Integrand size = 31, antiderivative size = 254 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {(b (A-B)+a (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(b (A-B)+a (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a (A-B)-b (A+B)) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 (a A-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d} \]
[Out]
Time = 0.34 (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 = {3673, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {(a (A+B)+b (A-B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a (A+B)+b (A-B)) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 (a B+A b) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 (a A-b B) \sqrt {\tan (c+d x)}}{d}+\frac {(a (A-B)-b (A+B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a (A-B)-b (A+B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d} \]
[In]
[Out]
Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3609
Rule 3615
Rule 3673
Rubi steps \begin{align*} \text {integral}& = \frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\int \tan ^{\frac {3}{2}}(c+d x) (a A-b B+(A b+a B) \tan (c+d x)) \, dx \\ & = \frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\int \sqrt {\tan (c+d x)} (-A b-a B+(a A-b B) \tan (c+d x)) \, dx \\ & = \frac {2 (a A-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\int \frac {-a A+b B-(A b+a B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = \frac {2 (a A-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 \text {Subst}\left (\int \frac {-a A+b B+(-A b-a B) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {2 (a A-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\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-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {(b (A-B)+a (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)) \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 {\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 {\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)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a (A-B)-b (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-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {(b (A-B)+a (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 {(b (A-B)+a (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 {(b (A-B)+a (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(b (A-B)+a (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a (A-B)-b (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-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.53 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {15 \sqrt [4]{-1} (a-i b) (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+15 \sqrt [4]{-1} (a+i b) (A+i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {\tan (c+d x)} \left (15 (a A-b B)+5 (A b+a B) \tan (c+d x)+3 b B \tan ^2(c+d x)\right )}{15 d} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {2 B b \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {2 A b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {2 B a \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a A \left (\sqrt {\tan }\left (d x +c \right )\right )-2 \left (\sqrt {\tan }\left (d x +c \right )\right ) B b +\frac {\left (-a A +B b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-A b -B a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(253\) |
default | \(\frac {\frac {2 B b \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {2 A b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {2 B a \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a A \left (\sqrt {\tan }\left (d x +c \right )\right )-2 \left (\sqrt {\tan }\left (d x +c \right )\right ) B b +\frac {\left (-a A +B b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-A b -B a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(253\) |
parts | \(\frac {\left (A b +B a \right ) \left (\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {B b \left (\frac {2 \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-2 \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {a A \left (2 \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(323\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 2252 vs. \(2 (216) = 432\).
Time = 0.38 (sec) , antiderivative size = 2252, normalized size of antiderivative = 8.87 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right ) \tan ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \]
[In]
[Out]
none
Time = 0.39 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.83 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {24 \, B b \tan \left (d x + c\right )^{\frac {5}{2}} - 30 \, \sqrt {2} {\left ({\left (A + B\right )} a + {\left (A - B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 30 \, \sqrt {2} {\left ({\left (A + B\right )} a + {\left (A - B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 15 \, \sqrt {2} {\left ({\left (A - B\right )} a - {\left (A + B\right )} b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 15 \, \sqrt {2} {\left ({\left (A - B\right )} a - {\left (A + B\right )} b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 40 \, {\left (B a + A b\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + 120 \, {\left (A a - B b\right )} \sqrt {\tan \left (d x + c\right )}}{60 \, d} \]
[In]
[Out]
Timed out. \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
[In]
[Out]
Time = 14.41 (sec) , antiderivative size = 1492, normalized size of antiderivative = 5.87 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
[In]
[Out]