Integrand size = 21, antiderivative size = 299 \[ \int (a+b \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\frac {5 a^3 x}{16}-\frac {45}{8} a b^2 x+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {5 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^2 b \sin (c+d x)}{d}+\frac {5 b^3 \sin (c+d x)}{2 d}-\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a^2 b \sin ^3(c+d x)}{d}+\frac {5 b^3 \sin ^3(c+d x)}{6 d}-\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {3 a^2 b \sin ^5(c+d x)}{5 d}-\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {45 a b^2 \tan (c+d x)}{8 d}-\frac {15 a b^2 \sin ^2(c+d x) \tan (c+d x)}{8 d}-\frac {3 a b^2 \sin ^4(c+d x) \tan (c+d x)}{4 d}+\frac {b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d} \]
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Time = 0.43 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3957, 2991, 2715, 8, 2672, 308, 212, 2671, 294, 327, 209} \[ \int (a+b \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac {5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {5 a^3 x}{16}+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {3 a^2 b \sin ^5(c+d x)}{5 d}-\frac {a^2 b \sin ^3(c+d x)}{d}-\frac {3 a^2 b \sin (c+d x)}{d}+\frac {45 a b^2 \tan (c+d x)}{8 d}-\frac {3 a b^2 \sin ^4(c+d x) \tan (c+d x)}{4 d}-\frac {15 a b^2 \sin ^2(c+d x) \tan (c+d x)}{8 d}-\frac {45}{8} a b^2 x-\frac {5 b^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {5 b^3 \sin ^3(c+d x)}{6 d}+\frac {5 b^3 \sin (c+d x)}{2 d}+\frac {b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d} \]
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Rule 8
Rule 209
Rule 212
Rule 294
Rule 308
Rule 327
Rule 2671
Rule 2672
Rule 2715
Rule 2991
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x))^3 \sin ^3(c+d x) \tan ^3(c+d x) \, dx \\ & = -\int \left (-a^3 \sin ^6(c+d x)-3 a^2 b \sin ^5(c+d x) \tan (c+d x)-3 a b^2 \sin ^4(c+d x) \tan ^2(c+d x)-b^3 \sin ^3(c+d x) \tan ^3(c+d x)\right ) \, dx \\ & = a^3 \int \sin ^6(c+d x) \, dx+\left (3 a^2 b\right ) \int \sin ^5(c+d x) \tan (c+d x) \, dx+\left (3 a b^2\right ) \int \sin ^4(c+d x) \tan ^2(c+d x) \, dx+b^3 \int \sin ^3(c+d x) \tan ^3(c+d x) \, dx \\ & = -\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{6} \left (5 a^3\right ) \int \sin ^4(c+d x) \, dx+\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {x^6}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{d}+\frac {b^3 \text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {3 a b^2 \sin ^4(c+d x) \tan (c+d x)}{4 d}+\frac {b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d}+\frac {1}{8} \left (5 a^3\right ) \int \sin ^2(c+d x) \, dx+\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \left (-1-x^2-x^4+\frac {1}{1-x^2}\right ) \, dx,x,\sin (c+d x)\right )}{d}+\frac {\left (15 a b^2\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 d}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d} \\ & = -\frac {3 a^2 b \sin (c+d x)}{d}-\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a^2 b \sin ^3(c+d x)}{d}-\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {3 a^2 b \sin ^5(c+d x)}{5 d}-\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {15 a b^2 \sin ^2(c+d x) \tan (c+d x)}{8 d}-\frac {3 a b^2 \sin ^4(c+d x) \tan (c+d x)}{4 d}+\frac {b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d}+\frac {1}{16} \left (5 a^3\right ) \int 1 \, dx+\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}+\frac {\left (45 a b^2\right ) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 d}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\sin (c+d x)\right )}{2 d} \\ & = \frac {5 a^3 x}{16}+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {3 a^2 b \sin (c+d x)}{d}+\frac {5 b^3 \sin (c+d x)}{2 d}-\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a^2 b \sin ^3(c+d x)}{d}+\frac {5 b^3 \sin ^3(c+d x)}{6 d}-\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {3 a^2 b \sin ^5(c+d x)}{5 d}-\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {45 a b^2 \tan (c+d x)}{8 d}-\frac {15 a b^2 \sin ^2(c+d x) \tan (c+d x)}{8 d}-\frac {3 a b^2 \sin ^4(c+d x) \tan (c+d x)}{4 d}+\frac {b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d}-\frac {\left (45 a b^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 d}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d} \\ & = \frac {5 a^3 x}{16}-\frac {45}{8} a b^2 x+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {5 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^2 b \sin (c+d x)}{d}+\frac {5 b^3 \sin (c+d x)}{2 d}-\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a^2 b \sin ^3(c+d x)}{d}+\frac {5 b^3 \sin ^3(c+d x)}{6 d}-\frac {5 a^3 \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac {3 a^2 b \sin ^5(c+d x)}{5 d}-\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac {45 a b^2 \tan (c+d x)}{8 d}-\frac {15 a b^2 \sin ^2(c+d x) \tan (c+d x)}{8 d}-\frac {3 a b^2 \sin ^4(c+d x) \tan (c+d x)}{4 d}+\frac {b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(818\) vs. \(2(299)=598\).
