Optimal. Leaf size=161 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} c^2 f}-\frac {\text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{2 \sqrt {2} \sqrt {a} c^2 f}+\frac {3 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{2 a c^2 f}-\frac {\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a^2 c^2 f} \]
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Rubi [A]
time = 0.16, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3989, 3972,
491, 597, 536, 209} \begin {gather*} -\frac {\cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a^2 c^2 f}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a} c^2 f}-\frac {\text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{2 \sqrt {2} \sqrt {a} c^2 f}+\frac {3 \cot (e+f x) \sqrt {a \sec (e+f x)+a}}{2 a c^2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 491
Rule 536
Rule 597
Rule 3972
Rule 3989
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^2} \, dx &=\frac {\int \cot ^4(e+f x) (a+a \sec (e+f x))^{3/2} \, dx}{a^2 c^2}\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^2 c^2 f}\\ &=-\frac {\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a^2 c^2 f}-\frac {\text {Subst}\left (\int \frac {-9 a-3 a^2 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{3 a^2 c^2 f}\\ &=\frac {3 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{2 a c^2 f}-\frac {\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a^2 c^2 f}+\frac {\text {Subst}\left (\int \frac {-21 a^2-9 a^3 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{6 a^2 c^2 f}\\ &=\frac {3 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{2 a c^2 f}-\frac {\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a^2 c^2 f}+\frac {\text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{2 c^2 f}-\frac {2 \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^2 f}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} c^2 f}-\frac {\tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{2 \sqrt {2} \sqrt {a} c^2 f}+\frac {3 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{2 a c^2 f}-\frac {\cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a^2 c^2 f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 24.25, size = 5576, normalized size = 34.63 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(376\) vs.
\(2(136)=272\).
time = 0.25, size = 377, normalized size = 2.34
method | result | size |
default | \(\frac {\sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (12 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {2}+3 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-12 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )-3 \sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right )-22 \left (\cos ^{3}\left (f x +e \right )\right )-4 \left (\cos ^{2}\left (f x +e \right )\right )+18 \cos \left (f x +e \right )\right )}{12 c^{2} f \sin \left (f x +e \right )^{3} a}\) | \(377\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.86, size = 566, normalized size = 3.52 \begin {gather*} \left [-\frac {3 \, \sqrt {2} \sqrt {-a} {\left (\cos \left (f x + e\right ) - 1\right )} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 3 \, a \cos \left (f x + e\right )^{2} - 2 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 12 \, \sqrt {-a} {\left (\cos \left (f x + e\right ) - 1\right )} \log \left (-\frac {8 \, a \cos \left (f x + e\right )^{3} + 4 \, {\left (2 \, \cos \left (f x + e\right )^{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 ) - 7 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) - 4 \, {\left (11 \, \cos \left (f x + e\right )^{2} - 9 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{24 \, {\left (a c^{2} f \cos \left (f x + e\right ) - a c^{2} f\right )} \sin \left (f x + e\right )}, \frac {3 \, \sqrt {2} \sqrt {a} {\left (\cos \left (f x + e\right ) - 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 12 \, \sqrt {a} {\left (\cos \left (f x + e\right ) - 1\right )} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - a}\right ) \sin \left (f x + e\right ) + 2 \, {\left (11 \, \cos \left (f x + e\right )^{2} - 9 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{12 \, {\left (a c^{2} f \cos \left (f x + e\right ) - a c^{2} f\right )} \sin \left (f x + e\right )}\right ] \end {gather*}
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 \begin {gather*} \frac {\int \frac {1}{\sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} - 2 \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx}{c^{2}} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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