Optimal. Leaf size=74 \[ \frac {a^2 \tan (c+d x)}{d}+\frac {b (2 a+b) \tan ^5(c+d x)}{5 d}+\frac {a (a+2 b) \tan ^3(c+d x)}{3 d}+\frac {b^2 \tan ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3675, 373} \[ \frac {a^2 \tan (c+d x)}{d}+\frac {b (2 a+b) \tan ^5(c+d x)}{5 d}+\frac {a (a+2 b) \tan ^3(c+d x)}{3 d}+\frac {b^2 \tan ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3675
Rubi steps
\begin {align*} \int \sec ^4(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \left (a+b x^2\right )^2 \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2+a (a+2 b) x^2+b (2 a+b) x^4+b^2 x^6\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {a^2 \tan (c+d x)}{d}+\frac {a (a+2 b) \tan ^3(c+d x)}{3 d}+\frac {b (2 a+b) \tan ^5(c+d x)}{5 d}+\frac {b^2 \tan ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 83, normalized size = 1.12 \[ \frac {\tan (c+d x) \left (\left (35 a^2-14 a b+3 b^2\right ) \sec ^2(c+d x)+70 a^2+6 b (7 a-4 b) \sec ^4(c+d x)-28 a b+15 b^2 \sec ^6(c+d x)+6 b^2\right )}{105 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 94, normalized size = 1.27 \[ \frac {{\left (2 \, {\left (35 \, a^{2} - 14 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{6} + {\left (35 \, a^{2} - 14 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (7 \, a b - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, b^{2}\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.99, size = 80, normalized size = 1.08 \[ \frac {15 \, b^{2} \tan \left (d x + c\right )^{7} + 42 \, a b \tan \left (d x + c\right )^{5} + 21 \, b^{2} \tan \left (d x + c\right )^{5} + 35 \, a^{2} \tan \left (d x + c\right )^{3} + 70 \, a b \tan \left (d x + c\right )^{3} + 105 \, a^{2} \tan \left (d x + c\right )}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 111, normalized size = 1.50 \[ \frac {-a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+2 a b \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+b^{2} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 66, normalized size = 0.89 \[ \frac {15 \, b^{2} \tan \left (d x + c\right )^{7} + 21 \, {\left (2 \, a b + b^{2}\right )} \tan \left (d x + c\right )^{5} + 35 \, {\left (a^{2} + 2 \, a b\right )} \tan \left (d x + c\right )^{3} + 105 \, a^{2} \tan \left (d x + c\right )}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.28, size = 60, normalized size = 0.81 \[ \frac {a^2\,\mathrm {tan}\left (c+d\,x\right )+\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a+2\,b\right )}{3}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (2\,a+b\right )}{5}}{d} \]
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 \[ \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2} \sec ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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