Optimal. Leaf size=149 \[ \frac {2 \sqrt {a} g^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d f}-\frac {2 \sqrt {a} \sqrt {c} g^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d \sqrt {c+d} f} \]
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Rubi [A]
time = 0.37, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {4055, 3887,
214, 4050} \begin {gather*} \frac {2 \sqrt {a} g^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{d f}-\frac {2 \sqrt {a} \sqrt {c} g^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{d f \sqrt {c+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 3887
Rule 4050
Rule 4055
Rubi steps
\begin {align*} \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx &=\frac {g \int \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)} \, dx}{d}-\frac {(c g) \int \frac {\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx}{d}\\ &=-\frac {\left (2 a g^2\right ) \text {Subst}\left (\int \frac {1}{a-g x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d f}+\frac {\left (2 a c g^2\right ) \text {Subst}\left (\int \frac {1}{a c+a d-c g x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d f}\\ &=\frac {2 \sqrt {a} g^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d f}-\frac {2 \sqrt {a} \sqrt {c} g^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d \sqrt {c+d} f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.40, size = 427, normalized size = 2.87 \begin {gather*} \frac {\left (-2 i+\sqrt {2}\right ) g^2 \left (2 \sqrt {c+d} \text {ArcTan}\left (\frac {\cos \left (\frac {1}{4} (e+f x)\right )-\left (-1+\sqrt {2}\right ) \sin \left (\frac {1}{4} (e+f x)\right )}{\left (1+\sqrt {2}\right ) \cos \left (\frac {1}{4} (e+f x)\right )-\sin \left (\frac {1}{4} (e+f x)\right )}\right )+2 \sqrt {c+d} \text {ArcTan}\left (\frac {\cos \left (\frac {1}{4} (e+f x)\right )-\left (1+\sqrt {2}\right ) \sin \left (\frac {1}{4} (e+f x)\right )}{\left (-1+\sqrt {2}\right ) \cos \left (\frac {1}{4} (e+f x)\right )-\sin \left (\frac {1}{4} (e+f x)\right )}\right )+i \left (2 \sqrt {c+d} \log \left (\sqrt {2}+2 \sin \left (\frac {1}{2} (e+f x)\right )\right )-\sqrt {c+d} \log \left (2-\sqrt {2} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right )-\sqrt {c+d} \log \left (2+\sqrt {2} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right )+2 \sqrt {c} \log \left (\sqrt {2} \sqrt {c+d}-2 \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )\right )-2 \sqrt {c} \log \left (\sqrt {2} \sqrt {c+d}+2 \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )\right )\right )\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))}}{4 \left (i+\sqrt {2}\right ) d \sqrt {c+d} f \sqrt {g \sec (e+f x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(567\) vs.
\(2(117)=234\).
time = 13.20, size = 568, normalized size = 3.81
method | result | size |
default | \(\frac {2 \left (\frac {g}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \left (\cos ^{2}\left (f x +e \right )\right ) \left (-1+\cos \left (f x +e \right )\right )^{2} \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (\sqrt {\frac {c}{c -d}}\, \arctanh \left (\frac {\sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \left (1+\cos \left (f x +e \right )+\sin \left (f x +e \right )\right )}{2}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}-\sqrt {\frac {c}{c -d}}\, \arctanh \left (\frac {\sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \left (1+\cos \left (f x +e \right )-\sin \left (f x +e \right )\right )}{2}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}-c \ln \left (-\frac {2 \left (2 \sqrt {\frac {c}{c -d}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )-2 \sqrt {\frac {c}{c -d}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )-\sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \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 )}\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 \left (2 \sqrt {\frac {c}{c -d}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )-2 \sqrt {\frac {c}{c -d}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )+\sqrt {\left (c +d \right ) \left (c -d \right )}\, \cos \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 )}\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 )\right ) \left (c -d \right )}{f \sin \left (f x +e \right )^{4} \left (\frac {1}{\cos \left (f x +e \right )+1}\right )^{\frac {3}{2}} \sqrt {\frac {c}{c -d}}\, \left (-c +d +\sqrt {\left (c +d \right ) \left (c -d \right )}\right ) \left (c -d +\sqrt {\left (c +d \right ) \left (c -d \right )}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\) | \(568\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 8.05, size = 1196, normalized size = 8.03 \begin {gather*} \left [\frac {\sqrt {\frac {a c g}{c + d}} g \log \left (\frac {a c^{2} g \cos \left (f x + e\right )^{3} - {\left (7 \, a c^{2} + 6 \, a c d\right )} g \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (c^{2} + c d\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, c^{2} + 3 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a c g}{c + d}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + {\left (2 \, a c d + a d^{2}\right )} g \cos \left (f x + e\right ) + {\left (8 \, a c^{2} + 8 \, a c d + a d^{2}\right )} g}{c^{2} \cos \left (f x + e\right )^{3} + {\left (c^{2} + 2 \, c d\right )} \cos \left (f x + e\right )^{2} + d^{2} + {\left (2 \, c d + d^{2}\right )} \cos \left (f x + e\right )}\right ) + \sqrt {a g} g \log \left (\frac {a g \cos \left (f x + e\right )^{3} - 7 \, a g \cos \left (f x + e\right )^{2} - 4 \, \sqrt {a g} {\left (\cos \left (f x + e\right )^{2} - 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 {g}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 8 \, a g}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2}}\right )}{2 \, d f}, -\frac {2 \, \sqrt {-\frac {a c g}{c + d}} g \arctan \left (\frac {{\left (c \cos \left (f x + e\right )^{2} - {\left (2 \, c + d\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {a c g}{c + d}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}}}{2 \, a c g \sin \left (f x + e\right )}\right ) - \sqrt {a g} g \log \left (\frac {a g \cos \left (f x + e\right )^{3} - 7 \, a g \cos \left (f x + e\right )^{2} - 4 \, \sqrt {a g} {\left (\cos \left (f x + e\right )^{2} - 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 {g}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 8 \, a g}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2}}\right )}{2 \, d f}, \frac {2 \, \sqrt {-a g} g \arctan \left (\frac {2 \, \sqrt {-a g} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a g \cos \left (f x + e\right )^{2} - a g \cos \left (f x + e\right ) - 2 \, a g}\right ) + \sqrt {\frac {a c g}{c + d}} g \log \left (\frac {a c^{2} g \cos \left (f x + e\right )^{3} - {\left (7 \, a c^{2} + 6 \, a c d\right )} g \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (c^{2} + c d\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, c^{2} + 3 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a c g}{c + d}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + {\left (2 \, a c d + a d^{2}\right )} g \cos \left (f x + e\right ) + {\left (8 \, a c^{2} + 8 \, a c d + a d^{2}\right )} g}{c^{2} \cos \left (f x + e\right )^{3} + {\left (c^{2} + 2 \, c d\right )} \cos \left (f x + e\right )^{2} + d^{2} + {\left (2 \, c d + d^{2}\right )} \cos \left (f x + e\right )}\right )}{2 \, d f}, \frac {\sqrt {-a g} g \arctan \left (\frac {2 \, \sqrt {-a g} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a g \cos \left (f x + e\right )^{2} - a g \cos \left (f x + e\right ) - 2 \, a g}\right ) - \sqrt {-\frac {a c g}{c + d}} g \arctan \left (\frac {{\left (c \cos \left (f x + e\right )^{2} - {\left (2 \, c + d\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {a c g}{c + d}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}}}{2 \, a c g \sin \left (f x + e\right )}\right )}{d 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 {\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \left (g \sec {\left (e + f x \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 [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (\frac {g}{\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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