Integrand size = 31, antiderivative size = 254 \[ \int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {(a (A-B)-b (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(b (A-B)+a (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(b (A-B)+a (A+B)) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}} \]
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Time = 0.46 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {3672, 3610, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {(a (A-B)-b (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}-\frac {2 (a B+A b)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\frac {(a (A+B)+b (A-B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a (A+B)+b (A-B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3610
Rule 3615
Rule 3672
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\int \frac {A b+a B-(a A-b B) \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)} \, dx \\ & = -\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\int \frac {-a A+b B-(A b+a B) \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\int \frac {-A b-a B+(a A-b B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = -\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {-A b-a B+(a A-b B) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}-\frac {(b (A-B)+a (A+B)) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {(a (A-B)-b (A+B)) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\frac {(b (A-B)+a (A+B)) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {(b (A-B)+a (A+B)) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}+\frac {(a (A-B)-b (A+B)) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d} \\ & = \frac {(b (A-B)+a (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(b (A-B)+a (A+B)) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\frac {(a (A-B)-b (A+B)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a (A-B)-b (A+B)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d} \\ & = -\frac {(a (A-B)-b (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(b (A-B)+a (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(b (A-B)+a (A+B)) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}} \\ \end{align*}
Time = 1.32 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.78 \[ \int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {30 \sqrt {2} (a (A-B)-b (A+B)) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )-15 \sqrt {2} (b (A-B)+a (A+B)) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )+\frac {24 a A}{\tan ^{\frac {5}{2}}(c+d x)}+\frac {40 (A b+a B)}{\tan ^{\frac {3}{2}}(c+d x)}-\frac {120 (a A-b B)}{\sqrt {\tan (c+d x)}}}{60 d} \]
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Time = 0.04 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (A b +B a \right )}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (-a A +B b \right )}{\sqrt {\tan \left (d x +c \right )}}-\frac {2 a A}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}+\frac {\left (-A b -B a \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (a A -B b \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(240\) |
default | \(\frac {-\frac {2 \left (A b +B a \right )}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (-a A +B b \right )}{\sqrt {\tan \left (d x +c \right )}}-\frac {2 a A}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}+\frac {\left (-A b -B a \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (a A -B b \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(240\) |
parts | \(\frac {\left (A b +B a \right ) \left (-\frac {2}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {B b \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {2}{\sqrt {\tan \left (d x +c \right )}}\right )}{d}+\frac {a A \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {2}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}+\frac {2}{\sqrt {\tan \left (d x +c \right )}}\right )}{d}\) | \(323\) |
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Leaf count of result is larger than twice the leaf count of optimal. 2290 vs. \(2 (216) = 432\).
Time = 0.40 (sec) , antiderivative size = 2290, normalized size of antiderivative = 9.02 \[ \int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )}{\tan ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx \]
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Time = 0.47 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=\frac {30 \, \sqrt {2} {\left ({\left (A - B\right )} a - {\left (A + B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 30 \, \sqrt {2} {\left ({\left (A - B\right )} a - {\left (A + B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 15 \, \sqrt {2} {\left ({\left (A + B\right )} a + {\left (A - B\right )} b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 15 \, \sqrt {2} {\left ({\left (A + B\right )} a + {\left (A - B\right )} b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \frac {8 \, {\left (15 \, {\left (A a - B b\right )} \tan \left (d x + c\right )^{2} - 3 \, A a - 5 \, {\left (B a + A b\right )} \tan \left (d x + c\right )\right )}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{60 \, d} \]
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Timed out. \[ \int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Time = 14.25 (sec) , antiderivative size = 1473, normalized size of antiderivative = 5.80 \[ \int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx=\text {Too large to display} \]
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