Optimal. Leaf size=168 \[ \frac {10 \sqrt {2} a^3 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{c^{3/2} f}-\frac {10 a^3 \tan (e+f x)}{c f \sqrt {c-c \sec (e+f x)}}-\frac {5 \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{3 c f \sqrt {c-c \sec (e+f x)}}-\frac {a \tan (e+f x) (a \sec (e+f x)+a)^2}{f (c-c \sec (e+f x))^{3/2}} \]
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Rubi [A] time = 0.34, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3957, 3956, 3795, 203} \[ \frac {10 \sqrt {2} a^3 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{c^{3/2} f}-\frac {10 a^3 \tan (e+f x)}{c f \sqrt {c-c \sec (e+f x)}}-\frac {5 \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{3 c f \sqrt {c-c \sec (e+f x)}}-\frac {a \tan (e+f x) (a \sec (e+f x)+a)^2}{f (c-c \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3795
Rule 3956
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^{3/2}} \, dx &=-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {(5 a) \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt {c-c \sec (e+f x)}} \, dx}{2 c}\\ &=-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f \sqrt {c-c \sec (e+f x)}}-\frac {\left (5 a^2\right ) \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{\sqrt {c-c \sec (e+f x)}} \, dx}{c}\\ &=-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {10 a^3 \tan (e+f x)}{c f \sqrt {c-c \sec (e+f x)}}-\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f \sqrt {c-c \sec (e+f x)}}-\frac {\left (10 a^3\right ) \int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}} \, dx}{c}\\ &=-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {10 a^3 \tan (e+f x)}{c f \sqrt {c-c \sec (e+f x)}}-\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f \sqrt {c-c \sec (e+f x)}}+\frac {\left (20 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{2 c+x^2} \, dx,x,\frac {c \tan (e+f x)}{\sqrt {c-c \sec (e+f x)}}\right )}{c f}\\ &=\frac {10 \sqrt {2} a^3 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{c^{3/2} f}-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac {10 a^3 \tan (e+f x)}{c f \sqrt {c-c \sec (e+f x)}}-\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 3.10, size = 324, normalized size = 1.93 \[ -\frac {a^3 \csc \left (\frac {e}{2}\right ) e^{-\frac {1}{2} i (e+f x)} \tan ^3\left (\frac {1}{2} (e+f x)\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (\sec (e+f x)+1)^3 \left (-\frac {i \left (-1+e^{i e}\right ) e^{\frac {i f x}{2}} \left (-24 e^{i (e+f x)}+34 e^{2 i (e+f x)}-24 e^{3 i (e+f x)}+19 e^{4 i (e+f x)}+19\right ) \sqrt {\sec (e+f x)}}{2 \left (-1+e^{i (e+f x)}\right )^2 \left (1+e^{2 i (e+f x)}\right )}-15 \sin \left (\frac {e}{2}\right ) \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt {1+e^{2 i (e+f x)}} \sec \left (\frac {1}{2} (e+f x)\right ) \tanh ^{-1}\left (\frac {1+e^{i (e+f x)}}{\sqrt {2} \sqrt {1+e^{2 i (e+f x)}}}\right )\right )}{3 c f (\sec (e+f x)-1) \sec ^{\frac {3}{2}}(e+f x) \sqrt {c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 432, normalized size = 2.57 \[ \left [\frac {15 \, \sqrt {2} {\left (a^{3} c \cos \left (f x + e\right )^{2} - a^{3} c \cos \left (f x + e\right )\right )} \sqrt {-\frac {1}{c}} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{c}} + {\left (3 \, \cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \, {\left (19 \, a^{3} \cos \left (f x + e\right )^{3} + 7 \, a^{3} \cos \left (f x + e\right )^{2} - 13 \, a^{3} \cos \left (f x + e\right ) - a^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3 \, {\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}, -\frac {2 \, {\left (\frac {15 \, \sqrt {2} {\left (a^{3} c \cos \left (f x + e\right )^{2} - a^{3} c \cos \left (f x + e\right )\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right )}{\sqrt {c}} - {\left (19 \, a^{3} \cos \left (f x + e\right )^{3} + 7 \, a^{3} \cos \left (f x + e\right )^{2} - 13 \, a^{3} \cos \left (f x + e\right ) - a^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}\right )}}{3 \, {\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}\right ] \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.73, size = 157, normalized size = 0.93 \[ -\frac {a^{3} \left (15 \left (\cos ^{2}\left (f x +e \right )\right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}-15 \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}+38 \left (\cos ^{2}\left (f x +e \right )\right )-24 \cos \left (f x +e \right )-2\right ) \sin \left (f x +e \right )}{3 f \cos \left (f x +e \right )^{3} \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{3} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^3}{\cos \left (e+f\,x\right )\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ a^{3} \left (\int \frac {\sec {\left (e + f x \right )}}{- c \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{- c \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {3 \sec ^{3}{\left (e + f x \right )}}{- c \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{- c \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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