Optimal. Leaf size=185 \[ \frac {2 c^4 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}-\frac {16 \sqrt {2} c^4 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {14 c^4 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3989, 3972,
490, 596, 536, 209} \begin {gather*} \frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac {2 c^4 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a} f}-\frac {16 \sqrt {2} c^4 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a} f}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a \sec (e+f x)+a)^{3/2}}+\frac {14 c^4 \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 490
Rule 536
Rule 596
Rule 3972
Rule 3989
Rubi steps
\begin {align*} \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx &=\left (a^4 c^4\right ) \int \frac {\tan ^8(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx\\ &=-\frac {\left (2 a^4 c^4\right ) \text {Subst}\left (\int \frac {x^8}{\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 )}{f}\\ &=\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}+\frac {\left (2 a^2 c^4\right ) \text {Subst}\left (\int \frac {x^4 \left (10+15 a x^2\right )}{\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 )}{5 f}\\ &=-\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}-\frac {\left (2 c^4\right ) \text {Subst}\left (\int \frac {x^2 \left (90 a+105 a^2 x^2\right )}{\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 )}{15 f}\\ &=\frac {14 c^4 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}+\frac {\left (2 c^4\right ) \text {Subst}\left (\int \frac {210 a^2+225 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 )}{15 a^2 f}\\ &=\frac {14 c^4 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}-\frac {\left (2 c^4\right ) \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 )}{f}+\frac {\left (32 c^4\right ) \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 )}{f}\\ &=\frac {2 c^4 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}-\frac {16 \sqrt {2} c^4 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {14 c^4 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.59, size = 153, normalized size = 0.83 \begin {gather*} \frac {c^4 \cot \left (\frac {1}{2} (e+f x)\right ) \left (100-155 \cos (e+f x)+96 \cos (2 (e+f x))-41 \cos (3 (e+f x))+20 \text {ArcTan}\left (\sqrt {-1+\sec (e+f x)}\right ) \cos ^3(e+f x) \sqrt {-1+\sec (e+f x)}-160 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {-1+\sec (e+f x)}}{\sqrt {2}}\right ) \cos ^3(e+f x) \sqrt {-1+\sec (e+f x)}\right ) \sec ^3(e+f x)}{10 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. \(543\) vs.
\(2(162)=324\).
time = 0.26, size = 544, normalized size = 2.94
method | result | size |
default | \(-\frac {c^{4} \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (5 \sin \left (f x +e \right ) \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 (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {5}{2}} \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}+10 \sin \left (f x +e \right ) \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 (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {2}+80 \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 (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {5}{2}} \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+5 \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 ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {5}{2}} \sin \left (f x +e \right )+160 \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 (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {5}{2}} \cos \left (f x +e \right ) \sin \left (f x +e \right )+80 \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 (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {5}{2}} \sin \left (f x +e \right )+328 \left (\cos ^{3}\left (f x +e \right )\right )-384 \left (\cos ^{2}\left (f x +e \right )\right )+64 \cos \left (f x +e \right )-8\right )}{20 f \cos \left (f x +e \right )^{2} \sin \left (f x +e \right ) a}\) | \(544\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.98, size = 597, normalized size = 3.23 \begin {gather*} \left [\frac {40 \, \sqrt {2} {\left (a c^{4} \cos \left (f x + e\right )^{3} + a c^{4} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 5 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + c^{4} \cos \left (f x + e\right )^{2}\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 ) + 2 \, {\left (41 \, c^{4} \cos \left (f x + e\right )^{2} - 7 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{5 \, {\left (a f \cos \left (f x + e\right )^{3} + a f \cos \left (f x + e\right )^{2}\right )}}, -\frac {2 \, {\left (5 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + c^{4} \cos \left (f x + e\right )^{2}\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 ) - {\left (41 \, c^{4} \cos \left (f x + e\right )^{2} - 7 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - \frac {40 \, \sqrt {2} {\left (a c^{4} \cos \left (f x + e\right )^{3} + a c^{4} \cos \left (f x + e\right )^{2}\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 )}{\sqrt {a}}\right )}}{5 \, {\left (a f \cos \left (f x + e\right )^{3} + a f \cos \left (f x + e\right )^{2}\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^{4} \left (\int \left (- \frac {4 \sec {\left (e + f x \right )}}{\sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {6 \sec ^{2}{\left (e + f x \right )}}{\sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \left (- \frac {4 \sec ^{3}{\left (e + f x \right )}}{\sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \frac {1}{\sqrt {a \sec {\left (e + f x \right )} + a}}\, 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 {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^4}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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