Time = 7.09 (sec) , antiderivative size = 818, normalized size of antiderivative = 2.74 \[ \int (a+b \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\frac {5 a \left (a^2-18 b^2\right ) (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{16 d (b+a \cos (c+d x))^3}+\frac {\left (-6 a^2 b+5 b^3\right ) \cos ^3(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^3}{2 d (b+a \cos (c+d x))^3}+\frac {\left (6 a^2 b-5 b^3\right ) \cos ^3(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^3}{2 d (b+a \cos (c+d x))^3}+\frac {b^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d (b+a \cos (c+d x))^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {3 a b^2 \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin \left (\frac {1}{2} (c+d x)\right )}{d (b+a \cos (c+d x))^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {b^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d (b+a \cos (c+d x))^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {3 a b^2 \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin \left (\frac {1}{2} (c+d x)\right )}{d (b+a \cos (c+d x))^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {3 b \left (-11 a^2+6 b^2\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{8 d (b+a \cos (c+d x))^3}-\frac {3 a \left (5 a^2-32 b^2\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (2 (c+d x))}{64 d (b+a \cos (c+d x))^3}-\frac {b \left (-21 a^2+4 b^2\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (3 (c+d x))}{48 d (b+a \cos (c+d x))^3}+\frac {3 a \left (a^2-2 b^2\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (4 (c+d x))}{64 d (b+a \cos (c+d x))^3}-\frac {3 a^2 b \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (5 (c+d x))}{80 d (b+a \cos (c+d x))^3}-\frac {a^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (6 (c+d x))}{192 d (b+a \cos (c+d x))^3} \]
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Time = 2.57 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(234\) |
default | \(\frac {a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(234\) |
parts | \(\frac {a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {b^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )}{d}+\frac {3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(242\) |
parallelrisch | \(\frac {-5760 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}-\frac {5 b^{2}}{6}\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+5760 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}-\frac {5 b^{2}}{6}\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+600 a d x \left (a^{2}-18 b^{2}\right ) \cos \left (2 d x +2 c \right )+\left (-405 a^{3}+8550 a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (-3156 a^{2} b +2000 b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (-140 a^{3}+1260 a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (348 a^{2} b -80 b^{3}\right ) \sin \left (5 d x +5 c \right )+\left (35 a^{3}-90 a \,b^{2}\right ) \sin \left (6 d x +6 c \right )-36 a^{2} b \sin \left (7 d x +7 c \right )-5 a^{3} \sin \left (8 d x +8 c \right )+\left (-3540 a^{2} b +4000 b^{3}\right ) \sin \left (d x +c \right )+600 a d x \left (a^{2}-18 b^{2}\right )}{1920 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(285\) |
risch | \(\frac {5 a^{3} x}{16}-\frac {45 a \,b^{2} x}{8}-\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{3}}{128 d}-\frac {9 i {\mathrm e}^{i \left (d x +c \right )} b^{3}}{8 d}+\frac {7 i {\mathrm e}^{-3 i \left (d x +c \right )} a^{2} b}{32 d}+\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )} a^{3}}{128 d}-\frac {7 i {\mathrm e}^{3 i \left (d x +c \right )} a^{2} b}{32 d}+\frac {33 i {\mathrm e}^{i \left (d x +c \right )} a^{2} b}{16 d}-\frac {i b^{2} \left (b \,{\mathrm e}^{3 i \left (d x +c \right )}-6 a \,{\mathrm e}^{2 i \left (d x +c \right )}-b \,{\mathrm e}^{i \left (d x +c \right )}-6 a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {9 i {\mathrm e}^{-i \left (d x +c \right )} b^{3}}{8 d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} a \,b^{2}}{4 d}-\frac {33 i {\mathrm e}^{-i \left (d x +c \right )} a^{2} b}{16 d}+\frac {i {\mathrm e}^{3 i \left (d x +c \right )} b^{3}}{24 d}-\frac {i {\mathrm e}^{-3 i \left (d x +c \right )} b^{3}}{24 d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} a \,b^{2}}{4 d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}-\frac {3 a^{2} b \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{3}}{64 d}-\frac {3 \sin \left (4 d x +4 c \right ) a \,b^{2}}{32 d}\) | \(454\) |
norman | \(\frac {\left (\frac {5}{16} a^{3}-\frac {45}{8} a \,b^{2}\right ) x +\left (-\frac {25}{8} a^{3}+\frac {225}{4} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-\frac {5}{4} a^{3}+\frac {45}{2} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {5}{4} a^{3}+\frac {45}{2} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {5}{4} a^{3}-\frac {45}{2} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {5}{4} a^{3}-\frac {45}{2} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {5}{4} a^{3}-\frac {45}{2} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {5}{4} a^{3}-\frac {45}{2} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {5}{16} a^{3}-\frac {45}{8} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\frac {\left (5 a^{3}-48 a^{2} b -90 a \,b^{2}+40 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{8 d}-\frac {\left (5 a^{3}+48 a^{2} b -90 a \,b^{2}-40 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (55 a^{3}-624 a^{2} b -990 a \,b^{2}+520 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{24 d}-\frac {\left (55 a^{3}+624 a^{2} b -990 a \,b^{2}-520 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {\left (215 a^{3}-3984 a^{2} b -3870 a \,b^{2}+3320 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{120 d}-\frac {\left (215 a^{3}+3984 a^{2} b -3870 a \,b^{2}-3320 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d}-\frac {\left (2545 a^{3}-7824 a^{2} b +270 a \,b^{2}-1160 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}+\frac {\left (2545 a^{3}+7824 a^{2} b +270 a \,b^{2}+1160 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{120 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {b \left (6 a^{2}-5 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {b \left (6 a^{2}-5 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(616\) |
