Optimal. Leaf size=42 \[ \frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b}+\frac {2 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2715, 2719}
\begin {gather*} \frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b}+\frac {2 \sin (a+b x) \cos ^{\frac {3}{2}}(a+b x)}{5 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2715
Rule 2719
Rubi steps
\begin {align*} \int \cos ^{\frac {5}{2}}(a+b x) \, dx &=\frac {2 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{5 b}+\frac {3}{5} \int \sqrt {\cos (a+b x)} \, dx\\ &=\frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b}+\frac {2 \cos ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{5 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 40, normalized size = 0.95 \begin {gather*} \frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sqrt {\cos (a+b x)} \sin (2 (a+b x))}{5 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs.
\(2(62)=124\).
time = 0.03, size = 202, normalized size = 4.81
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (-8 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+8 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right )}{5 \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.12, size = 74, normalized size = 1.76 \begin {gather*} \frac {2 \, \cos \left (b x + a\right )^{\frac {3}{2}} \sin \left (b x + a\right ) + 3 i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - 3 i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{5 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.17, size = 42, normalized size = 1.00 \begin {gather*} -\frac {2\,{\cos \left (a+b\,x\right )}^{7/2}\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (a+b\,x\right )}^2\right )}{7\,b\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________