Optimal. Leaf size=167 \[ \frac {(8 a+5 b) \csc ^2(e+f x)}{8 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (8 a^2+24 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{7/2} f}+\frac {8 a^2+24 a b+15 b^2}{8 a^3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\csc ^4(e+f x)}{4 a f \sqrt {a+b \sin ^2(e+f x)}} \]
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Rubi [A] time = 0.16, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3194, 89, 78, 51, 63, 208} \[ \frac {8 a^2+24 a b+15 b^2}{8 a^3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (8 a^2+24 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{7/2} f}+\frac {(8 a+5 b) \csc ^2(e+f x)}{8 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\csc ^4(e+f x)}{4 a f \sqrt {a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 89
Rule 208
Rule 3194
Rubi steps
\begin {align*} \int \frac {\cot ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(1-x)^2}{x^3 (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=-\frac {\csc ^4(e+f x)}{4 a f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (-8 a-5 b)+2 a x}{x^2 (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{4 a f}\\ &=\frac {(8 a+5 b) \csc ^2(e+f x)}{8 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\csc ^4(e+f x)}{4 a f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (8 a^2+24 a b+15 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{16 a^2 f}\\ &=\frac {8 a^2+24 a b+15 b^2}{8 a^3 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(8 a+5 b) \csc ^2(e+f x)}{8 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\csc ^4(e+f x)}{4 a f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (8 a^2+24 a b+15 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{16 a^3 f}\\ &=\frac {8 a^2+24 a b+15 b^2}{8 a^3 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(8 a+5 b) \csc ^2(e+f x)}{8 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\csc ^4(e+f x)}{4 a f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (8 a^2+24 a b+15 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^3 b f}\\ &=-\frac {\left (8 a^2+24 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{7/2} f}+\frac {8 a^2+24 a b+15 b^2}{8 a^3 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(8 a+5 b) \csc ^2(e+f x)}{8 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\csc ^4(e+f x)}{4 a f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.33, size = 94, normalized size = 0.56 \[ \frac {\left (8 a^2+24 a b+15 b^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b \sin ^2(e+f x)}{a}+1\right )+a \csc ^2(e+f x) \left (-2 a \csc ^2(e+f x)+8 a+5 b\right )}{8 a^3 f \sqrt {a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 652, normalized size = 3.90 \[ \left [\frac {{\left ({\left (8 \, a^{2} b + 24 \, a b^{2} + 15 \, b^{3}\right )} \cos \left (f x + e\right )^{6} - {\left (8 \, a^{3} + 48 \, a^{2} b + 87 \, a b^{2} + 45 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - 8 \, a^{3} - 32 \, a^{2} b - 39 \, a b^{2} - 15 \, b^{3} + {\left (16 \, a^{3} + 72 \, a^{2} b + 102 \, a b^{2} + 45 \, b^{3}\right )} \cos \left (f x + e\right )^{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^{3} + 24 \, a^{2} b + 15 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + 14 \, a^{3} + 29 \, a^{2} b + 15 \, a b^{2} - {\left (24 \, a^{3} + 53 \, a^{2} b + 30 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{16 \, {\left (a^{4} b f \cos \left (f x + e\right )^{6} - {\left (a^{5} + 3 \, a^{4} b\right )} f \cos \left (f x + e\right )^{4} + {\left (2 \, a^{5} + 3 \, a^{4} b\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{5} + a^{4} b\right )} f\right )}}, \frac {{\left ({\left (8 \, a^{2} b + 24 \, a b^{2} + 15 \, b^{3}\right )} \cos \left (f x + e\right )^{6} - {\left (8 \, a^{3} + 48 \, a^{2} b + 87 \, a b^{2} + 45 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - 8 \, a^{3} - 32 \, a^{2} b - 39 \, a b^{2} - 15 \, b^{3} + {\left (16 \, a^{3} + 72 \, a^{2} b + 102 \, a b^{2} + 45 \, b^{3}\right )} \cos \left (f x + e\right )^{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^{3} + 24 \, a^{2} b + 15 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + 14 \, a^{3} + 29 \, a^{2} b + 15 \, a b^{2} - {\left (24 \, a^{3} + 53 \, a^{2} b + 30 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{8 \, {\left (a^{4} b f \cos \left (f x + e\right )^{6} - {\left (a^{5} + 3 \, a^{4} b\right )} f \cos \left (f x + e\right )^{4} + {\left (2 \, a^{5} + 3 \, a^{4} b\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{5} + a^{4} b\right )} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.45, size = 1150, normalized size = 6.89 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.01, size = 288, normalized size = 1.72 \[ \frac {1}{a f \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f \,a^{\frac {3}{2}}}+\frac {1}{f a \sin \left (f x +e \right )^{2} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}+\frac {3 b}{f \,a^{2} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}-\frac {3 b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f \,a^{\frac {5}{2}}}-\frac {1}{4 f a \sin \left (f x +e \right )^{4} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}+\frac {5 b}{8 f \,a^{2} \sin \left (f x +e \right )^{2} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}+\frac {15 b^{2}}{8 f \,a^{3} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}-\frac {15 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{8 f \,a^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 219, normalized size = 1.31 \[ -\frac {\frac {8 \, \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {3}{2}}} + \frac {24 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {5}{2}}} + \frac {15 \, b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {7}{2}}} - \frac {8}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a} - \frac {24 \, b}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{2}} - \frac {15 \, b^{2}}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{3}} - \frac {8}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a \sin \left (f x + e\right )^{2}} - \frac {5 \, b}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{2} \sin \left (f x + e\right )^{2}} + \frac {2}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a \sin \left (f x + e\right )^{4}}}{8 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.01 \[ \text {Hanged} \]
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 \[ \int \frac {\cot ^{5}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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