Integrand size = 21, antiderivative size = 121 \[ \int \frac {\sec ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {2 \left (3 a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^5 d}-\frac {3 a \tan (c+d x)}{b^4 d}+\frac {\tan ^2(c+d x)}{2 b^3 d}-\frac {\left (a^2+b^2\right )^2}{2 b^5 d (a+b \tan (c+d x))^2}+\frac {4 a \left (a^2+b^2\right )}{b^5 d (a+b \tan (c+d x))} \]
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Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3587, 711} \[ \int \frac {\sec ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (a^2+b^2\right )^2}{2 b^5 d (a+b \tan (c+d x))^2}+\frac {4 a \left (a^2+b^2\right )}{b^5 d (a+b \tan (c+d x))}+\frac {2 \left (3 a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^5 d}-\frac {3 a \tan (c+d x)}{b^4 d}+\frac {\tan ^2(c+d x)}{2 b^3 d} \]
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Rule 711
Rule 3587
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 {\text {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*}
Time = 4.39 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \frac {\sec ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {b^4 \sec ^4(c+d x)}{2 (a+b \tan (c+d x))^2}-2 a \left (-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)-\frac {a^2+b^2}{a+b \tan (c+d x)}\right )+2 \left (a^2+b^2\right ) \left (\log (a+b \tan (c+d x))+\frac {3 a^2-b^2+4 a b \tan (c+d x)}{2 (a+b \tan (c+d x))^2}\right )}{b^5 d} \]
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Time = 174.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+3 a \tan \left (d x +c \right )}{b^{4}}-\frac {a^{4}+2 a^{2} b^{2}+b^{4}}{2 b^{5} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (6 a^{2}+2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5}}+\frac {4 a \left (a^{2}+b^{2}\right )}{b^{5} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(115\) |
default | \(\frac {-\frac {-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+3 a \tan \left (d x +c \right )}{b^{4}}-\frac {a^{4}+2 a^{2} b^{2}+b^{4}}{2 b^{5} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (6 a^{2}+2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5}}+\frac {4 a \left (a^{2}+b^{2}\right )}{b^{5} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(115\) |
risch | \(\frac {-36 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+36 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+4 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+12 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+12 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+36 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-24 a^{2} b -12 i a \,b^{2}+12 i a^{3}+4 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+4 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-4 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} b^{4} d}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) a^{2}}{b^{5} d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{3} d}-\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{b^{5} d}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{3} d}\) | \(346\) |
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Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (117) = 234\).
Time = 0.30 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.93 \[ \int \frac {\sec ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {24 \, a^{2} b^{2} \cos \left (d x + c\right )^{4} + b^{4} - 2 \, {\left (9 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left ({\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 2 \, {\left ({\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) - 4 \, {\left (a b^{3} \cos \left (d x + c\right ) + 3 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (2 \, a b^{6} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + b^{7} d \cos \left (d x + c\right )^{2} + {\left (a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
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\[ \int \frac {\sec ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\sec ^{6}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {7 \, a^{4} + 6 \, a^{2} b^{2} - b^{4} + 8 \, {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )}{b^{7} \tan \left (d x + c\right )^{2} + 2 \, a b^{6} \tan \left (d x + c\right ) + a^{2} b^{5}} + \frac {b \tan \left (d x + c\right )^{2} - 6 \, a \tan \left (d x + c\right )}{b^{4}} + \frac {4 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{5}}}{2 \, d} \]
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Time = 0.58 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \frac {\sec ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {4 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{5}} + \frac {b^{3} \tan \left (d x + c\right )^{2} - 6 \, a b^{2} \tan \left (d x + c\right )}{b^{6}} - \frac {18 \, a^{2} b^{2} \tan \left (d x + c\right )^{2} + 6 \, b^{4} \tan \left (d x + c\right )^{2} + 28 \, a^{3} b \tan \left (d x + c\right ) + 4 \, a b^{3} \tan \left (d x + c\right ) + 11 \, a^{4} + b^{4}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} b^{5}}}{2 \, d} \]
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Time = 4.73 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.18 \[ \int \frac {\sec ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {7\,a^4+6\,a^2\,b^2-b^4}{2\,b}+\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^3+4\,a\,b^2\right )}{d\,\left (a^2\,b^4+2\,a\,b^5\,\mathrm {tan}\left (c+d\,x\right )+b^6\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,b^3\,d}-\frac {3\,a\,\mathrm {tan}\left (c+d\,x\right )}{b^4\,d}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (6\,a^2+2\,b^2\right )}{b^5\,d} \]
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