Optimal. Leaf size=179 \[ -\frac {2 g^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {a} c f}+\frac {g^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {2} \sqrt {a} c f}+\frac {g^2 \cot (e+f x) \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}{a c f} \]
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Rubi [A] time = 0.36, antiderivative size = 242, normalized size of antiderivative = 1.35, number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3964, 98, 157, 63, 217, 203, 93, 205} \[ \frac {2 g^{5/2} \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {g \sec (e+f x)}}{\sqrt {g} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c} f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {g^{5/2} \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {g \sec (e+f x)}}{\sqrt {g} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {2} \sqrt {c} f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {g^2 \tan (e+f x) \sqrt {g \sec (e+f x)}}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 98
Rule 157
Rule 203
Rule 205
Rule 217
Rule 3964
Rubi steps
\begin {align*} \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx &=-\frac {(a c g \tan (e+f x)) \operatorname {Subst}\left (\int \frac {(g x)^{3/2}}{(a+a x) (c-c x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {g^2 \sqrt {g \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {(g \tan (e+f x)) \operatorname {Subst}\left (\int \frac {\frac {1}{2} a c g^2+a c g^2 x}{\sqrt {g x} (a+a x) \sqrt {c-c x}} \, dx,x,\sec (e+f x)\right )}{c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {g^2 \sqrt {g \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {\left (g^3 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {g x} \sqrt {c-c x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\left (a g^3 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {g x} (a+a x) \sqrt {c-c x}} \, dx,x,\sec (e+f x)\right )}{2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {g^2 \sqrt {g \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {\left (2 g^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {c x^2}{g}}} \, dx,x,\sqrt {g \sec (e+f x)}\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\left (a g^3 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{a g+2 a c x^2} \, dx,x,\frac {\sqrt {g \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {g^2 \sqrt {g \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}-\frac {g^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {g \sec (e+f x)}}{\sqrt {g} \sqrt {c-c \sec (e+f x)}}\right ) \tan (e+f x)}{\sqrt {2} \sqrt {c} f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {\left (2 g^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {c x^2}{g}} \, dx,x,\frac {\sqrt {g \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {g^2 \sqrt {g \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {2 g^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {g \sec (e+f x)}}{\sqrt {g} \sqrt {c-c \sec (e+f x)}}\right ) \tan (e+f x)}{\sqrt {c} f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {g^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {g \sec (e+f x)}}{\sqrt {g} \sqrt {c-c \sec (e+f x)}}\right ) \tan (e+f x)}{\sqrt {2} \sqrt {c} f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 2.29, size = 328, normalized size = 1.83 \[ -\frac {\sin ^3(e+f x) \sqrt {\sec (e+f x)+1} (g \sec (e+f x))^{5/2} \left (8 \sqrt {\sec (e+f x)} \sqrt {\sec (e+f x)+1}+\sqrt {\tan ^2(e+f x)} \left (16 \log (\sec (e+f x)+1)-16 \log \left (\sec ^{\frac {3}{2}}(e+f x)+\sqrt {\sec (e+f x)}+\sqrt {\tan ^2(e+f x)} \sqrt {\sec (e+f x)+1}\right )+\sqrt {2} \left (\log \left (-3 \sec ^2(e+f x)-2 \sec (e+f x)-2 \sqrt {2} \sqrt {\tan ^2(e+f x)} \sqrt {\sec (e+f x)+1} \sqrt {\sec (e+f x)}+1\right )-\log \left (-3 \sec ^2(e+f x)-2 \sec (e+f x)+2 \sqrt {2} \sqrt {\tan ^2(e+f x)} \sqrt {\sec (e+f x)+1} \sqrt {\sec (e+f x)}+1\right )\right )\right )\right )}{8 c f (\cos (e+f x)-1) (\cos (e+f x)+1)^2 (\sec (e+f x)-1) \sec ^{\frac {5}{2}}(e+f x) \sqrt {a (\sec (e+f x)+1)}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.60, size = 569, normalized size = 3.18 \[ \left [\frac {\sqrt {2} a g^{2} \sqrt {\frac {g}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {g}{a}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - g \cos \left (f x + e\right )^{2} + 2 \, g \cos \left (f x + e\right ) + 3 \, g}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 2 \, a g^{2} \sqrt {\frac {g}{a}} \log \left (\frac {g \cos \left (f x + e\right )^{3} + 4 \, {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {g}{a}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, g \cos \left (f x + e\right )^{2} + 8 \, g}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2}}\right ) \sin \left (f x + e\right ) + 4 \, g^{2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{4 \, a c f \sin \left (f x + e\right )}, -\frac {\sqrt {2} a g^{2} \sqrt {-\frac {g}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {-\frac {g}{a}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{g \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \, a g^{2} \sqrt {-\frac {g}{a}} \arctan \left (\frac {2 \, \sqrt {-\frac {g}{a}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{g \cos \left (f x + e\right )^{2} - g \cos \left (f x + e\right ) - 2 \, g}\right ) \sin \left (f x + e\right ) - 2 \, g^{2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{2 \, a c f \sin \left (f x + e\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\left (g \sec \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {a \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) - c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.95, size = 294, normalized size = 1.64 \[ \frac {\left (\cos \left (f x +e \right ) \sqrt {2}\, \arcsinh \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-\sqrt {2}\, \arcsinh \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+2 \sin \left (f x +e \right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}+2 \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1+\sin \left (f x +e \right )\right )}{2}\right ) \cos \left (f x +e \right )+2 \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \left (-\cos \left (f x +e \right )-1+\sin \left (f x +e \right )\right )}{2}\right ) \cos \left (f x +e \right )-2 \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1+\sin \left (f x +e \right )\right )}{2}\right )-2 \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \left (-\cos \left (f x +e \right )-1+\sin \left (f x +e \right )\right )}{2}\right )\right ) \left (\frac {g}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \left (-1+\cos \left (f x +e \right )\right )^{2} \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}}{2 c f \left (\frac {1}{1+\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \sin \left (f x +e \right )^{6} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.22, size = 1400, normalized size = 7.82 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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