Optimal. Leaf size=42 \[ -\frac {(a+a \sec (e+f x))^{5/2} \tan (e+f x)}{6 f (c-c \sec (e+f x))^{7/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {4035}
\begin {gather*} -\frac {\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{6 f (c-c \sec (e+f x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4035
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx &=-\frac {(a+a \sec (e+f x))^{5/2} \tan (e+f x)}{6 f (c-c \sec (e+f x))^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 76, normalized size = 1.81 \begin {gather*} \frac {a^2 (5+3 \cos (2 (e+f x))) \csc ^5\left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))}}{48 c^3 f \sqrt {c-c \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. \(74\) vs.
\(2(36)=72\).
time = 2.91, size = 75, normalized size = 1.79
method | result | size |
default | \(-\frac {\left (\sin ^{5}\left (f x +e \right )\right ) \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, a^{2}}{6 f \left (-1+\cos \left (f x +e \right )\right )^{2} \cos \left (f x +e \right )^{3} \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}}}\) | \(75\) |
risch | \(\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (3 \,{\mathrm e}^{5 i \left (f x +e \right )}+10 \,{\mathrm e}^{3 i \left (f x +e \right )}+3 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(133\) |
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. 1949 vs.
\(2 (39) = 78\).
time = 0.59, size = 1949, normalized size = 46.40 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (39) = 78\).
time = 2.34, size = 136, normalized size = 3.24 \begin {gather*} \frac {{\left (3 \, a^{2} \cos \left (f x + e\right )^{3} + a^{2} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )} \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 [A]
time = 1.82, size = 65, normalized size = 1.55 \begin {gather*} -\frac {{\left (a^{2} - \frac {a^{2}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}\right )} a^{2}}{6 \, \sqrt {-a c} c^{3} f {\left | a \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.04, size = 199, normalized size = 4.74 \begin {gather*} -\frac {\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}\,\left (\frac {a^2\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,52{}\mathrm {i}}{3\,c^4\,f}+\frac {a^2\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,4{}\mathrm {i}}{c^4\,f}\right )}{{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,28{}\mathrm {i}-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )\,28{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,12{}\mathrm {i}-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )\,2{}\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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