Optimal. Leaf size=54 \[ -\frac {(a+b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}+\frac {(a+2 b) \cos (x)}{b^2}-\frac {\cos ^3(x)}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3190, 390, 205} \[ \frac {(a+2 b) \cos (x)}{b^2}-\frac {(a+b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}-\frac {\cos ^3(x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 390
Rule 3190
Rubi steps
\begin {align*} \int \frac {\sin ^5(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a+b x^2} \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {a+2 b}{b^2}+\frac {x^2}{b}+\frac {a^2+2 a b+b^2}{b^2 \left (a+b x^2\right )}\right ) \, dx,x,\cos (x)\right )\\ &=\frac {(a+2 b) \cos (x)}{b^2}-\frac {\cos ^3(x)}{3 b}-\frac {(a+b)^2 \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\cos (x)\right )}{b^2}\\ &=-\frac {(a+b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}+\frac {(a+2 b) \cos (x)}{b^2}-\frac {\cos ^3(x)}{3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.18, size = 116, normalized size = 2.15 \[ \frac {3 \sqrt {b} (4 a+7 b) \cos (x)-\frac {12 (a+b)^2 \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {a+b} \tan \left (\frac {x}{2}\right )}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {12 (a+b)^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \tan \left (\frac {x}{2}\right )+\sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a}}-b^{3/2} \cos (3 x)}{12 b^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.30, size = 152, normalized size = 2.81 \[ \left [-\frac {2 \, a b^{2} \cos \relax (x)^{3} + 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a b} \log \left (-\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {-a b} \cos \relax (x) - a}{b \cos \relax (x)^{2} + a}\right ) - 6 \, {\left (a^{2} b + 2 \, a b^{2}\right )} \cos \relax (x)}{6 \, a b^{3}}, -\frac {a b^{2} \cos \relax (x)^{3} + 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} \cos \relax (x)}{a}\right ) - 3 \, {\left (a^{2} b + 2 \, a b^{2}\right )} \cos \relax (x)}{3 \, a b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 59, normalized size = 1.09 \[ -\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {b \cos \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} - \frac {b^{2} \cos \relax (x)^{3} - 3 \, a b \cos \relax (x) - 6 \, b^{2} \cos \relax (x)}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 86, normalized size = 1.59 \[ -\frac {\cos ^{3}\relax (x )}{3 b}+\frac {a \cos \relax (x )}{b^{2}}+\frac {2 \cos \relax (x )}{b}-\frac {\arctan \left (\frac {\cos \relax (x ) b}{\sqrt {a b}}\right ) a^{2}}{b^{2} \sqrt {a b}}-\frac {2 \arctan \left (\frac {\cos \relax (x ) b}{\sqrt {a b}}\right ) a}{b \sqrt {a b}}-\frac {\arctan \left (\frac {\cos \relax (x ) b}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.83, size = 53, normalized size = 0.98 \[ -\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {b \cos \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} - \frac {b \cos \relax (x)^{3} - 3 \, {\left (a + 2 \, b\right )} \cos \relax (x)}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 65, normalized size = 1.20 \[ \cos \relax (x)\,\left (\frac {a}{b^2}+\frac {2}{b}\right )-\frac {{\cos \relax (x)}^3}{3\,b}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\cos \relax (x)\,{\left (a+b\right )}^2}{\sqrt {a}\,\left (a^2+2\,a\,b+b^2\right )}\right )\,{\left (a+b\right )}^2}{\sqrt {a}\,b^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________