Optimal. Leaf size=126 \[ -\frac {(a-b)^{3/2} \text {ArcTan}\left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {(3 a-2 b) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{2 d}-\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3742, 427, 537,
223, 212, 385, 209} \begin {gather*} -\frac {(a-b)^{3/2} \text {ArcTan}\left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d}-\frac {\sqrt {b} (3 a-2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 223
Rule 385
Rule 427
Rule 537
Rule 3742
Rubi steps
\begin {align*} \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d}-\frac {\text {Subst}\left (\int \frac {a (2 a-b)+(3 a-2 b) b x^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d}-\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{d}-\frac {((3 a-2 b) b) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d}-\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {((3 a-2 b) b) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{2 d}\\ &=-\frac {(a-b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {(3 a-2 b) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{2 d}-\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 143, normalized size = 1.13 \begin {gather*} \frac {2 (a-b)^{3/2} \text {ArcTan}\left (\frac {\sqrt {b}+\sqrt {b} \cot ^2(c+d x)-\cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{\sqrt {a-b}}\right )-b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}+(3 a-2 b) \sqrt {b} \log \left (-\sqrt {b} \cot (c+d x)+\sqrt {a+b \cot ^2(c+d x)}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(328\) vs.
\(2(108)=216\).
time = 0.23, size = 329, normalized size = 2.61
method | result | size |
derivativedivides | \(\frac {-b^{2} \left (\frac {\cot \left (d x +c \right ) \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}{2 b}-\frac {a \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}\right )}{2 b^{\frac {3}{2}}}-\frac {\ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}\right )}{\sqrt {b}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{b^{2} \left (a -b \right )}\right )-2 a b \left (\frac {\ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}\right )}{\sqrt {b}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{b^{2} \left (a -b \right )}\right )-\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{b^{2} \left (a -b \right )}}{d}\) | \(329\) |
default | \(\frac {-b^{2} \left (\frac {\cot \left (d x +c \right ) \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}{2 b}-\frac {a \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}\right )}{2 b^{\frac {3}{2}}}-\frac {\ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}\right )}{\sqrt {b}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{b^{2} \left (a -b \right )}\right )-2 a b \left (\frac {\ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}\right )}{\sqrt {b}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{b^{2} \left (a -b \right )}\right )-\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{b^{2} \left (a -b \right )}}{d}\) | \(329\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 258 vs.
\(2 (108) = 216\).
time = 3.49, size = 1071, normalized size = 8.50 \begin {gather*} \left [-\frac {2 \, {\left (a - b\right )} \sqrt {-a + b} \log \left (-{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right ) + b\right ) \sin \left (2 \, d x + 2 \, c\right ) + {\left (3 \, a - 2 \, b\right )} \sqrt {b} \log \left (\frac {{\left (a - 2 \, b\right )} \cos \left (2 \, d x + 2 \, c\right ) - 2 \, \sqrt {b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right ) - a - 2 \, b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}\right ) \sin \left (2 \, d x + 2 \, c\right ) + 2 \, {\left (b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}}}{4 \, d \sin \left (2 \, d x + 2 \, c\right )}, \frac {{\left (3 \, a - 2 \, b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right )}{b \cos \left (2 \, d x + 2 \, c\right ) + b}\right ) \sin \left (2 \, d x + 2 \, c\right ) - {\left (a - b\right )} \sqrt {-a + b} \log \left (-{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right ) + b\right ) \sin \left (2 \, d x + 2 \, c\right ) - {\left (b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}}}{2 \, d \sin \left (2 \, d x + 2 \, c\right )}, -\frac {4 \, {\left (a - b\right )}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right )}{{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) + a - b}\right ) \sin \left (2 \, d x + 2 \, c\right ) + {\left (3 \, a - 2 \, b\right )} \sqrt {b} \log \left (\frac {{\left (a - 2 \, b\right )} \cos \left (2 \, d x + 2 \, c\right ) - 2 \, \sqrt {b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right ) - a - 2 \, b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}\right ) \sin \left (2 \, d x + 2 \, c\right ) + 2 \, {\left (b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}}}{4 \, d \sin \left (2 \, d x + 2 \, c\right )}, -\frac {2 \, {\left (a - b\right )}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right )}{{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) + a - b}\right ) \sin \left (2 \, d x + 2 \, c\right ) - {\left (3 \, a - 2 \, b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}} \sin \left (2 \, d x + 2 \, c\right )}{b \cos \left (2 \, d x + 2 \, c\right ) + b}\right ) \sin \left (2 \, d x + 2 \, c\right ) + {\left (b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a - b}{\cos \left (2 \, d x + 2 \, c\right ) - 1}}}{2 \, d \sin \left (2 \, d x + 2 \, c\right )}\right ] \end {gather*}
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 \begin {gather*} \int \left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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