Optimal. Leaf size=47 \[ \frac {3 (c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3516, 3462,
3460, 2718} \begin {gather*} \frac {3 (c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2718
Rule 3460
Rule 3462
Rule 3516
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{5/3}} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{(e x)^{5/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x)^{2/3} \text {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{x^{5/3}} \, dx,x,c+d x\right )}{d e (e (c+d x))^{2/3}}\\ &=-\frac {\left (3 (c+d x)^{2/3}\right ) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=\frac {3 (c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 44, normalized size = 0.94 \begin {gather*} \frac {3 (c+d x)^{5/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d (e (c+d x))^{5/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {5}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 30, normalized size = 0.64 \begin {gather*} \frac {3 \, \cos \left (\frac {{\left (d x + c\right )}^{\frac {2}{3}} a + b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) e^{\left (-\frac {5}{3}\right )}}{2 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (22) = 44\).
time = 0.37, size = 48, normalized size = 1.02 \begin {gather*} \frac {3 \, {\left (d x + c\right )} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right ) e^{\left (-\frac {5}{3}\right )}}{2 \, {\left (b d^{2} x + b c d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {5}{3}}}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{5/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________