Optimal. Leaf size=107 \[ -\frac {1}{8 x^4}+\frac {3}{8 b^2 x^2}+\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x^3}-\frac {3 \cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{4 b^3 x}+\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{8 b^4}-\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3460, 3392, 30,
3391} \begin {gather*} \frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{8 b^4}-\frac {3 \sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{4 b^3 x}-\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x^3}+\frac {3}{8 b^2 x^2}-\frac {1}{8 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 3391
Rule 3392
Rule 3460
Rubi steps
\begin {align*} \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^5} \, dx &=-\text {Subst}\left (\int x^3 \sin ^2(a+b x) \, dx,x,\frac {1}{x}\right )\\ &=\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x^3}-\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2}-\frac {1}{2} \text {Subst}\left (\int x^3 \, dx,x,\frac {1}{x}\right )+\frac {3 \text {Subst}\left (\int x \sin ^2(a+b x) \, dx,x,\frac {1}{x}\right )}{2 b^2}\\ &=-\frac {1}{8 x^4}+\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x^3}-\frac {3 \cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{4 b^3 x}+\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{8 b^4}-\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2}+\frac {3 \text {Subst}\left (\int x \, dx,x,\frac {1}{x}\right )}{4 b^2}\\ &=-\frac {1}{8 x^4}+\frac {3}{8 b^2 x^2}+\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x^3}-\frac {3 \cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{4 b^3 x}+\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{8 b^4}-\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.13, size = 65, normalized size = 0.61 \begin {gather*} -\frac {3 \left (-2 b^2 x^2+x^4\right ) \cos \left (2 \left (a+\frac {b}{x}\right )\right )+2 b \left (b^3+\left (-2 b^2 x+3 x^3\right ) \sin \left (2 \left (a+\frac {b}{x}\right )\right )\right )}{16 b^4 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(333\) vs.
\(2(95)=190\).
time = 0.07, size = 334, normalized size = 3.12
method | result | size |
risch | \(-\frac {1}{8 x^{4}}+\frac {3 \left (2 b^{2}-x^{2}\right ) \cos \left (\frac {2 a x +2 b}{x}\right )}{16 x^{2} b^{4}}+\frac {\left (2 b^{2}-3 x^{2}\right ) \sin \left (\frac {2 a x +2 b}{x}\right )}{8 b^{3} x^{3}}\) | \(67\) |
norman | \(\frac {-\frac {1}{8}+\frac {x \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{b}-\frac {\left (\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{4}-\frac {\left (\tan ^{4}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{8}+\frac {3 x^{2}}{8 b^{2}}-\frac {3 x^{3} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{2 b^{3}}+\frac {3 x^{3} \left (\tan ^{3}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{2 b^{3}}-\frac {9 x^{2} \left (\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{4 b^{2}}+\frac {3 x^{2} \left (\tan ^{4}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{8 b^{2}}-\frac {x \left (\tan ^{3}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{b}+\frac {3 x^{4} \left (\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{2 b^{4}}}{\left (1+\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )^{2} x^{4}}\) | \(200\) |
derivativedivides | \(-\frac {-a^{3} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )+3 a^{2} \left (\left (a +\frac {b}{x}\right ) \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right )^{2}}{4}+\frac {\left (\sin ^{2}\left (a +\frac {b}{x}\right )\right )}{4}\right )-3 a \left (\left (a +\frac {b}{x}\right )^{2} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right ) \left (\cos ^{2}\left (a +\frac {b}{x}\right )\right )}{2}+\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{4}+\frac {b}{4 x}+\frac {a}{4}-\frac {\left (a +\frac {b}{x}\right )^{3}}{3}\right )+\left (a +\frac {b}{x}\right )^{3} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {3 \left (a +\frac {b}{x}\right )^{2} \left (\cos ^{2}\left (a +\frac {b}{x}\right )\right )}{4}+\frac {3 \left (a +\frac {b}{x}\right ) \left (\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {b}{2 x}+\frac {a}{2}\right )}{2}-\frac {3 \left (a +\frac {b}{x}\right )^{2}}{8}-\frac {3 \left (\sin ^{2}\left (a +\frac {b}{x}\right )\right )}{8}-\frac {3 \left (a +\frac {b}{x}\right )^{4}}{8}}{b^{4}}\) | \(334\) |
