Optimal. Leaf size=125 \[ \frac {2 \sqrt {x}}{a^2}+\frac {4 b \left (2 a^2-b^2\right ) \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2} d}-\frac {2 b^2 \cot \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \]
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Rubi [A]
time = 0.14, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4290, 3870,
4004, 3916, 2739, 632, 212} \begin {gather*} \frac {4 b \left (2 a^2-b^2\right ) \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 d \left (a^2-b^2\right )^{3/2}}-\frac {2 b^2 \cot \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d \sqrt {x}\right )\right )}+\frac {2 \sqrt {x}}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 2739
Rule 3870
Rule 3916
Rule 4004
Rule 4290
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {1}{(a+b \csc (c+d x))^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b^2 \cot \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d \sqrt {x}\right )\right )}-\frac {2 \text {Subst}\left (\int \frac {-a^2+b^2+a b \csc (c+d x)}{a+b \csc (c+d x)} \, dx,x,\sqrt {x}\right )}{a \left (a^2-b^2\right )}\\ &=\frac {2 \sqrt {x}}{a^2}-\frac {2 b^2 \cot \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d \sqrt {x}\right )\right )}-\frac {\left (2 b \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {\csc (c+d x)}{a+b \csc (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right )}\\ &=\frac {2 \sqrt {x}}{a^2}-\frac {2 b^2 \cot \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d \sqrt {x}\right )\right )}-\frac {\left (2 \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a \sin (c+d x)}{b}} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2-b^2\right )}\\ &=\frac {2 \sqrt {x}}{a^2}-\frac {2 b^2 \cot \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d \sqrt {x}\right )\right )}-\frac {\left (4 \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}+x^2} \, dx,x,\tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )}{a^2 \left (a^2-b^2\right ) d}\\ &=\frac {2 \sqrt {x}}{a^2}-\frac {2 b^2 \cot \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d \sqrt {x}\right )\right )}+\frac {\left (8 \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (1-\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}+2 \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )}{a^2 \left (a^2-b^2\right ) d}\\ &=\frac {2 \sqrt {x}}{a^2}+\frac {4 b \left (2 a^2-b^2\right ) \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}+\tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2} d}-\frac {2 b^2 \cot \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d \sqrt {x}\right )\right )}\\ \end {align*}
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Mathematica [A]
time = 0.78, size = 172, normalized size = 1.38 \begin {gather*} \frac {2 \csc \left (c+d \sqrt {x}\right ) \left (\frac {a b^2 \cot \left (c+d \sqrt {x}\right )}{(-a+b) (a+b)}+\left (c+d \sqrt {x}\right ) \left (a+b \csc \left (c+d \sqrt {x}\right )\right )-\frac {2 b \left (-2 a^2+b^2\right ) \text {ArcTan}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {-a^2+b^2}}\right ) \left (a+b \csc \left (c+d \sqrt {x}\right )\right )}{\left (-a^2+b^2\right )^{3/2}}\right ) \left (b+a \sin \left (c+d \sqrt {x}\right )\right )}{a^2 d \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 179, normalized size = 1.43
method | result | size |
derivativedivides | \(\frac {-\frac {4 b \left (\frac {\frac {a^{2} \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a b}{2 a^{2}-2 b^{2}}}{\frac {b \left (\tan ^{2}\left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{2}+a \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+\frac {b}{2}}+\frac {2 \left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {2 b \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (2 a^{2}-2 b^{2}\right ) \sqrt {-a^{2}+b^{2}}}\right )}{a^{2}}+\frac {4 \arctan \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{a^{2}}}{d}\) | \(179\) |
default | \(\frac {-\frac {4 b \left (\frac {\frac {a^{2} \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a b}{2 a^{2}-2 b^{2}}}{\frac {b \left (\tan ^{2}\left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{2}+a \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+\frac {b}{2}}+\frac {2 \left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {2 b \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (2 a^{2}-2 b^{2}\right ) \sqrt {-a^{2}+b^{2}}}\right )}{a^{2}}+\frac {4 \arctan \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{a^{2}}}{d}\) | \(179\) |
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. 254 vs.
\(2 (112) = 224\).
time = 3.76, size = 576, normalized size = 4.61 \begin {gather*} \left [\frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \sin \left (d \sqrt {x} + c\right ) + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x} - 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d \sqrt {x} + c\right ) + {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}} \sin \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {a^{2} - b^{2}}\right )} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d \sqrt {x} + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} a \cos \left (d \sqrt {x} + c\right ) + a^{2} + b^{2} + 2 \, {\left (\sqrt {a^{2} - b^{2}} b \cos \left (d \sqrt {x} + c\right ) + a b\right )} \sin \left (d \sqrt {x} + c\right )}{a^{2} \cos \left (d \sqrt {x} + c\right )^{2} - 2 \, a b \sin \left (d \sqrt {x} + c\right ) - a^{2} - b^{2}}\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sin \left (d \sqrt {x} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d}, \frac {2 \, {\left ({\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \sin \left (d \sqrt {x} + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x} + {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}} \sin \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {-a^{2} + b^{2}}\right )} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} b \sin \left (d \sqrt {x} + c\right ) + \sqrt {-a^{2} + b^{2}} a}{{\left (a^{2} - b^{2}\right )} \cos \left (d \sqrt {x} + c\right )}\right ) - {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sin \left (d \sqrt {x} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d}\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}{\sqrt {x} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 174, normalized size = 1.39 \begin {gather*} -\frac {4 \, {\left (2 \, a^{2} b - b^{3}\right )} {\left (\pi \left \lfloor \frac {d \sqrt {x} + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} d - a^{2} b^{2} d\right )} \sqrt {-a^{2} + b^{2}}} - \frac {4 \, {\left (a b \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right ) + b^{2}\right )}}{{\left (a^{3} d - a b^{2} d\right )} {\left (b \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right ) + b\right )}} + \frac {2 \, {\left (d \sqrt {x} + c\right )}}{a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.35, size = 2737, normalized size = 21.90 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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