Integrand size = 33, antiderivative size = 392 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\sqrt {a} \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]
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Time = 1.08 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3662, 3690, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\sqrt {a} \left (a^3 B+a^2 A b+5 a b^2 B-3 A b^3\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d \left (a^2+b^2\right )^2} \]
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Rule 65
Rule 210
Rule 211
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rule 3662
Rule 3690
Rule 3715
Rule 3734
Rubi steps \begin{align*} \text {integral}& = \int \frac {B+A \cot (c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2} \, dx \\ & = \frac {a (A b-a B) \sqrt {\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\int \frac {\frac {1}{2} \left (-a A b-a^2 B-2 b^2 B\right )-b (A b-a B) \cot (c+d x)+\frac {1}{2} a (A b-a B) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{b \left (a^2+b^2\right )} \\ & = \frac {a (A b-a B) \sqrt {\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\int \frac {-b \left (2 a A b-a^2 B+b^2 B\right )+b \left (a^2 A-A b^2+2 a b B\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{b \left (a^2+b^2\right )^2}+\frac {\left (a \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{2 b \left (a^2+b^2\right )^2} \\ & = \frac {a (A b-a B) \sqrt {\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {2 \text {Subst}\left (\int \frac {b \left (2 a A b-a^2 B+b^2 B\right )-b \left (a^2 A-A b^2+2 a b B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{b \left (a^2+b^2\right )^2 d}+\frac {\left (a \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{2 b \left (a^2+b^2\right )^2 d} \\ & = \frac {a (A b-a B) \sqrt {\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\left (a \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{b \left (a^2+b^2\right )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d} \\ & = -\frac {\sqrt {a} \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d} \\ & = -\frac {\sqrt {a} \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d} \\ & = \frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\sqrt {a} \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \sqrt {\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \\ \end{align*}
Time = 2.68 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.87 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-2 \sqrt {2} \left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )+\frac {4 \sqrt {a} \left (a^2+b^2\right ) (A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2}}+\frac {8 \sqrt {a} \left (-2 A b^3+a \left (a^2+3 b^2\right ) B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2}}+\sqrt {2} \left (a^2 (A-B)+b^2 (-A+B)+2 a b (A+B)\right ) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )+\frac {4 a \left (a^2+b^2\right ) (A b-a B) \sqrt {\tan (c+d x)}}{b (a+b \tan (c+d x))}\right )}{4 \left (a^2+b^2\right )^2 d} \]
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Time = 0.44 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {\left (2 a b A -B \,a^{2}+B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}+\frac {\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 a \left (-\frac {\left (A \,a^{2} b +A \,b^{3}-B \,a^{3}-B a \,b^{2}\right ) \sqrt {\cot \left (d x +c \right )}}{2 b \left (b +a \cot \left (d x +c \right )\right )}+\frac {\left (A \,a^{2} b -3 A \,b^{3}+B \,a^{3}+5 B a \,b^{2}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{2 b \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(336\) |
default | \(\frac {-\frac {2 \left (\frac {\left (2 a b A -B \,a^{2}+B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}+\frac {\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}{1+\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{8}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 a \left (-\frac {\left (A \,a^{2} b +A \,b^{3}-B \,a^{3}-B a \,b^{2}\right ) \sqrt {\cot \left (d x +c \right )}}{2 b \left (b +a \cot \left (d x +c \right )\right )}+\frac {\left (A \,a^{2} b -3 A \,b^{3}+B \,a^{3}+5 B a \,b^{2}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{2 b \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(336\) |
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Leaf count of result is larger than twice the leaf count of optimal. 5899 vs. \(2 (353) = 706\).
Time = 19.64 (sec) , antiderivative size = 11824, normalized size of antiderivative = 30.16 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int \frac {A + B \tan {\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2} \cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
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Time = 0.73 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.92 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (B a^{4} + A a^{3} b + 5 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {4 \, {\left (B a^{2} - A a b\right )}}{{\left (a^{2} b^{2} + b^{4} + \frac {a^{3} b + a b^{3}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]
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\[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int { \frac {B \tan \left (d x + c\right ) + A}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \]
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