Integrand size = 21, antiderivative size = 159 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {3 a \left (2 a^2-b^2\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {3 a \left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a \left (2 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d} \]
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Time = 0.19 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3593, 757, 794, 201, 221} \[ \int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {3 a \left (2 a^2-b^2\right ) \sec (c+d x) \text {arcsinh}(\tan (c+d x))}{16 d \sqrt {\sec ^2(c+d x)}}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}+\frac {a \left (2 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac {3 a \left (2 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d} \]
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Rule 201
Rule 221
Rule 757
Rule 794
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {\sec (c+d x) \text {Subst}\left (\int (a+x)^3 \left (1+\frac {x^2}{b^2}\right )^{3/2} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt {\sec ^2(c+d x)}} \\ & = \frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {(b \sec (c+d x)) \text {Subst}\left (\int (a+x) \left (-2+\frac {7 a^2}{b^2}+\frac {9 a x}{b^2}\right ) \left (1+\frac {x^2}{b^2}\right )^{3/2} \, dx,x,b \tan (c+d x)\right )}{7 d \sqrt {\sec ^2(c+d x)}} \\ & = \frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}-\frac {\left (\left (\frac {9 a}{b^2}-\frac {6 a \left (-2+\frac {7 a^2}{b^2}\right )}{b^2}\right ) b^3 \sec (c+d x)\right ) \text {Subst}\left (\int \left (1+\frac {x^2}{b^2}\right )^{3/2} \, dx,x,b \tan (c+d x)\right )}{42 d \sqrt {\sec ^2(c+d x)}} \\ & = \frac {a \left (2 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}-\frac {\left (\left (\frac {9 a}{b^2}-\frac {6 a \left (-2+\frac {7 a^2}{b^2}\right )}{b^2}\right ) b^3 \sec (c+d x)\right ) \text {Subst}\left (\int \sqrt {1+\frac {x^2}{b^2}} \, dx,x,b \tan (c+d x)\right )}{56 d \sqrt {\sec ^2(c+d x)}} \\ & = \frac {3 a \left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a \left (2 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}-\frac {\left (\left (\frac {9 a}{b^2}-\frac {6 a \left (-2+\frac {7 a^2}{b^2}\right )}{b^2}\right ) b^3 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{112 d \sqrt {\sec ^2(c+d x)}} \\ & = \frac {3 a \left (2 a^2-b^2\right ) \text {arcsinh}(\tan (c+d x)) \sec (c+d x)}{16 d \sqrt {\sec ^2(c+d x)}}+\frac {3 a \left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a \left (2 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac {b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(637\) vs. \(2(159)=318\).
Time = 3.32 (sec) , antiderivative size = 637, normalized size of antiderivative = 4.01 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {\sec ^7(c+d x) \left (10752 a^2 b+1536 b^3+3584 \left (3 a^2 b-b^3\right ) \cos (2 (c+d x))-4410 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2205 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-1470 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+735 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-210 a^3 \cos (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+105 a b^2 \cos (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-3675 a \left (2 a^2-b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+4410 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2205 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+1470 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-735 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+210 a^3 \cos (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 a b^2 \cos (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+4340 a^3 \sin (2 (c+d x))+6790 a b^2 \sin (2 (c+d x))+2800 a^3 \sin (4 (c+d x))-1400 a b^2 \sin (4 (c+d x))+420 a^3 \sin (6 (c+d x))-210 a b^2 \sin (6 (c+d x))\right )}{35840 d} \]
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Time = 31.31 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.56
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {3 a^{2} b}{5 \cos \left (d x +c \right )^{5}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )}{d}\) | \(248\) |
default | \(\frac {a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {3 a^{2} b}{5 \cos \left (d x +c \right )^{5}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )}{d}\) | \(248\) |
risch | \(-\frac {{\mathrm e}^{i \left (d x +c \right )} \left (-1400 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+700 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+1400 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}-105 i a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+210 i a^{3} {\mathrm e}^{12 i \left (d x +c \right )}-210 i a^{3}-5376 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+1792 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-10752 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-1536 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+105 i a \,b^{2}-700 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-5376 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+1792 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-2170 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3395 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+2170 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}-3395 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}\right )}{280 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{16 d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{16 d}\) | \(405\) |
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Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.07 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {105 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 160 \, b^{3} + 224 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 70 \, {\left (3 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{5} + 8 \, a b^{2} \cos \left (d x + c\right ) + 2 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{1120 \, d \cos \left (d x + c\right )^{7}} \]
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\[ \int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \sec ^{5}{\left (c + d x \right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.31 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {35 \, a b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 70 \, a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {672 \, a^{2} b}{\cos \left (d x + c\right )^{5}} - \frac {32 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} b^{3}}{\cos \left (d x + c\right )^{7}}}{1120 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (146) = 292\).
Time = 0.84 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.92 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {105 \, {\left (2 \, a^{3} - a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (2 \, a^{3} - a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (350 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 105 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 1680 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1540 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 1120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 630 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1085 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5040 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6720 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 630 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1085 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3696 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 448 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1540 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 672 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 224 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 350 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 336 \, a^{2} b + 32 \, b^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}}{560 \, d} \]
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Time = 8.11 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.66 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {3\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^2-b^2\right )}{8\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,a^3}{4}+\frac {3\,a\,b^2}{8}\right )+\frac {6\,a^2\,b}{5}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {11\,a\,b^2}{2}-3\,a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {11\,a\,b^2}{2}-3\,a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {5\,a^3}{4}+\frac {3\,a\,b^2}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {9\,a^3}{4}+\frac {31\,a\,b^2}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {9\,a^3}{4}+\frac {31\,a\,b^2}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (12\,a^2\,b-4\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {12\,a^2\,b}{5}-\frac {4\,b^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (18\,a^2\,b+4\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (24\,a^2\,b-8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {66\,a^2\,b}{5}+\frac {8\,b^3}{5}\right )-\frac {4\,b^3}{35}+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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