Optimal. Leaf size=203 \[ \frac {2 c^4 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} f}+\frac {12 \sqrt {2} c^4 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} f}-\frac {14 c^4 \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)}}+\frac {8 c^4 \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}-\frac {a c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^4(e+f x)}{f (a+a \sec (e+f x))^{5/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 203, 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,
481, 596, 536, 209} \begin {gather*} \frac {2 c^4 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{a^{3/2} f}+\frac {12 \sqrt {2} c^4 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{a^{3/2} f}+\frac {8 c^4 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}-\frac {14 c^4 \tan (e+f x)}{a f \sqrt {a \sec (e+f x)+a}}-\frac {a c^4 \sin (e+f x) \tan ^4(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{f (a \sec (e+f x)+a)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 481
Rule 536
Rule 596
Rule 3972
Rule 3989
Rubi steps
\begin {align*} \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{3/2}} \, dx &=\left (a^4 c^4\right ) \int \frac {\tan ^8(e+f x)}{(a+a \sec (e+f x))^{11/2}} \, dx\\ &=-\frac {\left (2 a^3 c^4\right ) \text {Subst}\left (\int \frac {x^8}{\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 {a c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^4(e+f x)}{f (a+a \sec (e+f x))^{5/2}}-\frac {\left (a c^4\right ) \text {Subst}\left (\int \frac {x^4 \left (10+8 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 )}{f}\\ &=\frac {8 c^4 \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}-\frac {a c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^4(e+f x)}{f (a+a \sec (e+f x))^{5/2}}+\frac {c^4 \text {Subst}\left (\int \frac {x^2 \left (48 a+42 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 )}{3 a f}\\ &=-\frac {14 c^4 \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)}}+\frac {8 c^4 \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}-\frac {a c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^4(e+f x)}{f (a+a \sec (e+f x))^{5/2}}-\frac {c^4 \text {Subst}\left (\int \frac {84 a^2+78 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 )}{3 a^3 f}\\ &=-\frac {14 c^4 \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)}}+\frac {8 c^4 \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}-\frac {a c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^4(e+f x)}{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 )}{a f}-\frac {\left (24 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 )}{a f}\\ &=\frac {2 c^4 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} f}+\frac {12 \sqrt {2} c^4 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} f}-\frac {14 c^4 \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)}}+\frac {8 c^4 \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}-\frac {a c^4 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^4(e+f x)}{f (a+a \sec (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.56, size = 196, normalized size = 0.97 \begin {gather*} \frac {c^4 \csc \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \left (-22+20 \cos (e+f x)-26 \cos (2 (e+f x))+28 \cos (3 (e+f x))+6 \text {ArcTan}\left (\sqrt {-1+\sec (e+f x)}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {3}{2} (e+f x)\right )\right )^2 \sqrt {-1+\sec (e+f x)}+36 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {-1+\sec (e+f x)}}{\sqrt {2}}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {3}{2} (e+f x)\right )\right )^2 \sqrt {-1+\sec (e+f x)}\right ) \sec ^2(e+f x)}{12 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. \(551\) vs.
\(2(179)=358\).
time = 0.24, size = 552, normalized size = 2.72
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 (-1+\cos \left (f x +e \right )\right ) \left (3 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {3}{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 {2}+6 \cos \left (f x +e \right ) \sin \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {3}{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 {2}-36 \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {3}{2}} \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 )+3 \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 {3}{2}} \sin \left (f x +e \right )-72 \sin \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {3}{2}} \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 )-36 \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 {3}{2}} \sin \left (f x +e \right )+112 \left (\cos ^{3}\left (f x +e \right )\right )-52 \left (\cos ^{2}\left (f x +e \right )\right )-64 \cos \left (f x +e \right )+4\right )}{6 f \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a^{2}}\) | \(552\) |
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 = 3.66, size = 685, normalized size = 3.37 \begin {gather*} \left [\frac {18 \, \sqrt {2} {\left (a c^{4} \cos \left (f x + e\right )^{3} + 2 \, a c^{4} \cos \left (f x + e\right )^{2} + a c^{4} \cos \left (f x + e\right )\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 ) - 3 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + 2 \, c^{4} \cos \left (f x + e\right )^{2} + c^{4} \cos \left (f x + e\right )\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 (28 \, c^{4} \cos \left (f x + e\right )^{2} + 15 \, 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 )}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{3} + 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f \cos \left (f x + e\right )\right )}}, -\frac {2 \, {\left (3 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + 2 \, c^{4} \cos \left (f x + e\right )^{2} + c^{4} \cos \left (f x + e\right )\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 (28 \, c^{4} \cos \left (f x + e\right )^{2} + 15 \, 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 {18 \, \sqrt {2} {\left (a c^{4} \cos \left (f x + e\right )^{3} + 2 \, a c^{4} \cos \left (f x + e\right )^{2} + a c^{4} \cos \left (f x + e\right )\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 )}}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{3} + 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f \cos \left (f x + e\right )\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 )}}{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 + \int \frac {6 \sec ^{2}{\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 {4 \sec ^{3}{\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}}\right )\, dx + \int \frac {\sec ^{4}{\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 \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}}\, 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 = -0.00 \begin {gather*} \int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^4}{{\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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