Optimal. Leaf size=126 \[ \frac {(8 a+3 b) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac {\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f} \]
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Rubi [A] time = 0.13, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3194, 89, 78, 63, 208} \[ -\frac {\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}+\frac {(8 a+3 b) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 89
Rule 208
Rule 3194
Rubi steps
\begin {align*} \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(1-x)^2}{x^3 \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (-8 a-3 b)+2 a x}{x^2 \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{4 a f}\\ &=\frac {(8 a+3 b) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{16 a^2 f}\\ &=\frac {(8 a+3 b) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{8 a^2 b f}\\ &=-\frac {\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}+\frac {(8 a+3 b) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 101, normalized size = 0.80 \[ \frac {\sqrt {a} \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)} \left (-2 a \csc ^2(e+f x)+8 a+3 b\right )-\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 388, normalized size = 3.08 \[ \left [\frac {{\left ({\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, {\left ({\left (8 \, a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{2} - 6 \, a^{2} - 3 \, a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{16 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f\right )}}, \frac {{\left ({\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - {\left ({\left (8 \, a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{2} - 6 \, a^{2} - 3 \, a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{8 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.03, size = 898, normalized size = 7.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.97, size = 219, normalized size = 1.74 \[ -\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{\sqrt {a}\, f}+\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{f a \sin \left (f x +e \right )^{2}}-\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f \,a^{\frac {3}{2}}}-\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{4 f a \sin \left (f x +e \right )^{4}}+\frac {3 b \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{8 f \,a^{2} \sin \left (f x +e \right )^{2}}-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{8 f \,a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 158, normalized size = 1.25 \[ -\frac {\frac {8 \, \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{\sqrt {a}} + \frac {8 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {3}{2}}} + \frac {3 \, b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {5}{2}}} - \frac {8 \, \sqrt {b \sin \left (f x + e\right )^{2} + a}}{a \sin \left (f x + e\right )^{2}} - \frac {3 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} b}{a^{2} \sin \left (f x + e\right )^{2}} + \frac {2 \, \sqrt {b \sin \left (f x + e\right )^{2} + a}}{a \sin \left (f x + e\right )^{4}}}{8 \, f} \]
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 \[ \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^5}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \]
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 \[ \int \frac {\cot ^{5}{\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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