Integrand size = 21, antiderivative size = 235 \[ \int \frac {\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \text {arcsinh}(\tan (c+d x)) \sec (c+d x)}{8 b^6 d \sqrt {\sec ^2(c+d x)}}+\frac {5 a \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \sec (c+d x)}{b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d} \]
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Time = 0.34 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3593, 747, 829, 858, 221, 739, 212} \[ \int \frac {\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {5 a \left (a^2+b^2\right )^{3/2} \sec (c+d x) \text {arctanh}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right )}{b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d}+\frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \sec (c+d x) \text {arcsinh}(\tan (c+d x))}{8 b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))} \]
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Rule 212
Rule 221
Rule 739
Rule 747
Rule 829
Rule 858
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {\sec (c+d x) \text {Subst}\left (\int \frac {\left (1+\frac {x^2}{b^2}\right )^{5/2}}{(a+x)^2} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt {\sec ^2(c+d x)}} \\ & = -\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}+\frac {(5 \sec (c+d x)) \text {Subst}\left (\int \frac {x \left (1+\frac {x^2}{b^2}\right )^{3/2}}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^3 d \sqrt {\sec ^2(c+d x)}} \\ & = -\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}+\frac {(5 \sec (c+d x)) \text {Subst}\left (\int \frac {\left (-\frac {a}{b^2}+\frac {\left (4 a^2+3 b^2\right ) x}{b^4}\right ) \sqrt {1+\frac {x^2}{b^2}}}{a+x} \, dx,x,b \tan (c+d x)\right )}{4 b d \sqrt {\sec ^2(c+d x)}} \\ & = -\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d}+\frac {(5 b \sec (c+d x)) \text {Subst}\left (\int \frac {-\frac {a \left (4 a^2+5 b^2\right )}{b^6}+\frac {\left (8 a^4+12 a^2 b^2+3 b^4\right ) x}{b^8}}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{8 d \sqrt {\sec ^2(c+d x)}} \\ & = -\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d}-\frac {\left (5 a \left (a^2+b^2\right )^2 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{b^7 d \sqrt {\sec ^2(c+d x)}}+\frac {\left (5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \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 )}{8 b^7 d \sqrt {\sec ^2(c+d x)}} \\ & = \frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \text {arcsinh}(\tan (c+d x)) \sec (c+d x)}{8 b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d}+\frac {\left (5 a \left (a^2+b^2\right )^2 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a^2}{b^2}-x^2} \, dx,x,\frac {1-\frac {a \tan (c+d x)}{b}}{\sqrt {\sec ^2(c+d x)}}\right )}{b^7 d \sqrt {\sec ^2(c+d x)}} \\ & = \frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \text {arcsinh}(\tan (c+d x)) \sec (c+d x)}{8 b^6 d \sqrt {\sec ^2(c+d x)}}+\frac {5 a \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b \left (1-\frac {a \tan (c+d x)}{b}\right )}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \sec (c+d x)}{b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.76 (sec) , antiderivative size = 1152, normalized size of antiderivative = 4.90 \[ \int \frac {\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {(a-i b)^2 (a+i b)^2 \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))}{b^5 d (a+b \tan (c+d x))^2}-\frac {a \left (12 a^2+13 b^2\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{3 b^5 d (a+b \tan (c+d x))^2}+\frac {10 i a (a+i b) (i a+b) \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sqrt {a^2+b^2} \left (-b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 \cos \left (\frac {1}{2} (c+d x)\right )+b^2 \cos \left (\frac {1}{2} (c+d x)\right )}\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{b^6 d (a+b \tan (c+d x))^2}-\frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{8 b^6 d (a+b \tan (c+d x))^2}+\frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{8 b^6 d (a+b \tan (c+d x))^2}+\frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{16 b^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 (a+b \tan (c+d x))^2}+\frac {\left (36 a^2-8 a b+21 b^2\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{48 b^4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a+b \tan (c+d x))^2}-\frac {a \sec ^2(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{3 b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a+b \tan (c+d x))^2}-\frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{16 b^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 (a+b \tan (c+d x))^2}+\frac {a \sec ^2(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{3 b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a+b \tan (c+d x))^2}+\frac {\left (-36 a^2-8 a b-21 b^2\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{48 b^4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a+b \tan (c+d x))^2}+\frac {\sec ^2(c+d x) \left (-12 a^3 \sin \left (\frac {1}{2} (c+d x)\right )-13 a b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{3 b^5 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^2}+\frac {\sec ^2(c+d x) \left (12 a^3 \sin \left (\frac {1}{2} (c+d x)\right )+13 a b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{3 b^5 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(453\) vs. \(2(219)=438\).
