Optimal. Leaf size=78 \[ -\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {a \sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f} \]
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Rubi [A]
time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3273, 52, 65,
214} \begin {gather*} -\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {a \sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 3273
Rubi steps
\begin {align*} \int \cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac {a \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=\frac {a \sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac {a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=\frac {a \sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac {a^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{b f}\\ &=-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {a \sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 69, normalized size = 0.88 \begin {gather*} \frac {-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a+b \sin ^2(e+f x)} \left (4 a+b \sin ^2(e+f x)\right )}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 9.76, size = 86, normalized size = 1.10
method | result | size |
default | \(\frac {\frac {b \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{3}+\frac {4 a \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{3}-a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 64, normalized size = 0.82 \begin {gather*} -\frac {3 \, a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right ) - {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} a}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.87, size = 175, normalized size = 2.24 \begin {gather*} \left [\frac {3 \, a^{\frac {3}{2}} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, {\left (b \cos \left (f x + e\right )^{2} - 4 \, a - b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{6 \, f}, \frac {3 \, \sqrt {-a} a \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - {\left (b \cos \left (f x + e\right )^{2} - 4 \, a - b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{3 \, f}\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 \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \cot {\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 71, normalized size = 0.91 \begin {gather*} \frac {\frac {3 \, a^{2} \arctan \left (\frac {\sqrt {b \sin \left (f x + e\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} + 3 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} a}{3 \, f} \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 \mathrm {cot}\left (e+f\,x\right )\,{\left (b\,{\sin \left (e+f\,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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