Optimal. Leaf size=36 \[ \frac {a \left (a+b \cos ^2(x)\right )^4}{8 b^2}-\frac {\left (a+b \cos ^2(x)\right )^5}{10 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4335, 266, 43} \[ \frac {a \left (a+b \cos ^2(x)\right )^4}{8 b^2}-\frac {\left (a+b \cos ^2(x)\right )^5}{10 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rule 4335
Rubi steps
\begin {align*} \int \cos ^3(x) \left (a+b \cos ^2(x)\right )^3 \sin (x) \, dx &=-\operatorname {Subst}\left (\int x^3 \left (a+b x^2\right )^3 \, dx,x,\cos (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^3 \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^3}{b}+\frac {(a+b x)^4}{b}\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=\frac {a \left (a+b \cos ^2(x)\right )^4}{8 b^2}-\frac {\left (a+b \cos ^2(x)\right )^5}{10 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.28, size = 137, normalized size = 3.81 \[ \frac {1}{32} \left (-4 a^3 \cos (2 x)-a^3 \cos (4 x)-12 a^2 b \cos ^4(x)-4 a^2 b \cos (3 x) \cos ^3(x)-8 a b^2 \cos ^6(x)-\frac {1}{32} a b^2 (48 \cos (2 x)+36 \cos (4 x)+16 \cos (6 x)+3 \cos (8 x))-2 b^3 \cos ^8(x)-\frac {1}{320} b^3 (140 \cos (2 x)+100 \cos (4 x)+50 \cos (6 x)+15 \cos (8 x)+2 \cos (10 x))\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 39, normalized size = 1.08 \[ -\frac {1}{10} \, b^{3} \cos \relax (x)^{10} - \frac {3}{8} \, a b^{2} \cos \relax (x)^{8} - \frac {1}{2} \, a^{2} b \cos \relax (x)^{6} - \frac {1}{4} \, a^{3} \cos \relax (x)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 39, normalized size = 1.08 \[ -\frac {1}{10} \, b^{3} \cos \relax (x)^{10} - \frac {3}{8} \, a b^{2} \cos \relax (x)^{8} - \frac {1}{2} \, a^{2} b \cos \relax (x)^{6} - \frac {1}{4} \, a^{3} \cos \relax (x)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 40, normalized size = 1.11 \[ -\frac {b^{3} \left (\cos ^{10}\relax (x )\right )}{10}-\frac {3 a \,b^{2} \left (\cos ^{8}\relax (x )\right )}{8}-\frac {a^{2} b \left (\cos ^{6}\relax (x )\right )}{2}-\frac {a^{3} \left (\cos ^{4}\relax (x )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.39, size = 103, normalized size = 2.86 \[ \frac {1}{10} \, b^{3} \sin \relax (x)^{10} - \frac {1}{8} \, {\left (3 \, a b^{2} + 4 \, b^{3}\right )} \sin \relax (x)^{8} + \frac {1}{2} \, {\left (a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} \sin \relax (x)^{6} - \frac {1}{4} \, {\left (a^{3} + 6 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3}\right )} \sin \relax (x)^{4} + \frac {1}{2} \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sin \relax (x)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 39, normalized size = 1.08 \[ -\frac {a^3\,{\cos \relax (x)}^4}{4}-\frac {a^2\,b\,{\cos \relax (x)}^6}{2}-\frac {3\,a\,b^2\,{\cos \relax (x)}^8}{8}-\frac {b^3\,{\cos \relax (x)}^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 11.65, size = 46, normalized size = 1.28 \[ - \frac {a^{3} \cos ^{4}{\relax (x )}}{4} - \frac {a^{2} b \cos ^{6}{\relax (x )}}{2} - \frac {3 a b^{2} \cos ^{8}{\relax (x )}}{8} - \frac {b^{3} \cos ^{10}{\relax (x )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________