 d}\\ &=\frac {3 (2 a-b) b \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) \left (a-(a-b) \sin ^2(c+d x)\right ) \tan (c+d x)}{4 d}+\frac {\left (8 a^2-8 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{8 d}\\ &=\frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {3 (2 a-b) b \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^3(c+d x) \left (a-(a-b) \sin ^2(c+d x)\right ) \tan (c+d x)}{4 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 4.48, size = 347, normalized size = 3.61 \begin {gather*} \frac {\csc ^3(c+d x) \left (128 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {9}{2};\sin ^2(c+d x)\right ) \sin ^6(c+d x) \left (a+(-a+b) \sin ^2(c+d x)\right )^2+128 \, _4F_3\left (\frac {3}{2},2,2,2;1,1,\frac {9}{2};\sin ^2(c+d x)\right ) \sin ^6(c+d x) \left (\frac {1}{2} a^2 \cos ^2(c+d x) (9+5 \cos (2 (c+d x)))+b \sin ^2(c+d x) \left (7 a+5 a \cos (2 (c+d x))+5 b \sin ^2(c+d x)\right )\right )+35 \left (-3375 a^2+3 a (1969 a-1750 b) \sin ^2(c+d x)+\left (-3161 a^2+5108 a b-1947 b^2\right ) \sin ^4(c+d x)+485 (a-b)^2 \sin ^6(c+d x)+\frac {3 \tanh ^{-1}\left (\sqrt {\sin ^2(c+d x)}\right ) \left (1125 a^2-2 a (1172 a-875 b) \sin ^2(c+d x)+\left (1674 a^2-2286 a b+649 b^2\right ) \sin ^4(c+d x)+\left (-400 a^2+778 a b-378 b^2\right ) \sin ^6(c+d x)+9 (a-b)^2 \sin ^8(c+d x)\right )}{\sqrt {\sin ^2(c+d x)}}\right )\right )}{6720 d} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.20, size = 146, normalized size = 1.52
method | result | size |
derivativedivides | \(\frac {b^{2} \left (\frac {\sin ^{5}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{5}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+2 a b \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(146\) |
default | \(\frac {b^{2} \left (\frac {\sin ^{5}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{5}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+2 a b \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(146\) |
risch | \(\frac {i b \left (-8 a \,{\mathrm e}^{7 i \left (d x +c \right )}+5 b \,{\mathrm e}^{7 i \left (d x +c \right )}-8 a \,{\mathrm e}^{5 i \left (d x +c \right )}-3 b \,{\mathrm e}^{5 i \left (d x +c \right )}+8 a \,{\mathrm e}^{3 i \left (d x +c \right )}+3 b \,{\mathrm e}^{3 i \left (d x +c \right )}+8 a \,{\mathrm e}^{i \left (d x +c \right )}-5 b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a b}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{8 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a b}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{8 d}\) | \(241\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 119, normalized size = 1.24 \begin {gather*} \frac {{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left ({\left (8 \, a b - 5 \, b^{2}\right )} \sin \left (d x + c\right )^{3} - {\left (8 \, a b - 3 \, b^{2}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.66, size = 116, normalized size = 1.21 \begin {gather*} \frac {{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left ({\left (8 \, a b - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, b^{2}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \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 ^{2}{\left (c + d x \right )}\right )^{2} \sec {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.74, size = 120, normalized size = 1.25 \begin {gather*} \frac {{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (8 \, a b \sin \left (d x + c\right )^{3} - 5 \, b^{2} \sin \left (d x + c\right )^{3} - 8 \, a b \sin \left (d x + c\right ) + 3 \, b^{2} \sin \left (d x + c\right )\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.57, size = 177, normalized size = 1.84 \begin {gather*} \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^2-2\,a\,b+\frac {3\,b^2}{4}\right )}{d}+\frac {\left (2\,a\,b-\frac {3\,b^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {11\,b^2}{4}-2\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {11\,b^2}{4}-2\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a\,b-\frac {3\,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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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