Time = 152.14 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.93
method | result | size |
derivativedivides | \(\frac {-\frac {1}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {4 a -3 b}{6 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {12 a^{2}-8 a b +11 b^{2}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (40 a^{4}+60 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{6}}-\frac {32 a^{3}-12 a^{2} b +40 a \,b^{2}-9 b^{3}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {1}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-4 a -3 b}{6 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-12 a^{2}-8 a b -11 b^{2}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-40 a^{4}-60 a^{2} b^{2}-15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{6}}-\frac {-32 a^{3}-12 a^{2} b -40 a \,b^{2}-9 b^{3}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\frac {2 \left (\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a}-\frac {10 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}}{b^{6}}}{d}\) | \(454\) |
default | \(\frac {-\frac {1}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {4 a -3 b}{6 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {12 a^{2}-8 a b +11 b^{2}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (40 a^{4}+60 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{6}}-\frac {32 a^{3}-12 a^{2} b +40 a \,b^{2}-9 b^{3}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {1}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-4 a -3 b}{6 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-12 a^{2}-8 a b -11 b^{2}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-40 a^{4}-60 a^{2} b^{2}-15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{6}}-\frac {-32 a^{3}-12 a^{2} b -40 a \,b^{2}-9 b^{3}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\frac {2 \left (\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a}-\frac {10 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}}{b^{6}}}{d}\) | \(454\) |
risch | \(-\frac {120 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+60 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+180 a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+640 a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+920 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+640 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+180 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+190 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-75 i a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-120 i a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}-60 i a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}+75 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-190 i a \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+120 a^{4} {\mathrm e}^{9 i \left (d x +c \right )}+45 b^{4} {\mathrm e}^{9 i \left (d x +c \right )}+480 a^{4} {\mathrm e}^{7 i \left (d x +c \right )}+120 b^{4} {\mathrm e}^{7 i \left (d x +c \right )}+720 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}+54 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+480 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}+120 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+120 a^{4} {\mathrm e}^{i \left (d x +c \right )}+45 b^{4} {\mathrm e}^{i \left (d x +c \right )}}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right ) d \,b^{5}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{4}}{d \,b^{6}}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{2 d \,b^{4}}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d \,b^{2}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{4}}{d \,b^{6}}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{2 d \,b^{4}}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d \,b^{2}}+\frac {5 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (i a -b \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\right )}{d \,b^{6}}-\frac {5 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (i a -b \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\right )}{d \,b^{6}}\) | \(676\) |
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Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (220) = 440\).
Time = 0.42 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.01 \[ \int \frac {\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {12 \, b^{5} - 30 \, {\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2} + 120 \, {\left ({\left (a^{4} + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{5} + {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 15 \, {\left ({\left (8 \, a^{5} + 12 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (8 \, a^{5} + 12 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \, {\left (2 \, a b^{4} \cos \left (d x + c\right ) + 3 \, {\left (4 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{48 \, {\left (a b^{6} d \cos \left (d x + c\right )^{5} + b^{7} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\sec ^{7}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 827 vs. \(2 (220) = 440\).
Time = 0.32 (sec) , antiderivative size = 827, normalized size of antiderivative = 3.52 \[ \int \frac {\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (220) = 440\).
Time = 0.53 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.26 \[ \int \frac {\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {15 \, {\left (8 \, a^{4} + 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{6}} - \frac {15 \, {\left (8 \, a^{4} + 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{6}} + \frac {120 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} + \frac {48 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )} a b^{5}} + \frac {2 \, {\left (36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 27 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 144 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 288 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 304 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, a^{3} - 112 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} b^{5}}}{24 \, d} \]
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Time = 7.56 (sec) , antiderivative size = 2654, normalized size of antiderivative = 11.29 \[ \int \frac {\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
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