Optimal. Leaf size=96 \[ \frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {3 b (2 a-b) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b \tan (c+d x) \sec ^3(c+d x) \left (a-(a-b) \sin ^2(c+d x)\right )}{4 d} \]
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Rubi [A] time = 0.09, 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 = {3676, 413, 385, 206} \[ \frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {3 b (2 a-b) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b \tan (c+d x) \sec ^3(c+d x) \left (a-(a-b) \sin ^2(c+d x)\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 413
Rule 3676
Rubi steps
\begin {align*} \int \sec (c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {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 {\operatorname {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 ) \operatorname {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] time = 7.33, size = 347, normalized size = 3.61 \[ \frac {\csc ^3(c+d x) \left (128 \sin ^6(c+d x) \left (\frac {1}{2} a^2 (5 \cos (2 (c+d x))+9) \cos ^2(c+d x)+b \sin ^2(c+d x) \left (5 a \cos (2 (c+d x))+7 a+5 b \sin ^2(c+d x)\right )\right ) \, _4F_3\left (\frac {3}{2},2,2,2;1,1,\frac {9}{2};\sin ^2(c+d x)\right )+128 \sin ^6(c+d x) \left ((b-a) \sin ^2(c+d x)+a\right )^2 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {9}{2};\sin ^2(c+d x)\right )+35 \left (\left (-3161 a^2+5108 a b-1947 b^2\right ) \sin ^4(c+d x)+\frac {3 \tanh ^{-1}\left (\sqrt {\sin ^2(c+d x)}\right ) \left (\left (-400 a^2+778 a b-378 b^2\right ) \sin ^6(c+d x)+\left (1674 a^2-2286 a b+649 b^2\right ) \sin ^4(c+d x)+1125 a^2+9 (a-b)^2 \sin ^8(c+d x)-2 a (1172 a-875 b) \sin ^2(c+d x)\right )}{\sqrt {\sin ^2(c+d x)}}-3375 a^2+485 (a-b)^2 \sin ^6(c+d x)+3 a (1969 a-1750 b) \sin ^2(c+d x)\right )\right )}{6720 d} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 116, normalized size = 1.21 \[ \frac {{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left ({\left (8 \, a b - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, b^{2}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.70, size = 120, normalized size = 1.25 \[ \frac {{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (8 \, a b \sin \left (d x + c\right )^{3} - 5 \, b^{2} \sin \left (d x + c\right )^{3} - 8 \, a b \sin \left (d x + c\right ) + 3 \, b^{2} \sin \left (d x + c\right )\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 178, normalized size = 1.85 \[ \frac {b^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {b^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d}-\frac {3 b^{2} \sin \left (d x +c \right )}{8 d}+\frac {3 b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {a b \left (\sin ^{3}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {a b \sin \left (d x +c \right )}{d}-\frac {a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 119, normalized size = 1.24 \[ \frac {{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left ({\left (8 \, a b - 5 \, b^{2}\right )} \sin \left (d x + c\right )^{3} - {\left (8 \, a b - 3 \, b^{2}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \]
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 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^2-2\,a\,b+\frac {3\,b^2}{4}\right )}{d}+\frac {\left (2\,a\,b-\frac {3\,b^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {11\,b^2}{4}-2\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {11\,b^2}{4}-2\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a\,b-\frac {3\,b^2}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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 \[ \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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