Optimal. Leaf size=230 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} c f}-\frac {71 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{32 \sqrt {2} a^{5/2} c f}-\frac {7 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{32 a^3 c f}+\frac {13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{32 a^3 c f}+\frac {\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{16 a^3 c f} \]
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Rubi [A]
time = 0.21, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3989, 3972,
483, 593, 597, 536, 209} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{a^{5/2} c f}-\frac {71 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{32 \sqrt {2} a^{5/2} c f}-\frac {7 \cot (e+f x) \sqrt {a \sec (e+f x)+a}}{32 a^3 c f}+\frac {\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {a \sec (e+f x)+a}}{16 a^3 c f}+\frac {13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {a \sec (e+f x)+a}}{32 a^3 c f} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 483
Rule 536
Rule 593
Rule 597
Rule 3972
Rule 3989
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))} \, dx &=-\frac {\int \frac {\cot ^2(e+f x)}{(a+a \sec (e+f x))^{3/2}} \, dx}{a c}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^3 c f}\\ &=\frac {\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{16 a^3 c f}+\frac {\text {Subst}\left (\int \frac {3 a-5 a^2 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{4 a^4 c f}\\ &=\frac {13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{32 a^3 c f}+\frac {\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{16 a^3 c f}+\frac {\text {Subst}\left (\int \frac {-7 a^2-39 a^3 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 )}{16 a^5 c f}\\ &=-\frac {7 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{32 a^3 c f}+\frac {13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{32 a^3 c f}+\frac {\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{16 a^3 c f}-\frac {\text {Subst}\left (\int \frac {57 a^3-7 a^4 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 )}{32 a^5 c f}\\ &=-\frac {7 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{32 a^3 c f}+\frac {13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{32 a^3 c f}+\frac {\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{16 a^3 c 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 )}{a^2 c f}+\frac {71 \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 )}{32 a^2 c f}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} c f}-\frac {71 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{32 \sqrt {2} a^{5/2} c f}-\frac {7 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{32 a^3 c f}+\frac {13 \cos (e+f x) \cot (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{32 a^3 c f}+\frac {\cos ^2(e+f x) \cot (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {a+a \sec (e+f x)}}{16 a^3 c f}\\ \end {align*}
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Mathematica [A]
time = 1.51, size = 158, normalized size = 0.69 \begin {gather*} \frac {\left (13+24 \cos (e+f x)+27 \cos (2 (e+f x))+512 \text {ArcTan}\left (\sqrt {-1+\sec (e+f x)}\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {-1+\sec (e+f x)}-284 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {-1+\sec (e+f x)}}{\sqrt {2}}\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {-1+\sec (e+f x)}\right ) \tan ^3\left (\frac {1}{2} (e+f x)\right )}{64 a^2 c f (-1+\cos (e+f x))^2 \sqrt {a (1+\sec (e+f x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs.
\(2(199)=398\).
time = 0.24, size = 545, normalized size = 2.37
method | result | size |
default | \(\frac {\sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (-1+\cos \left (f x +e \right )\right )^{2} \left (-64 \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}-71 \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 )-128 \sin \left (f x +e \right ) \cos \left (f x +e \right ) \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 ) \sqrt {2}-142 \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 ) \cos \left (f x +e \right ) \sin \left (f x +e \right )-64 \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 )+54 \left (\cos ^{3}\left (f x +e \right )\right )-71 \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 )+24 \left (\cos ^{2}\left (f x +e \right )\right )-14 \cos \left (f x +e \right )\right )}{64 c f \sin \left (f x +e \right )^{5} a^{3}}\) | \(545\) |
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 = 3.69, size = 662, normalized size = 2.88 \begin {gather*} \left [-\frac {71 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {-a} \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 ) + 64 \, {\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {-a} \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 (27 \, \cos \left (f x + e\right )^{3} + 12 \, \cos \left (f x + e\right )^{2} - 7 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{128 \, {\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} c f\right )} \sin \left (f x + e\right )}, \frac {71 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {a} \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 ) + 64 \, {\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {a} \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 (27 \, \cos \left (f x + e\right )^{3} + 12 \, \cos \left (f x + e\right )^{2} - 7 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{64 \, {\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} c 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}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{3}{\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} - a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} - a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx}{c} \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 = -0.00 \begin {gather*} \int \frac {1}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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