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Time = 0.30 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.81 \[ \int (a+b \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\frac {75 \, {\left (a^{3} - 18 \, a b^{2}\right )} d x \cos \left (d x + c\right )^{2} + 60 \, {\left (6 \, a^{2} b - 5 \, b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, {\left (6 \, a^{2} b - 5 \, b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (40 \, a^{3} \cos \left (d x + c\right )^{7} + 144 \, a^{2} b \cos \left (d x + c\right )^{6} - 10 \, {\left (13 \, a^{3} - 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 16 \, {\left (33 \, a^{2} b - 5 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - 720 \, a b^{2} \cos \left (d x + c\right ) + 15 \, {\left (11 \, a^{3} - 54 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 120 \, b^{3} + 16 \, {\left (69 \, a^{2} b - 35 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{2}} \]
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Timed out. \[ \int (a+b \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.81 \[ \int (a+b \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 96 \, {\left (6 \, \sin \left (d x + c\right )^{5} + 10 \, \sin \left (d x + c\right )^{3} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 30 \, \sin \left (d x + c\right )\right )} a^{2} b - 360 \, {\left (15 \, d x + 15 \, c - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a b^{2} + 80 \, {\left (4 \, \sin \left (d x + c\right )^{3} - \frac {6 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 24 \, \sin \left (d x + c\right )\right )} b^{3}}{960 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (273) = 546\).
Time = 0.44 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.88 \[ \int (a+b \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\frac {75 \, {\left (a^{3} - 18 \, a b^{2}\right )} {\left (d x + c\right )} + 120 \, {\left (6 \, a^{2} b - 5 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 120 \, {\left (6 \, a^{2} b - 5 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {240 \, {\left (6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {2 \, {\left (75 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 720 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 630 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 480 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 425 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4560 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2610 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2720 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 990 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12384 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1980 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5760 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 990 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12384 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1980 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5760 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 425 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4560 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2610 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2720 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 75 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 720 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 630 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
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Time = 14.78 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.25 \[ \int (a+b \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\frac {7\,b^3\,\sin \left (c+d\,x\right )}{3\,d}+\frac {5\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8\,d}-\frac {5\,b^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {13\,a^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{24\,d}-\frac {a^3\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{6\,d}+\frac {b^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}-\frac {b^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}-\frac {11\,a^3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{16\,d}-\frac {23\,a^2\,b\,\sin \left (c+d\,x\right )}{5\,d}-\frac {45\,a\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {6\,a^2\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {27\,a\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d}+\frac {3\,a\,b^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {11\,a^2\,b\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{5\,d}-\frac {3\,a\,b^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d}-\frac {3\,a^2\,b\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{5\,d} \]
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