Optimal. Leaf size=140 \[ \frac {\csc (e+f x) \sqrt {a \sec (e+f x)+a}}{a c f \sqrt {\sec (e+f x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x) \sqrt {\sec (e+f x)}}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {2} \sqrt {a} c f}-\frac {2 \sinh ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a} c f} \]
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Rubi [A] time = 0.28, antiderivative size = 213, normalized size of antiderivative = 1.52, number of steps used = 8, number of rules used = 8, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3964, 98, 157, 63, 217, 203, 93, 205} \[ -\frac {\sin (e+f x) \sec ^{\frac {3}{2}}(e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))}+\frac {2 \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {c} f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {\tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {\sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {2} \sqrt {c} f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}} \]
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 {\sec ^{\frac {5}{2}}(e+f x)}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx &=-\frac {(a c \tan (e+f x)) \operatorname {Subst}\left (\int \frac {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 {\sec ^{\frac {3}{2}}(e+f x) \sin (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {\frac {a c}{2}+a c x}{\sqrt {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 {\sec ^{\frac {3}{2}}(e+f x) \sin (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {1}{\sqrt {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 {(a \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {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 {\sec ^{\frac {3}{2}}(e+f x) \sin (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {(2 \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-c x^2}} \, dx,x,\sqrt {\sec (e+f x)}\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {(a \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{a+2 a c x^2} \, dx,x,\frac {\sqrt {\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 {\sec ^{\frac {3}{2}}(e+f x) \sin (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {\sec (e+f x)}}{\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 {(2 \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{1+c x^2} \, dx,x,\frac {\sqrt {\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 {\sec ^{\frac {3}{2}}(e+f x) \sin (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sec (e+f x)}}{\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 {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {\sec (e+f x)}}{\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 [B] time = 6.46, size = 724, normalized size = 5.17 \[ \frac {\sin (e+f x) \sin ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \cos (e+f x) (\sec (e+f x)+1)^{3/2} \sqrt {\sec ^2(e+f x)-1} \left (\log \left (-3 \sec ^2(e+f x)-2 \sqrt {2} \sqrt {\sec (e+f x)+1} \sqrt {\sec ^2(e+f x)-1} \sqrt {\sec (e+f x)}-2 \sec (e+f x)+1\right )-\log \left (-3 \sec ^2(e+f x)+2 \sqrt {2} \sqrt {\sec (e+f x)+1} \sqrt {\sec ^2(e+f x)-1} \sqrt {\sec (e+f x)}-2 \sec (e+f x)+1\right )\right )}{2 f (\cos (e+f x)+1) \sqrt {2-2 \cos ^2(e+f x)} \sqrt {1-\cos ^2(e+f x)} \sqrt {a (\sec (e+f x)+1)} (c-c \sec (e+f x))}+\frac {\sin (e+f x) \sin ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \cos (e+f x) (\sec (e+f x)+1)^{3/2} \sqrt {\sec ^2(e+f x)-1} \left (\sqrt {2} \left (\log \left (-3 \sec ^2(e+f x)+2 \sqrt {2} \sqrt {\sec (e+f x)+1} \sqrt {\sec ^2(e+f x)-1} \sqrt {\sec (e+f x)}-2 \sec (e+f x)+1\right )-\log \left (-3 \sec ^2(e+f x)-2 \sqrt {2} \sqrt {\sec (e+f x)+1} \sqrt {\sec ^2(e+f x)-1} \sqrt {\sec (e+f x)}-2 \sec (e+f x)+1\right )\right )+8 \log \left (\sec ^{\frac {3}{2}}(e+f x)+\sqrt {\sec (e+f x)+1} \sqrt {\sec ^2(e+f x)-1}+\sqrt {\sec (e+f x)}\right )-8 \log (\sec (e+f x)+1)\right )}{2 f (\cos (e+f x)+1) \left (1-\cos ^2(e+f x)\right ) \sqrt {a (\sec (e+f x)+1)} (c-c \sec (e+f x))}+\frac {\sin ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^{\frac {3}{2}}(e+f x) \sqrt {\sec (e+f x)+1} \sqrt {(\cos (e+f x)+1) \sec (e+f x)} \left (\frac {\csc \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {\sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \sec \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {2 \cot (e)}{f}\right )}{\sqrt {a (\sec (e+f x)+1)} (c-c \sec (e+f x))} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 462, normalized size = 3.30 \[ \left [\frac {\sqrt {2} \sqrt {a} \log \left (-\frac {\cos \left (f x + e\right )^{2} - \frac {2 \, \sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\cos \left (f x + e\right )} \sin \left (f x + e\right )}{\sqrt {a}} - 2 \, \cos \left (f x + e\right ) - 3}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 2 \, \sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} + \frac {4 \, {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{\sqrt {\cos \left (f x + e\right )}} + 8 \, a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2}}\right ) \sin \left (f x + e\right ) + 4 \, \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\cos \left (f x + e\right )}}{4 \, a c f \sin \left (f x + e\right )}, -\frac {\sqrt {2} a \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a}} \sqrt {\cos \left (f x + e\right )}}{\sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \, \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\cos \left (f x + e\right )} \sin \left (f x + e\right )}{a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) - 2 \, a}\right ) \sin \left (f x + e\right ) - 2 \, \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\cos \left (f x + e\right )}}{2 \, a c f \sin \left (f x + e\right )}\right ] \]
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 \[ \int -\frac {\sec \left (f x + e\right )^{\frac {5}{2}}}{\sqrt {a \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) - c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.88, size = 317, normalized size = 2.26 \[ \frac {\left (\frac {1}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1+\sin \left (f x +e \right )\right ) \sqrt {2}}{4}\right ) \cos \left (f x +e \right )-\sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1-\sin \left (f x +e \right )\right ) \sqrt {2}}{4}\right ) \cos \left (f x +e \right )-\sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1+\sin \left (f x +e \right )\right ) \sqrt {2}}{4}\right )+\sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1-\sin \left (f x +e \right )\right ) \sqrt {2}}{4}\right )-\arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2}{1+\cos \left (f x +e \right )}}}{2}\right ) \cos \left (f x +e \right )+\sin \left (f x +e \right ) \sqrt {-\frac {2}{1+\cos \left (f x +e \right )}}+\arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2}{1+\cos \left (f x +e \right )}}}{2}\right )\right )}{c f \sqrt {-\frac {2}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.06, size = 1310, normalized size = 9.36 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]
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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