Optimal. Leaf size=408 \[ \frac {b \left (a^2+b^2\right ) \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \cot (c+d x)}+\sqrt {a^2+b^2}+a+b \cot (c+d x)\right )}{2 \sqrt {2} d \sqrt {\sqrt {a^2+b^2}+a}}-\frac {b \left (a^2+b^2\right ) \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \cot (c+d x)}+\sqrt {a^2+b^2}+a+b \cot (c+d x)\right )}{2 \sqrt {2} d \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a-\sqrt {a^2+b^2}}}-\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.51, antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3528, 12, 3485, 700, 1129, 634, 618, 206, 628} \[ \frac {b \left (a^2+b^2\right ) \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \cot (c+d x)}+\sqrt {a^2+b^2}+a+b \cot (c+d x)\right )}{2 \sqrt {2} d \sqrt {\sqrt {a^2+b^2}+a}}-\frac {b \left (a^2+b^2\right ) \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \cot (c+d x)}+\sqrt {a^2+b^2}+a+b \cot (c+d x)\right )}{2 \sqrt {2} d \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a-\sqrt {a^2+b^2}}}-\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 618
Rule 628
Rule 634
Rule 700
Rule 1129
Rule 3485
Rule 3528
Rubi steps
\begin {align*} \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx &=-\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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*}
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Mathematica [C] time = 1.94, size = 178, normalized size = 0.44 \[ \frac {\sin ^2(c+d x) (b \cot (c+d x)-a) (a+b \cot (c+d x)) \left (3 i \sqrt {a-i b} \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )-3 i \sqrt {a+i b} \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )+2 b (a+b \cot (c+d x))^{3/2}\right )}{3 a^2 d \sin ^2(c+d x)-3 b^2 d \cos ^2(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cot \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (b \cot \left (d x + c\right ) - a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.51, size = 972, normalized size = 2.38 \[ -\frac {2 b \left (a +b \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3} \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 )}{4 d b}+\frac {b \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \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 )}{4 d}+\frac {b \,a^{2} \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 {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \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 ) a^{2}}{4 d b}-\frac {b \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \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 )}{4 d}+\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 {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3} \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 )}{4 d b}-\frac {b \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \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 )}{4 d}+\frac {b \,a^{2} \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 {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \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 ) a^{2}}{4 d b}+\frac {b \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \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 )}{4 d}+\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}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.96, size = 2529, normalized size = 6.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int a^{2} \sqrt {a + b \cot {\left (c + d x \right )}}\, dx - \int \left (- b^{2} \sqrt {a + b \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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