Optimal. Leaf size=174 \[ -\frac {15 a^3 \text {ArcTan}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{2 \sqrt {2} c^{5/2} f}-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}+\frac {15 a^3 \tan (e+f x)}{4 c^2 f \sqrt {c-c \sec (e+f x)}} \]
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Rubi [A]
time = 0.24, antiderivative size = 174, 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 = {4042, 4041,
3880, 209} \begin {gather*} -\frac {15 a^3 \text {ArcTan}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{2 \sqrt {2} c^{5/2} f}+\frac {15 a^3 \tan (e+f x)}{4 c^2 f \sqrt {c-c \sec (e+f x)}}+\frac {5 \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{4 c f (c-c \sec (e+f x))^{3/2}}-\frac {a \tan (e+f x) (a \sec (e+f x)+a)^2}{2 f (c-c \sec (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 3880
Rule 4041
Rule 4042
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^{5/2}} \, dx &=-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}-\frac {(5 a) \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^{3/2}} \, dx}{4 c}\\ &=-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}+\frac {\left (15 a^2\right ) \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{\sqrt {c-c \sec (e+f x)}} \, dx}{8 c^2}\\ &=-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}+\frac {15 a^3 \tan (e+f x)}{4 c^2 f \sqrt {c-c \sec (e+f x)}}+\frac {\left (15 a^3\right ) \int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}} \, dx}{4 c^2}\\ &=-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}+\frac {15 a^3 \tan (e+f x)}{4 c^2 f \sqrt {c-c \sec (e+f x)}}-\frac {\left (15 a^3\right ) \text {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 )}{2 c^2 f}\\ &=-\frac {15 a^3 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{2 \sqrt {2} c^{5/2} f}-\frac {a (a+a \sec (e+f x))^2 \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{4 c f (c-c \sec (e+f x))^{3/2}}+\frac {15 a^3 \tan (e+f x)}{4 c^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 = 4.59, size = 263, normalized size = 1.51 \begin {gather*} -\frac {a^3 e^{-\frac {1}{2} i (2 e+f x)} \sec \left (\frac {1}{2} (e+f x)\right ) (1+\sec (e+f x))^3 \left (120 e^{\frac {i e}{2}} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt {1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\frac {1+e^{i (e+f x)}}{\sqrt {2} \sqrt {1+e^{2 i (e+f x)}}}\right )+\left (25 \cos \left (\frac {3}{2} (e+f x)\right )-9 \cos \left (\frac {5}{2} (e+f x)\right )\right ) \csc ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \left (\cos \left (e+\frac {f x}{2}\right )+i \sin \left (e+\frac {f x}{2}\right )\right )\right ) \tan ^5\left (\frac {1}{2} (e+f x)\right )}{32 c^2 f (-1+\sec (e+f x))^2 \sqrt {\sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.52, size = 206, normalized size = 1.18
method | result | size |
default | \(-\frac {a^{3} \left (15 \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{2}\left (f x +e \right )\right )-30 \cos \left (f x +e \right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-18 \left (\cos ^{2}\left (f x +e \right )\right )+15 \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+34 \cos \left (f x +e \right )-8\right ) \sin \left (f x +e \right )}{4 f \cos \left (f x +e \right )^{3} \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}\) | \(206\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.43, size = 475, normalized size = 2.73 \begin {gather*} \left [-\frac {15 \, \sqrt {2} {\left (a^{3} \cos \left (f x + e\right )^{2} - 2 \, a^{3} \cos \left (f x + e\right ) + a^{3}\right )} \sqrt {-c} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {-c} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} + {\left (3 \, c \cos \left (f x + e\right ) + c\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 ) + 4 \, {\left (9 \, a^{3} \cos \left (f x + e\right )^{3} - 8 \, a^{3} \cos \left (f x + e\right )^{2} - 13 \, a^{3} \cos \left (f x + e\right ) + 4 \, a^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{8 \, {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )}, \frac {15 \, \sqrt {2} {\left (a^{3} \cos \left (f x + e\right )^{2} - 2 \, a^{3} \cos \left (f x + e\right ) + a^{3}\right )} \sqrt {c} \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 ) - 2 \, {\left (9 \, a^{3} \cos \left (f x + e\right )^{3} - 8 \, a^{3} \cos \left (f x + e\right )^{2} - 13 \, a^{3} \cos \left (f x + e\right ) + 4 \, a^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{4 \, {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\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*} a^{3} \left (\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {3 \sec ^{3}{\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 133, normalized size = 0.76 \begin {gather*} \frac {a^{3} {\left (\frac {15 \, \sqrt {2} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {8 \, \sqrt {2}}{\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{2}} + \frac {7 \, \sqrt {2} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} + 9 \, \sqrt {2} \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c}{c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}\right )}}{4 \, f} \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 )}^3}{\cos \left (e+f\,x\right )\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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