Optimal. Leaf size=65 \[ \frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {3}{8} a^2 \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} a \left (a \sec ^2(x)\right )^{3/2} \tan (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4207, 201, 223,
212} \begin {gather*} \frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {3}{8} a^2 \tan (x) \sqrt {a \sec ^2(x)}+\frac {1}{4} a \tan (x) \left (a \sec ^2(x)\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 4207
Rubi steps
\begin {align*} \int \left (a \sec ^2(x)\right )^{5/2} \, dx &=a \text {Subst}\left (\int \left (a+a x^2\right )^{3/2} \, dx,x,\tan (x)\right )\\ &=\frac {1}{4} a \left (a \sec ^2(x)\right )^{3/2} \tan (x)+\frac {1}{4} \left (3 a^2\right ) \text {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\tan (x)\right )\\ &=\frac {3}{8} a^2 \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} a \left (a \sec ^2(x)\right )^{3/2} \tan (x)+\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\tan (x)\right )\\ &=\frac {3}{8} a^2 \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} a \left (a \sec ^2(x)\right )^{3/2} \tan (x)+\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\tan (x)}{\sqrt {a \sec ^2(x)}}\right )\\ &=\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {3}{8} a^2 \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} a \left (a \sec ^2(x)\right )^{3/2} \tan (x)\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 72, normalized size = 1.11 \begin {gather*} \frac {1}{16} \cos ^5(x) \left (a \sec ^2(x)\right )^{5/2} \left (-6 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+6 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\frac {1}{2} \sec ^4(x) (11 \sin (x)+3 \sin (3 x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 66, normalized size = 1.02
method | result | size |
default | \(\frac {\left (3 \left (\cos ^{4}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1-\sin \left (x \right )}{\sin \left (x \right )}\right )-3 \left (\cos ^{4}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1+\sin \left (x \right )}{\sin \left (x \right )}\right )+3 \left (\cos ^{2}\left (x \right )\right ) \sin \left (x \right )+2 \sin \left (x \right )\right ) \cos \left (x \right ) \left (\frac {a}{\cos \left (x \right )^{2}}\right )^{\frac {5}{2}}}{8}\) | \(66\) |
risch | \(-\frac {i a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 i x}+11 \,{\mathrm e}^{4 i x}-11 \,{\mathrm e}^{2 i x}-3\right )}{4 \left ({\mathrm e}^{2 i x}+1\right )^{3}}+\frac {3 a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )}{4}-\frac {3 a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )}{4}\) | \(126\) |
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. 1111 vs.
\(2 (49) = 98\).
time = 0.85, size = 1111, normalized size = 17.09 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.96, size = 56, normalized size = 0.86 \begin {gather*} -\frac {{\left (3 \, a^{2} \cos \left (x\right )^{4} \log \left (-\frac {\sin \left (x\right ) - 1}{\sin \left (x\right ) + 1}\right ) - 2 \, {\left (3 \, a^{2} \cos \left (x\right )^{2} + 2 \, a^{2}\right )} \sin \left (x\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{2}}}}{16 \, \cos \left (x\right )^{3}} \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 \left (a \sec ^{2}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 67, normalized size = 1.03 \begin {gather*} \frac {1}{16} \, {\left (3 \, a^{2} \log \left (\sin \left (x\right ) + 1\right ) \mathrm {sgn}\left (\cos \left (x\right )\right ) - 3 \, a^{2} \log \left (-\sin \left (x\right ) + 1\right ) \mathrm {sgn}\left (\cos \left (x\right )\right ) - \frac {2 \, {\left (3 \, a^{2} \mathrm {sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right )^{3} - 5 \, a^{2} \mathrm {sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right )\right )}}{{\left (\sin \left (x\right )^{2} - 1\right )}^{2}}\right )} \sqrt {a} \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 = -0.02 \begin {gather*} \int {\left (\frac {a}{{\cos \left (x\right )}^2}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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