Optimal. Leaf size=135 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{(a-b)^{5/2} d}+\frac {b \cot (c+d x)}{3 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac {(5 a-2 b) b \cot (c+d x)}{3 a^2 (a-b)^2 d \sqrt {a+b \cot ^2(c+d x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3742, 425, 541,
12, 385, 209} \begin {gather*} \frac {b (5 a-2 b) \cot (c+d x)}{3 a^2 d (a-b)^2 \sqrt {a+b \cot ^2(c+d x)}}-\frac {\text {ArcTan}\left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d (a-b)^{5/2}}+\frac {b \cot (c+d x)}{3 a d (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 385
Rule 425
Rule 541
Rule 3742
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{5/2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {b \cot (c+d x)}{3 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {3 a-2 b-2 b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot (c+d x)\right )}{3 a (a-b) d}\\ &=\frac {b \cot (c+d x)}{3 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac {(5 a-2 b) b \cot (c+d x)}{3 a^2 (a-b)^2 d \sqrt {a+b \cot ^2(c+d x)}}-\frac {\text {Subst}\left (\int \frac {3 a^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{3 a^2 (a-b)^2 d}\\ &=\frac {b \cot (c+d x)}{3 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac {(5 a-2 b) b \cot (c+d x)}{3 a^2 (a-b)^2 d \sqrt {a+b \cot ^2(c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{(a-b)^2 d}\\ &=\frac {b \cot (c+d x)}{3 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac {(5 a-2 b) b \cot (c+d x)}{3 a^2 (a-b)^2 d \sqrt {a+b \cot ^2(c+d x)}}-\frac {\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 )}{(a-b)^2 d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{(a-b)^{5/2} d}+\frac {b \cot (c+d x)}{3 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac {(5 a-2 b) b \cot (c+d x)}{3 a^2 (a-b)^2 d \sqrt {a+b \cot ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 8.00, size = 367, normalized size = 2.72 \begin {gather*} -\frac {\cot ^5(c+d x) \left (24 (a-b)^3 \cos ^2(c+d x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a-b) \cos ^2(c+d x)}{a}\right ) \left (b+a \tan ^2(c+d x)\right )^2+24 (a-b)^3 \cos ^2(c+d x) \, _2F_1\left (2,2;\frac {9}{2};\frac {(a-b) \cos ^2(c+d x)}{a}\right ) \left (3 b^2+7 a b \tan ^2(c+d x)+4 a^2 \tan ^4(c+d x)\right )-\frac {35 a \left (8 b^2+20 a b \tan ^2(c+d x)+15 a^2 \tan ^4(c+d x)\right ) \left (-3 \text {ArcSin}\left (\sqrt {\frac {(a-b) \cos ^2(c+d x)}{a}}\right ) \left (b+a \tan ^2(c+d x)\right )^2+a \sec ^2(c+d x) \sqrt {\frac {(a-b) \cos ^4(c+d x) \left (b+a \tan ^2(c+d x)\right )}{a^2}} \left (4 b+a \left (-1+3 \tan ^2(c+d x)\right )\right )\right )}{\sqrt {\frac {(a-b) \cos ^4(c+d x) \left (b+a \tan ^2(c+d x)\right )}{a^2}}}\right )}{315 a^5 (a-b)^2 d \left (1+\cot ^2(c+d x)\right ) \sqrt {a+b \cot ^2(c+d x)} \left (1+\frac {b \cot ^2(c+d x)}{a}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.22, size = 162, normalized size = 1.20
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{\left (a -b \right )^{3} b^{2}}+\frac {b \left (\frac {\cot \left (d x +c \right )}{3 a \left (a +b \left (\cot ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}}}+\frac {2 \cot \left (d x +c \right )}{3 a^{2} \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{a -b}+\frac {b \cot \left (d x +c \right )}{\left (a -b \right )^{2} a \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}}{d}\) | \(162\) |
default | \(\frac {-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{\left (a -b \right )^{3} b^{2}}+\frac {b \left (\frac {\cot \left (d x +c \right )}{3 a \left (a +b \left (\cot ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}}}+\frac {2 \cot \left (d x +c \right )}{3 a^{2} \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{a -b}+\frac {b \cot \left (d x +c \right )}{\left (a -b \right )^{2} a \sqrt {a +b \left (\cot ^{2}\left (d x +c \right )\right )}}}{d}\) | \(162\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 423 vs.
