Optimal. Leaf size=65 \[ \frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {3}{8} a^2 \tan (x) \sqrt {a \sec ^2(x)}+\frac {1}{4} a \tan (x) \left (a \sec ^2(x)\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, 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 = {4122, 195, 217, 206} \[ \frac {3}{8} a^2 \tan (x) \sqrt {a \sec ^2(x)}+\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {1}{4} a \tan (x) \left (a \sec ^2(x)\right )^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 206
Rule 217
Rule 4122
Rubi steps
\begin {align*} \int \left (a \sec ^2(x)\right )^{5/2} \, dx &=a \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 72, normalized size = 1.11 \[ \frac {1}{16} \cos ^5(x) \left (a \sec ^2(x)\right )^{5/2} \left (\frac {1}{2} (11 \sin (x)+3 \sin (3 x)) \sec ^4(x)-6 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+6 \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 56, normalized size = 0.86 \[ -\frac {{\left (3 \, a^{2} \cos \relax (x)^{4} \log \left (-\frac {\sin \relax (x) - 1}{\sin \relax (x) + 1}\right ) - 2 \, {\left (3 \, a^{2} \cos \relax (x)^{2} + 2 \, a^{2}\right )} \sin \relax (x)\right )} \sqrt {\frac {a}{\cos \relax (x)^{2}}}}{16 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.41, size = 67, normalized size = 1.03 \[ \frac {1}{16} \, {\left (3 \, a^{2} \log \left (\sin \relax (x) + 1\right ) \mathrm {sgn}\left (\cos \relax (x)\right ) - 3 \, a^{2} \log \left (-\sin \relax (x) + 1\right ) \mathrm {sgn}\left (\cos \relax (x)\right ) - \frac {2 \, {\left (3 \, a^{2} \mathrm {sgn}\left (\cos \relax (x)\right ) \sin \relax (x)^{3} - 5 \, a^{2} \mathrm {sgn}\left (\cos \relax (x)\right ) \sin \relax (x)\right )}}{{\left (\sin \relax (x)^{2} - 1\right )}^{2}}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.36, size = 66, normalized size = 1.02 \[ -\frac {\left (3 \left (\cos ^{4}\relax (x )\right ) \ln \left (-\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}\right )-3 \left (\cos ^{4}\relax (x )\right ) \ln \left (-\frac {-\sin \relax (x )-1+\cos \relax (x )}{\sin \relax (x )}\right )-3 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )-2 \sin \relax (x )\right ) \cos \relax (x ) \left (\frac {a}{\cos \relax (x )^{2}}\right )^{\frac {5}{2}}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.00, size = 1111, normalized size = 17.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {a}{{\cos \relax (x)}^2}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sec ^{2}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________