Optimal. Leaf size=121 \[ -\frac {\left (a^2+b^2\right )^2}{2 b^5 d (a+b \tan (c+d x))^2}+\frac {4 a \left (a^2+b^2\right )}{b^5 d (a+b \tan (c+d x))}+\frac {2 \left (3 a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^5 d}-\frac {3 a \tan (c+d x)}{b^4 d}+\frac {\tan ^2(c+d x)}{2 b^3 d} \]
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Rubi [A] time = 0.10, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3506, 697} \[ -\frac {\left (a^2+b^2\right )^2}{2 b^5 d (a+b \tan (c+d x))^2}+\frac {4 a \left (a^2+b^2\right )}{b^5 d (a+b \tan (c+d x))}+\frac {2 \left (3 a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^5 d}-\frac {3 a \tan (c+d x)}{b^4 d}+\frac {\tan ^2(c+d x)}{2 b^3 d} \]
Antiderivative was successfully verified.
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Rule 697
Rule 3506
Rubi steps
\begin {align*} \int \frac {\sec ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+\frac {x^2}{b^2}\right )^2}{(a+x)^3} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {3 a}{b^4}+\frac {x}{b^4}+\frac {\left (a^2+b^2\right )^2}{b^4 (a+x)^3}-\frac {4 a \left (a^2+b^2\right )}{b^4 (a+x)^2}+\frac {2 \left (3 a^2+b^2\right )}{b^4 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {2 \left (3 a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^5 d}-\frac {3 a \tan (c+d x)}{b^4 d}+\frac {\tan ^2(c+d x)}{2 b^3 d}-\frac {\left (a^2+b^2\right )^2}{2 b^5 d (a+b \tan (c+d x))^2}+\frac {4 a \left (a^2+b^2\right )}{b^5 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [A] time = 3.48, size = 140, normalized size = 1.16 \[ \frac {-2 a \left (-\frac {a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)\right )+2 \left (a^2+b^2\right ) \left (\frac {3 a^2+4 a b \tan (c+d x)-b^2}{2 (a+b \tan (c+d x))^2}+\log (a+b \tan (c+d x))\right )+\frac {b^4 \sec ^4(c+d x)}{2 (a+b \tan (c+d x))^2}}{b^5 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 354, normalized size = 2.93 \[ \frac {24 \, a^{2} b^{2} \cos \left (d x + c\right )^{4} + b^{4} - 2 \, {\left (9 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left ({\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 2 \, {\left ({\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) - 4 \, {\left (a b^{3} \cos \left (d x + c\right ) + 3 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (2 \, a b^{6} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + b^{7} d \cos \left (d x + c\right )^{2} + {\left (a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.04, size = 140, normalized size = 1.16 \[ \frac {\frac {4 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{5}} + \frac {b^{3} \tan \left (d x + c\right )^{2} - 6 \, a b^{2} \tan \left (d x + c\right )}{b^{6}} - \frac {18 \, a^{2} b^{2} \tan \left (d x + c\right )^{2} + 6 \, b^{4} \tan \left (d x + c\right )^{2} + 28 \, a^{3} b \tan \left (d x + c\right ) + 4 \, a b^{3} \tan \left (d x + c\right ) + 11 \, a^{4} + b^{4}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} b^{5}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 184, normalized size = 1.52 \[ \frac {\tan ^{2}\left (d x +c \right )}{2 b^{3} d}-\frac {3 a \tan \left (d x +c \right )}{b^{4} d}+\frac {6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2}}{d \,b^{5}}+\frac {2 \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} d}+\frac {4 a^{3}}{d \,b^{5} \left (a +b \tan \left (d x +c \right )\right )}+\frac {4 a}{b^{3} d \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{4}}{2 d \,b^{5} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {a^{2}}{d \,b^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {1}{2 b d \left (a +b \tan \left (d x +c \right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 128, normalized size = 1.06 \[ \frac {\frac {7 \, a^{4} + 6 \, a^{2} b^{2} - b^{4} + 8 \, {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )}{b^{7} \tan \left (d x + c\right )^{2} + 2 \, a b^{6} \tan \left (d x + c\right ) + a^{2} b^{5}} + \frac {b \tan \left (d x + c\right )^{2} - 6 \, a \tan \left (d x + c\right )}{b^{4}} + \frac {4 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{5}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.73, size = 143, normalized size = 1.18 \[ \frac {\frac {7\,a^4+6\,a^2\,b^2-b^4}{2\,b}+\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^3+4\,a\,b^2\right )}{d\,\left (a^2\,b^4+2\,a\,b^5\,\mathrm {tan}\left (c+d\,x\right )+b^6\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,b^3\,d}-\frac {3\,a\,\mathrm {tan}\left (c+d\,x\right )}{b^4\,d}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (6\,a^2+2\,b^2\right )}{b^5\,d} \]
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 \[ \int \frac {\sec ^{6}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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