Optimal. Leaf size=98 \[ \frac {c^2 \tan (e+f x) \log (\cos (e+f x)+1)}{a f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {2 c^2 \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2} \sqrt {c-c \sec (e+f x)}} \]
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Rubi [A] time = 0.20, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3908, 3911, 31} \[ \frac {c^2 \tan (e+f x) \log (\cos (e+f x)+1)}{a f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {2 c^2 \tan (e+f x)}{f (a \sec (e+f x)+a)^{3/2} \sqrt {c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3908
Rule 3911
Rubi steps
\begin {align*} \int \frac {(c-c \sec (e+f x))^{3/2}}{(a+a \sec (e+f x))^{3/2}} \, dx &=-\frac {2 c^2 \tan (e+f x)}{f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}+\frac {c \int \frac {\sqrt {c-c \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx}{a}\\ &=-\frac {2 c^2 \tan (e+f x)}{f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}+\frac {\left (c^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{a+a x} \, dx,x,\cos (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {2 c^2 \tan (e+f x)}{f (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)}}+\frac {c^2 \log (1+\cos (e+f x)) \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 1.14, size = 114, normalized size = 1.16 \[ \frac {i c \cot \left (\frac {1}{2} (e+f x)\right ) \sqrt {c-c \sec (e+f x)} \left (2 i \log \left (1+e^{i (e+f x)}\right )+\left (f x+2 i \log \left (1+e^{i (e+f x)}\right )\right ) \cos (e+f x)+f x+2 i\right )}{a f (\cos (e+f x)+1) \sqrt {a (\sec (e+f x)+1)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sec \left (f x + e\right ) + a} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.19, size = 106, normalized size = 1.08 \[ -\frac {\left (\cos \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+\cos \left (f x +e \right )+\ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-1\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}}{f \sin \left (f x +e \right )^{3} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 70, normalized size = 0.71 \[ \frac {\frac {c^{\frac {3}{2}} \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}{\sqrt {-a} a} - \frac {c^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{\sqrt {-a} a {\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{f} \]
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 \[ \int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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