default | \(-\frac {-a^{3} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )+3 a^{2} \left (\left (a +\frac {b}{x}\right ) \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right )^{2}}{4}+\frac {\left (\sin ^{2}\left (a +\frac {b}{x}\right )\right )}{4}\right )-3 a \left (\left (a +\frac {b}{x}\right )^{2} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right ) \left (\cos ^{2}\left (a +\frac {b}{x}\right )\right )}{2}+\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{4}+\frac {b}{4 x}+\frac {a}{4}-\frac {\left (a +\frac {b}{x}\right )^{3}}{3}\right )+\left (a +\frac {b}{x}\right )^{3} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {3 \left (a +\frac {b}{x}\right )^{2} \left (\cos ^{2}\left (a +\frac {b}{x}\right )\right )}{4}+\frac {3 \left (a +\frac {b}{x}\right ) \left (\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {b}{2 x}+\frac {a}{2}\right )}{2}-\frac {3 \left (a +\frac {b}{x}\right )^{2}}{8}-\frac {3 \left (\sin ^{2}\left (a +\frac {b}{x}\right )\right )}{8}-\frac {3 \left (a +\frac {b}{x}\right )^{4}}{8}}{b^{4}}\) | \(334\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.33, size = 68, normalized size = 0.64 \begin {gather*} -\frac {{\left ({\left (\Gamma \left (4, \frac {2 i \, b}{x}\right ) + \Gamma \left (4, -\frac {2 i \, b}{x}\right )\right )} \cos \left (2 \, a\right ) - {\left (i \, \Gamma \left (4, \frac {2 i \, b}{x}\right ) - i \, \Gamma \left (4, -\frac {2 i \, b}{x}\right )\right )} \sin \left (2 \, a\right )\right )} x^{4} + 8 \, b^{4}}{64 \, b^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 90, normalized size = 0.84 \begin {gather*} -\frac {2 \, b^{4} + 6 \, b^{2} x^{2} - 3 \, x^{4} - 6 \, {\left (2 \, b^{2} x^{2} - x^{4}\right )} \cos \left (\frac {a x + b}{x}\right )^{2} - 4 \, {\left (2 \, b^{3} x - 3 \, b x^{3}\right )} \cos \left (\frac {a x + b}{x}\right ) \sin \left (\frac {a x + b}{x}\right )}{16 \, b^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 726 vs.
\(2 (92) = 184\).
time = 3.43, size = 726, normalized size = 6.79 \begin {gather*} \begin {cases} - \frac {b^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} - \frac {2 b^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} - \frac {b^{4}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} - \frac {8 b^{3} x \tan ^{3}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} + \frac {8 b^{3} x \tan {\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} + \frac {3 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} - \frac {18 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} + \frac {3 b^{2} x^{2}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} + \frac {12 b x^{3} \tan ^{3}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} - \frac {12 b x^{3} \tan {\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} + \frac {12 x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{8 b^{4} x^{4} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 16 b^{4} x^{4} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{4} x^{4}} & \text {for}\: b \neq 0 \\- \frac {\sin ^{2}{\left (a \right )}}{4 x^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs.
\(2 (95) = 190\).
time = 3.76, size = 255, normalized size = 2.38 \begin {gather*} -\frac {4 \, a^{3} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) - \frac {8 \, {\left (a x + b\right )} a^{3}}{x} - 6 \, a^{2} \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) - \frac {12 \, {\left (a x + b\right )} a^{2} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} + \frac {12 \, {\left (a x + b\right )}^{2} a^{2}}{x^{2}} + \frac {12 \, {\left (a x + b\right )} a \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} - 6 \, a \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) + \frac {12 \, {\left (a x + b\right )}^{2} a \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x^{2}} - \frac {8 \, {\left (a x + b\right )}^{3} a}{x^{3}} - \frac {6 \, {\left (a x + b\right )}^{2} \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x^{2}} - \frac {4 \, {\left (a x + b\right )}^{3} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x^{3}} + \frac {6 \, {\left (a x + b\right )} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} + \frac {2 \, {\left (a x + b\right )}^{4}}{x^{4}} + 3 \, \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{16 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.72, size = 84, normalized size = 0.79 \begin {gather*} -\frac {3\,\cos \left (2\,a+\frac {2\,b}{x}\right )}{16\,b^4}-\frac {\frac {b^4}{8}-\frac {3\,b^2\,x^2\,\cos \left (2\,a+\frac {2\,b}{x}\right )}{8}+\frac {3\,b\,x^3\,\sin \left (2\,a+\frac {2\,b}{x}\right )}{8}-\frac {b^3\,x\,\sin \left (2\,a+\frac {2\,b}{x}\right )}{4}}{b^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________