Integrand size = 28, antiderivative size = 30 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {(b+a \cot (c+d x))^5 \tan ^5(c+d x)}{5 b d} \]
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Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3167, 37} \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {\tan ^5(c+d x) (a \cot (c+d x)+b)^5}{5 b d} \]
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Rule 37
Rule 3167
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(b+a x)^4}{x^6} \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {(b+a \cot (c+d x))^5 \tan ^5(c+d x)}{5 b d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(30)=60\).
Time = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.43 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {\tan (c+d x) \left (5 a^4+10 a^3 b \tan (c+d x)+10 a^2 b^2 \tan ^2(c+d x)+5 a b^3 \tan ^3(c+d x)+b^4 \tan ^4(c+d x)\right )}{5 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(28)=56\).
Time = 1.31 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.20
method | result | size |
derivativedivides | \(\frac {a^{4} \tan \left (d x +c \right )+\frac {2 a^{3} b}{\cos \left (d x +c \right )^{2}}+\frac {2 a^{2} b^{2} \sin \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3}}+\frac {a \,b^{3} \sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{4}}+\frac {b^{4} \sin \left (d x +c \right )^{5}}{5 \cos \left (d x +c \right )^{5}}}{d}\) | \(96\) |
default | \(\frac {a^{4} \tan \left (d x +c \right )+\frac {2 a^{3} b}{\cos \left (d x +c \right )^{2}}+\frac {2 a^{2} b^{2} \sin \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3}}+\frac {a \,b^{3} \sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{4}}+\frac {b^{4} \sin \left (d x +c \right )^{5}}{5 \cos \left (d x +c \right )^{5}}}{d}\) | \(96\) |
parts | \(\frac {a^{4} \tan \left (d x +c \right )}{d}+\frac {b^{4} \sin \left (d x +c \right )^{5}}{5 d \cos \left (d x +c \right )^{5}}+\frac {2 a^{3} b \sec \left (d x +c \right )^{2}}{d}+\frac {4 a \,b^{3} \left (\frac {\sec \left (d x +c \right )^{4}}{4}-\frac {\sec \left (d x +c \right )^{2}}{2}\right )}{d}+\frac {2 a^{2} b^{2} \sin \left (d x +c \right )^{3}}{d \cos \left (d x +c \right )^{3}}\) | \(113\) |
parallelrisch | \(-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a^{4}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a^{3} b +\left (-4 a^{4}+8 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (12 a^{3} b -8 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (6 a^{4}-16 a^{2} b^{2}+\frac {16}{5} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-12 a^{3} b +8 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-4 a^{4}+8 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} b +a^{4}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(225\) |
risch | \(\frac {2 i \left (-60 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+20 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+5 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}-30 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+5 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+20 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+20 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+20 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-60 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+20 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-20 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+30 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-40 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+10 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-60 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}-20 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+20 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-20 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+5 a^{4}-10 a^{2} b^{2}+b^{4}\right )}{5 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}\) | \(317\) |
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.63 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {5 \, a b^{3} \cos \left (d x + c\right ) + 10 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left ({\left (5 \, a^{4} - 10 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + b^{4} + 2 \, {\left (5 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{5 \, d \cos \left (d x + c\right )^{5}} \]
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Timed out. \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (28) = 56\).
Time = 0.24 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.43 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {b^{4} \tan \left (d x + c\right )^{5} + 10 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} + 5 \, a^{4} \tan \left (d x + c\right ) + \frac {5 \, {\left (2 \, \sin \left (d x + c\right )^{2} - 1\right )} a b^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \frac {10 \, a^{3} b}{\sin \left (d x + c\right )^{2} - 1}}{5 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (28) = 56\).
Time = 0.44 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.43 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {b^{4} \tan \left (d x + c\right )^{5} + 5 \, a b^{3} \tan \left (d x + c\right )^{4} + 10 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} + 10 \, a^{3} b \tan \left (d x + c\right )^{2} + 5 \, a^{4} \tan \left (d x + c\right )}{5 \, d} \]
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Time = 22.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.63 \[ \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {\frac {b^4\,\sin \left (c+d\,x\right )}{5}-{\cos \left (c+d\,x\right )}^3\,\left (2\,a\,b^3-2\,a^3\,b\right )-{\cos \left (c+d\,x\right )}^2\,\left (\frac {2\,b^4\,\sin \left (c+d\,x\right )}{5}-2\,a^2\,b^2\,\sin \left (c+d\,x\right )\right )+{\cos \left (c+d\,x\right )}^4\,\left (\sin \left (c+d\,x\right )\,a^4-2\,\sin \left (c+d\,x\right )\,a^2\,b^2+\frac {\sin \left (c+d\,x\right )\,b^4}{5}\right )+a\,b^3\,\cos \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^5} \]
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