Optimal. Leaf size=113 \[ \frac {2 c \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} f}-\frac {3 c \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {2} a^{3/2} f}-\frac {c \tan (e+f x)}{f (a+a \sec (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 130, normalized size of antiderivative = 1.15, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3989, 3972,
482, 536, 209} \begin {gather*} \frac {2 c \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{a^{3/2} f}-\frac {3 c \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {2} a^{3/2} f}-\frac {c \sin (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{2 a f \sqrt {a \sec (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 482
Rule 536
Rule 3972
Rule 3989
Rubi steps
\begin {align*} \int \frac {c-c \sec (e+f x)}{(a+a \sec (e+f x))^{3/2}} \, dx &=-\left ((a c) \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx\right )\\ &=\frac {(2 c) \text {Subst}\left (\int \frac {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 )}{f}\\ &=-\frac {c \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{2 a f \sqrt {a+a \sec (e+f x)}}-\frac {c \text {Subst}\left (\int \frac {1-a 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 )}{a f}\\ &=-\frac {c \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{2 a f \sqrt {a+a \sec (e+f x)}}-\frac {(2 c) \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 f}+\frac {(3 c) \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 )}{a f}\\ &=\frac {2 c \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} f}-\frac {3 c \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {2} a^{3/2} f}-\frac {c \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{2 a f \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.06, size = 130, normalized size = 1.15 \begin {gather*} \frac {c \cot \left (\frac {1}{2} (e+f x)\right ) \left (\sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (-1+\cos (e+f x)+2 \text {ArcTan}\left (\sqrt {-1+\sec (e+f x)}\right ) (1+\cos (e+f x)) \sqrt {-1+\sec (e+f x)}\right )-3 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {-1+\sec (e+f x)}}{\sqrt {2}}\right ) \sqrt {-1+\sec (e+f x)}\right )}{2 a f \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. \(370\) vs.
\(2(96)=192\).
time = 0.16, size = 371, normalized size = 3.28
method | result | size |
default | \(-\frac {c \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (2 \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}+3 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right )+2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )+3 \sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right )-2 \left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )\right )}{2 f \left (\cos \left (f x +e \right )+1\right ) \sin \left (f x +e \right ) a^{2}}\) | \(371\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 228 vs.
\(2 (102) = 204\).
time = 4.32, size = 548, normalized size = 4.85 \begin {gather*} \left [-\frac {4 \, c \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 \, \sqrt {2} {\left (c \cos \left (f x + e\right )^{2} + 2 \, c \cos \left (f x + e\right ) + c\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 ) + 4 \, {\left (c \cos \left (f x + e\right )^{2} + 2 \, c \cos \left (f x + e\right ) + c\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} + 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 ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right )}{4 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )}}, -\frac {2 \, c \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 \, \sqrt {2} {\left (c \cos \left (f x + e\right )^{2} + 2 \, c \cos \left (f x + e\right ) + c\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 ) + 4 \, {\left (c \cos \left (f x + e\right )^{2} + 2 \, c \cos \left (f x + e\right ) + c\right )} \sqrt {a} \arctan \left (\frac {\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 )}{2 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\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*} - c \left (\int \frac {\sec {\left (e + f x \right )}}{a \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \left (- \frac {1}{a \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a \sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx\right ) \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 {c-\frac {c}{\cos \left (e+f\,x\right )}}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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