Optimal. Leaf size=141 \[ \frac {4 a^3 \log (1-\sec (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {2 a^2 \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}+\frac {a (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}} \]
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Rubi [A]
time = 0.27, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {4040, 4037}
\begin {gather*} \frac {4 a^3 \tan (e+f x) \log (1-\sec (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {2 a^2 \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{f \sqrt {c-c \sec (e+f x)}}+\frac {a \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4037
Rule 4040
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx &=\frac {a (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}+(2 a) \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{\sqrt {c-c \sec (e+f x)}} \, dx\\ &=\frac {2 a^2 \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}+\frac {a (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}+\left (4 a^2\right ) \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}} \, dx\\ &=\frac {4 a^3 \log (1-\sec (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {2 a^2 \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}+\frac {a (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.61, size = 328, normalized size = 2.33 \begin {gather*} \frac {4 \sqrt {2} e^{\frac {1}{2} i (e+f x)} \sqrt {\frac {\left (1+e^{i (e+f x)}\right )^2}{1+e^{2 i (e+f x)}}} \left (2 \log \left (1-e^{i (e+f x)}\right )-\log \left (1+e^{2 i (e+f x)}\right )\right ) \sqrt {\sec (e+f x)} (a (1+\sec (e+f x)))^{5/2} \sin \left (\frac {e}{2}+\frac {f x}{2}\right )}{\left (1+e^{i (e+f x)}\right ) \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} f (1+\sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}}+\frac {\sec (e+f x) \sqrt {(1+\cos (e+f x)) \sec (e+f x)} (a (1+\sec (e+f x)))^{5/2} \left (\frac {5 \sec \left (\frac {e}{2}+\frac {f x}{2}\right )}{2 f}+\frac {\cos \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec (e+f x)}{f}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}\right )}{(1+\sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.84, size = 189, normalized size = 1.34
method | result | size |
default | \(\frac {\left (8 \ln \left (-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right )-16 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+8 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\frac {\cos \left (f x +e \right )-1+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-5 \left (\cos ^{2}\left (f x +e \right )\right )-6 \cos \left (f x +e \right )-1\right ) \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, a^{2}}{2 f \sin \left (f x +e \right ) \cos \left (f x +e \right ) c}\) | \(189\) |
risch | \(-\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (3 \,{\mathrm e}^{2 i \left (f x +e \right )}+{\mathrm e}^{i \left (f x +e \right )}+3\right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}-{\mathrm e}^{i \left (f x +e \right )}\right )}{\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}-\frac {8 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}+\frac {4 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(350\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 791 vs.
\(2 (137) = 274\).
time = 0.59, size = 791, normalized size = 5.61 \begin {gather*} -\frac {2 \, {\left (a^{2} \cos \left (2 \, f x + 2 \, e\right ) \sin \left (4 \, f x + 4 \, e\right ) - a^{2} \cos \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - a^{2} \sin \left (2 \, f x + 2 \, e\right ) + 2 \, {\left (a^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + a^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a^{2} \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2} + 2 \, {\left (2 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \cos \left (4 \, f x + 4 \, e\right )\right )} \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 4 \, {\left (a^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + a^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a^{2} \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2} + 2 \, {\left (2 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \cos \left (4 \, f x + 4 \, e\right )\right )} \arctan \left (\sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 1\right ) + 3 \, {\left (a^{2} \sin \left (4 \, f x + 4 \, e\right ) + 2 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 3 \, {\left (a^{2} \sin \left (4 \, f x + 4 \, e\right ) + 2 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 3 \, {\left (a^{2} \cos \left (4 \, f x + 4 \, e\right ) + 2 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 3 \, {\left (a^{2} \cos \left (4 \, f x + 4 \, e\right ) + 2 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a} \sqrt {c}}{{\left (c \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, c \cos \left (2 \, f x + 2 \, e\right )^{2} + c \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, c \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, c \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, {\left (2 \, c \cos \left (2 \, f x + 2 \, e\right ) + c\right )} \cos \left (4 \, f x + 4 \, e\right ) + 4 \, c \cos \left (2 \, f x + 2 \, e\right ) + c\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\cos \left (e+f\,x\right )\,\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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