Optimal. Leaf size=110 \[ \frac {2 a^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c f}+\frac {2 a^{3/2} (c-d) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}\right )}{c \sqrt {d} \sqrt {c+d} f} \]
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Rubi [A]
time = 0.16, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4012, 3859,
209, 4052, 211} \begin {gather*} \frac {2 a^{3/2} (c-d) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}\right )}{c \sqrt {d} f \sqrt {c+d}}+\frac {2 a^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{c f} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 3859
Rule 4012
Rule 4052
Rubi steps
\begin {align*} \int \frac {(a+a \sec (e+f x))^{3/2}}{c+d \sec (e+f x)} \, dx &=\frac {a \int \sqrt {a+a \sec (e+f x)} \, dx}{c}+\frac {(a c-a d) \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx}{c}\\ &=-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c f}-\frac {\left (2 a^2 (c-d)\right ) \text {Subst}\left (\int \frac {1}{a c+a d+d x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c f}\\ &=\frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c f}+\frac {2 a^{3/2} (c-d) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}\right )}{c \sqrt {d} \sqrt {c+d} f}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 135, normalized size = 1.23 \begin {gather*} \frac {\sqrt {2} a \left (\sqrt {d} \sqrt {c+d} \text {ArcSin}\left (\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right )+(c-d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d} \sqrt {\cos (e+f x)}}\right )\right ) \sqrt {\cos (e+f x)} \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))}}{c \sqrt {d} \sqrt {c+d} f} \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. \(867\) vs.
\(2(90)=180\).
time = 4.65, size = 868, normalized size = 7.89
method | result | size |
default | \(-\frac {\left (2 \sqrt {\frac {d}{c -d}}\, \sqrt {\left (c +d \right ) \left (c -d \right )}\, \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 )+\ln \left (-\frac {2 \left (\sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )-\sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )-c \sin \left (f x +e \right )+d \sin \left (f x +e \right )+\sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \left (f x +e \right )-\sqrt {\left (c +d \right ) \left (c -d \right )}\right )}{c \cos \left (f x +e \right )-d \cos \left (f x +e \right )-\sqrt {\left (c +d \right ) \left (c -d \right )}\, \sin \left (f x +e \right )-c +d}\right ) c -\ln \left (-\frac {2 \left (\sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )-\sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )-c \sin \left (f x +e \right )+d \sin \left (f x +e \right )+\sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \left (f x +e \right )-\sqrt {\left (c +d \right ) \left (c -d \right )}\right )}{c \cos \left (f x +e \right )-d \cos \left (f x +e \right )-\sqrt {\left (c +d \right ) \left (c -d \right )}\, \sin \left (f x +e \right )-c +d}\right ) d -\ln \left (\frac {-2 \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )+2 \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )+2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \left (f x +e \right )+2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )-2 \sqrt {\left (c +d \right ) \left (c -d \right )}}{\sqrt {\left (c +d \right ) \left (c -d \right )}\, \sin \left (f x +e \right )+c \cos \left (f x +e \right )-d \cos \left (f x +e \right )-c +d}\right ) c +\ln \left (\frac {-2 \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )+2 \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )+2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \left (f x +e \right )+2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )-2 \sqrt {\left (c +d \right ) \left (c -d \right )}}{\sqrt {\left (c +d \right ) \left (c -d \right )}\, \sin \left (f x +e \right )+c \cos \left (f x +e \right )-d \cos \left (f x +e \right )-c +d}\right ) d \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \sqrt {2}\, a}{2 f \sqrt {\left (c +d \right ) \left (c -d \right )}\, c \sqrt {\frac {d}{c -d}}}\) | \(868\) |
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 = 4.01, size = 777, normalized size = 7.06 \begin {gather*} \left [-\frac {{\left (a c - a d\right )} \sqrt {-\frac {a}{c d + d^{2}}} \log \left (\frac {2 \, {\left (c d + d^{2}\right )} \sqrt {-\frac {a}{c d + d^{2}}} \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 ) + {\left (a c + 2 \, a d\right )} \cos \left (f x + e\right )^{2} - a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{c \cos \left (f x + e\right )^{2} + {\left (c + d\right )} \cos \left (f x + e\right ) + d}\right ) - \sqrt {-a} 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 )}{c f}, -\frac {2 \, a^{\frac {3}{2}} \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 (a c - a d\right )} \sqrt {-\frac {a}{c d + d^{2}}} \log \left (\frac {2 \, {\left (c d + d^{2}\right )} \sqrt {-\frac {a}{c d + d^{2}}} \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 ) + {\left (a c + 2 \, a d\right )} \cos \left (f x + e\right )^{2} - a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{c \cos \left (f x + e\right )^{2} + {\left (c + d\right )} \cos \left (f x + e\right ) + d}\right )}{c f}, -\frac {2 \, {\left (a c - a d\right )} \sqrt {\frac {a}{c d + d^{2}}} \arctan \left (\frac {{\left (c + d\right )} \sqrt {\frac {a}{c d + d^{2}}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a \sin \left (f x + e\right )}\right ) - \sqrt {-a} 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 )}{c f}, -\frac {2 \, {\left ({\left (a c - a d\right )} \sqrt {\frac {a}{c d + d^{2}}} \arctan \left (\frac {{\left (c + d\right )} \sqrt {\frac {a}{c d + d^{2}}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a \sin \left (f x + e\right )}\right ) + a^{\frac {3}{2}} \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 )\right )}}{c f}\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*} \int \frac {\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}{c + d \sec {\left (e + f x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 285 vs.
\(2 (90) = 180\).
time = 1.19, size = 285, normalized size = 2.59 \begin {gather*} \frac {\sqrt {2} \sqrt {-a} a^{3} {\left (\frac {2 \, \sqrt {2} {\left (c - d\right )} \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} c - {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} d + a c + 3 \, a d\right )}}{4 \, \sqrt {-c d - d^{2}} a}\right )}{\sqrt {-c d - d^{2}} a^{2} c} - \frac {\sqrt {2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right )}{a c {\left | a \right |}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{2 \, 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 )}^{3/2}}{c+\frac {d}{\cos \left (e+f\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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