\(2 (121) = 242\).
time = 3.34, size = 898, normalized size = 6.65 \begin {gather*} \left [-\frac {3 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt {-a + b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, d x + 2 \, c\right ) - b\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 ) + a^{2} - 2 \, b^{2} + 4 \, {\left (a b - b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )\right ) - 8 \, {\left (3 \, a^{3} b - 2 \, a^{2} b^{2} - 2 \, a b^{3} + b^{4} - {\left (3 \, a^{3} b - 7 \, a^{2} b^{2} + 5 \, a b^{3} - b^{4}\right )} \cos \left (2 \, d x + 2 \, c\right )\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}} \sin \left (2 \, d x + 2 \, c\right )}{12 \, {\left ({\left (a^{7} - 5 \, a^{6} b + 10 \, a^{5} b^{2} - 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - a^{2} b^{5}\right )} d \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, {\left (a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} + a^{2} b^{5}\right )} d \cos \left (2 \, d x + 2 \, c\right ) + {\left (a^{7} - a^{6} b - 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4} - a^{2} b^{5}\right )} d\right )}}, -\frac {3 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt {a - b} \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 ) - b}\right ) - 4 \, {\left (3 \, a^{3} b - 2 \, a^{2} b^{2} - 2 \, a b^{3} + b^{4} - {\left (3 \, a^{3} b - 7 \, a^{2} b^{2} + 5 \, a b^{3} - b^{4}\right )} \cos \left (2 \, d x + 2 \, c\right )\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}} \sin \left (2 \, d x + 2 \, c\right )}{6 \, {\left ({\left (a^{7} - 5 \, a^{6} b + 10 \, a^{5} b^{2} - 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - a^{2} b^{5}\right )} d \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, {\left (a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} + a^{2} b^{5}\right )} d \cos \left (2 \, d x + 2 \, c\right ) + {\left (a^{7} - a^{6} b - 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4} - a^{2} b^{5}\right )} d\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 \frac {1}{\left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1160 vs.
\(2 (121) = 242\).
time = 1.49, size = 1160, normalized size = 8.59 \begin {gather*} -\frac {\frac {{\left ({\left (\frac {{\left (5 \, a^{9} b^{2} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 42 \, a^{8} b^{3} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 156 \, a^{7} b^{4} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 336 \, a^{6} b^{5} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 462 \, a^{5} b^{6} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 420 \, a^{4} b^{7} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 252 \, a^{3} b^{8} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 96 \, a^{2} b^{9} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 21 \, a b^{10} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 2 \, b^{11} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{12} - 10 \, a^{11} b + 45 \, a^{10} b^{2} - 120 \, a^{9} b^{3} + 210 \, a^{8} b^{4} - 252 \, a^{7} b^{5} + 210 \, a^{6} b^{6} - 120 \, a^{5} b^{7} + 45 \, a^{4} b^{8} - 10 \, a^{3} b^{9} + a^{2} b^{10}} + \frac {3 \, {\left (8 \, a^{10} b \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 73 \, a^{9} b^{2} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 298 \, a^{8} b^{3} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 716 \, a^{7} b^{4} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 1120 \, a^{6} b^{5} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 1190 \, a^{5} b^{6} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 868 \, a^{4} b^{7} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 428 \, a^{3} b^{8} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 136 \, a^{2} b^{9} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 25 \, a b^{10} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 2 \, b^{11} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )\right )}}{a^{12} - 10 \, a^{11} b + 45 \, a^{10} b^{2} - 120 \, a^{9} b^{3} + 210 \, a^{8} b^{4} - 252 \, a^{7} b^{5} + 210 \, a^{6} b^{6} - 120 \, a^{5} b^{7} + 45 \, a^{4} b^{8} - 10 \, a^{3} b^{9} + a^{2} b^{10}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {3 \, {\left (8 \, a^{10} b \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 73 \, a^{9} b^{2} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 298 \, a^{8} b^{3} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 716 \, a^{7} b^{4} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 1120 \, a^{6} b^{5} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 1190 \, a^{5} b^{6} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 868 \, a^{4} b^{7} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 428 \, a^{3} b^{8} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 136 \, a^{2} b^{9} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 25 \, a b^{10} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 2 \, b^{11} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )\right )}}{a^{12} - 10 \, a^{11} b + 45 \, a^{10} b^{2} - 120 \, a^{9} b^{3} + 210 \, a^{8} b^{4} - 252 \, a^{7} b^{5} + 210 \, a^{6} b^{6} - 120 \, a^{5} b^{7} + 45 \, a^{4} b^{8} - 10 \, a^{3} b^{9} + a^{2} b^{10}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {5 \, a^{9} b^{2} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 42 \, a^{8} b^{3} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 156 \, a^{7} b^{4} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 336 \, a^{6} b^{5} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 462 \, a^{5} b^{6} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 420 \, a^{4} b^{7} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 252 \, a^{3} b^{8} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 96 \, a^{2} b^{9} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + 21 \, a b^{10} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 2 \, b^{11} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )}{a^{12} - 10 \, a^{11} b + 45 \, a^{10} b^{2} - 120 \, a^{9} b^{3} + 210 \, a^{8} b^{4} - 252 \, a^{7} b^{5} + 210 \, a^{6} b^{6} - 120 \, a^{5} b^{7} + 45 \, a^{4} b^{8} - 10 \, a^{3} b^{9} + a^{2} b^{10}}}{{\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b\right )}^{\frac {3}{2}}} - \frac {6 \, \arctan \left (-\frac {\sqrt {b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b} + \sqrt {b}}{2 \, \sqrt {a - b}}\right )}{{\left (a^{2} \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) - 2 \, a b \mathrm {sgn}\left (\sin \left (d x + c\right )\right ) + b^{2} \mathrm {sgn}\left (\sin \left (d x + c\right )\right )\right )} \sqrt {a - b}}}{3 \, d} \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 \frac {1}{{\left (b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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