Integrand size = 27, antiderivative size = 408 \[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\frac {b \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac {b \left (a^2+b^2\right ) \log \left (a+\sqrt {a^2+b^2}+b \cot (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \cot (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \left (a^2+b^2\right ) \log \left (a+\sqrt {a^2+b^2}+b \cot (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \cot (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d} \]
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Time = 0.61 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3609, 12, 3566, 714, 1143, 648, 632, 212, 642} \[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\frac {b \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a-\sqrt {a^2+b^2}}}+\frac {b \left (a^2+b^2\right ) \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \cot (c+d x)}+\sqrt {a^2+b^2}+a+b \cot (c+d x)\right )}{2 \sqrt {2} d \sqrt {\sqrt {a^2+b^2}+a}}-\frac {b \left (a^2+b^2\right ) \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \cot (c+d x)}+\sqrt {a^2+b^2}+a+b \cot (c+d x)\right )}{2 \sqrt {2} d \sqrt {\sqrt {a^2+b^2}+a}}-\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d} \]
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Rule 12
Rule 212
Rule 632
Rule 642
Rule 648
Rule 714
Rule 1143
Rule 3566
Rule 3609
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\int \left (-a^2-b^2\right ) \sqrt {a+b \cot (c+d x)} \, dx \\ & = -\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\left (-a^2-b^2\right ) \int \sqrt {a+b \cot (c+d x)} \, dx \\ & = -\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a+x}}{b^2+x^2} \, dx,x,b \cot (c+d x)\right )}{d} \\ & = -\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac {\left (2 b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a^2+b^2-2 a x^2+x^4} \, dx,x,\sqrt {a+b \cot (c+d x)}\right )}{d} \\ & = -\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \cot (c+d x)}\right )}{\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \cot (c+d x)}\right )}{\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ & = -\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \cot (c+d x)}\right )}{2 d}+\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \cot (c+d x)}\right )}{2 d}+\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \cot (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \cot (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ & = -\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac {b \left (a^2+b^2\right ) \log \left (a+\sqrt {a^2+b^2}+b \cot (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \cot (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \left (a^2+b^2\right ) \log \left (a+\sqrt {a^2+b^2}+b \cot (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \cot (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \cot (c+d x)}\right )}{d}-\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \cot (c+d x)}\right )}{d} \\ & = \frac {b \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac {b \left (a^2+b^2\right ) \log \left (a+\sqrt {a^2+b^2}+b \cot (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \cot (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \left (a^2+b^2\right ) \log \left (a+\sqrt {a^2+b^2}+b \cot (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \cot (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.62 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.44 \[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\frac {(-a+b \cot (c+d x)) (a+b \cot (c+d x)) \left (3 i \sqrt {a-i b} \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )-3 i \sqrt {a+i b} \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )+2 b (a+b \cot (c+d x))^{3/2}\right ) \sin ^2(c+d x)}{-3 b^2 d \cos ^2(c+d x)+3 a^2 d \sin ^2(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(985\) vs. \(2(333)=666\).
Time = 0.07 (sec) , antiderivative size = 986, normalized size of antiderivative = 2.42
method | result | size |
derivativedivides | \(-\frac {2 b \left (a +b \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}}{4 d b}+\frac {b \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d}-\frac {b \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b^{3} \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}}{4 d b}-\frac {b \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d}-\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}}{4 d b}-\frac {b \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b^{3} \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}}{4 d b}+\frac {b \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d}\) | \(986\) |
default | \(-\frac {2 b \left (a +b \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}}{4 d b}+\frac {b \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d}-\frac {b \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b^{3} \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}}{4 d b}-\frac {b \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d}-\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}}{4 d b}-\frac {b \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b^{3} \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}}{4 d b}+\frac {b \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d}\) | \(986\) |
parts | \(\text {Expression too large to display}\) | \(1656\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1148 vs. \(2 (335) = 670\).
Time = 0.29 (sec) , antiderivative size = 1148, normalized size of antiderivative = 2.81 \[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\text {Too large to display} \]
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\[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=- \int a^{2} \sqrt {a + b \cot {\left (c + d x \right )}}\, dx - \int \left (- b^{2} \sqrt {a + b \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )}\right )\, dx \]
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\[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\int { {\left (b \cot \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (b \cot \left (d x + c\right ) - a\right )} \,d x } \]
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\[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\int { {\left (b \cot \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (b \cot \left (d x + c\right ) - a\right )} \,d x } \]
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Time = 25.17 (sec) , antiderivative size = 2529, normalized size of antiderivative = 6.20 \[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\text {Too large to display} \]
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