Integrand size = 21, antiderivative size = 75 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {\left (a^2+b^2\right ) (a+b \tan (c+d x))^4}{4 b^3 d}-\frac {2 a (a+b \tan (c+d x))^5}{5 b^3 d}+\frac {(a+b \tan (c+d x))^6}{6 b^3 d} \]
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Time = 0.09 (sec) , antiderivative size = 75, 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} \[ \int \sec ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {\left (a^2+b^2\right ) (a+b \tan (c+d x))^4}{4 b^3 d}+\frac {(a+b \tan (c+d x))^6}{6 b^3 d}-\frac {2 a (a+b \tan (c+d x))^5}{5 b^3 d} \]
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Rule 711
Rule 3587
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x)^3 \left (1+\frac {x^2}{b^2}\right ) \, dx,x,b \tan (c+d x)\right )}{b d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\left (a^2+b^2\right ) (a+x)^3}{b^2}-\frac {2 a (a+x)^4}{b^2}+\frac {(a+x)^5}{b^2}\right ) \, dx,x,b \tan (c+d x)\right )}{b d} \\ & = \frac {\left (a^2+b^2\right ) (a+b \tan (c+d x))^4}{4 b^3 d}-\frac {2 a (a+b \tan (c+d x))^5}{5 b^3 d}+\frac {(a+b \tan (c+d x))^6}{6 b^3 d} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.72 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {(a+b \tan (c+d x))^4 \left (a^2+15 b^2-4 a b \tan (c+d x)+10 b^2 \tan ^2(c+d x)\right )}{60 b^3 d} \]
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Time = 16.39 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.69
method | result | size |
derivativedivides | \(\frac {-a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{4 \cos \left (d x +c \right )^{4}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(127\) |
default | \(\frac {-a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{4 \cos \left (d x +c \right )^{4}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(127\) |
risch | \(-\frac {4 \left (-15 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+45 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-45 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+15 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-50 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+30 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-90 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-10 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-60 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-45 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+15 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-30 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+18 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-5 i a^{3}+3 i a \,b^{2}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}\) | \(228\) |
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Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.40 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {10 \, b^{3} + 15 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, {\left (5 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 9 \, a b^{2} \cos \left (d x + c\right ) + {\left (5 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{6}} \]
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\[ \int \sec ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \sec ^{4}{\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.31 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {10 \, b^{3} \tan \left (d x + c\right )^{6} + 36 \, a b^{2} \tan \left (d x + c\right )^{5} + 90 \, a^{2} b \tan \left (d x + c\right )^{2} + 15 \, {\left (3 \, a^{2} b + b^{3}\right )} \tan \left (d x + c\right )^{4} + 60 \, a^{3} \tan \left (d x + c\right ) + 20 \, {\left (a^{3} + 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{3}}{60 \, d} \]
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Time = 0.75 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.49 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {10 \, b^{3} \tan \left (d x + c\right )^{6} + 36 \, a b^{2} \tan \left (d x + c\right )^{5} + 45 \, a^{2} b \tan \left (d x + c\right )^{4} + 15 \, b^{3} \tan \left (d x + c\right )^{4} + 20 \, a^{3} \tan \left (d x + c\right )^{3} + 60 \, a b^{2} \tan \left (d x + c\right )^{3} + 90 \, a^{2} b \tan \left (d x + c\right )^{2} + 60 \, a^{3} \tan \left (d x + c\right )}{60 \, d} \]
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Time = 4.00 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.29 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {a^3}{3}+a\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {3\,a^2\,b}{4}+\frac {b^3}{4}\right )+a^3\,\mathrm {tan}\left (c+d\,x\right )+\frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^6}{6}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}}{d} \]
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