Integrand size = 28, antiderivative size = 120 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {a^3 \tan (c+d x)}{d}+\frac {3 a^2 b \tan ^2(c+d x)}{2 d}+\frac {a \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac {3 a b^2 \tan ^5(c+d x)}{5 d}+\frac {b^3 \tan ^6(c+d x)}{6 d} \]
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Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3167, 908} \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {a^3 \tan (c+d x)}{d}+\frac {b \left (3 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac {a \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {3 a^2 b \tan ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan ^5(c+d x)}{5 d}+\frac {b^3 \tan ^6(c+d x)}{6 d} \]
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Rule 908
Rule 3167
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(b+a x)^3 \left (1+x^2\right )}{x^7} \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {b^3}{x^7}+\frac {3 a b^2}{x^6}+\frac {3 a^2 b+b^3}{x^5}+\frac {a^3+3 a b^2}{x^4}+\frac {3 a^2 b}{x^3}+\frac {a^3}{x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {a^3 \tan (c+d x)}{d}+\frac {3 a^2 b \tan ^2(c+d x)}{2 d}+\frac {a \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac {3 a b^2 \tan ^5(c+d x)}{5 d}+\frac {b^3 \tan ^6(c+d x)}{6 d} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.45 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {(a+b \tan (c+d x))^4 \left (a^2+15 b^2-4 a b \tan (c+d x)+10 b^2 \tan ^2(c+d x)\right )}{60 b^3 d} \]
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Time = 1.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.99
method | result | size |
parts | \(-\frac {a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{3} \left (\frac {\sec \left (d x +c \right )^{6}}{6}-\frac {\sec \left (d x +c \right )^{4}}{4}\right )}{d}+\frac {3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \sin \left (d x +c \right )^{3}}{15 \cos \left (d x +c \right )^{3}}\right )}{d}+\frac {3 a^{2} b \sec \left (d x +c \right )^{4}}{4 d}\) | \(119\) |
derivativedivides | \(\frac {-a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{4 \cos \left (d x +c \right )^{4}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \sin \left (d x +c \right )^{3}}{15 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{12 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(127\) |
default | \(\frac {-a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{4 \cos \left (d x +c \right )^{4}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \sin \left (d x +c \right )^{3}}{15 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{12 \cos \left (d x +c \right )^{4}}\right )}{d}\) | \(127\) |
risch | \(-\frac {4 \left (-15 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+45 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-45 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+15 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-50 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+30 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-90 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-10 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-60 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-45 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+15 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-30 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+18 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-5 i a^{3}+3 i a \,b^{2}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}\) | \(228\) |
parallelrisch | \(-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} a^{3}-45 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} a^{2} b -55 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a^{3}+60 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a \,b^{2}+90 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a^{2} b -30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b^{3}+90 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{3}-36 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a \,b^{2}-90 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{2} b -20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{3}-90 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}+36 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a \,b^{2}+90 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} b -30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{3}+55 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}-60 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{2}-45 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -15 a^{3}\right )}{15 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}\) | \(315\) |
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Time = 0.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.88 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {10 \, b^{3} + 15 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, {\left (5 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 9 \, a b^{2} \cos \left (d x + c\right ) + {\left (5 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{6}} \]
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Timed out. \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.02 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {20 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} + 12 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} a b^{2} - \frac {5 \, {\left (3 \, \sin \left (d x + c\right )^{2} - 1\right )} b^{3}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + \frac {45 \, a^{2} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{60 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.93 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {10 \, b^{3} \tan \left (d x + c\right )^{6} + 36 \, a b^{2} \tan \left (d x + c\right )^{5} + 45 \, a^{2} b \tan \left (d x + c\right )^{4} + 15 \, b^{3} \tan \left (d x + c\right )^{4} + 20 \, a^{3} \tan \left (d x + c\right )^{3} + 60 \, a b^{2} \tan \left (d x + c\right )^{3} + 90 \, a^{2} b \tan \left (d x + c\right )^{2} + 60 \, a^{3} \tan \left (d x + c\right )}{60 \, d} \]
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Time = 22.48 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.02 \[ \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {a^3\,\sin \left (c+d\,x\right )}{3}-\frac {a\,b^2\,\sin \left (c+d\,x\right )}{5}\right )+{\cos \left (c+d\,x\right )}^5\,\left (\frac {2\,a^3\,\sin \left (c+d\,x\right )}{3}-\frac {2\,a\,b^2\,\sin \left (c+d\,x\right )}{5}\right )+{\cos \left (c+d\,x\right )}^2\,\left (\frac {3\,a^2\,b}{4}-\frac {b^3}{4}\right )+\frac {b^3}{6}+\frac {3\,a\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{5}}{d\,{\cos \left (c+d\,x\right )}^6} \]
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