Optimal. Leaf size=69 \[ \frac {3 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{10 b}-\frac {\cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{10 b}-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x)}{14 b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4383, 2715,
2719} \begin {gather*} -\frac {\sin ^{\frac {7}{2}}(2 a+2 b x)}{14 b}+\frac {3 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{10 b}-\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{10 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2715
Rule 2719
Rule 4383
Rubi steps
\begin {align*} \int \sin ^2(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx &=-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x)}{14 b}+\frac {1}{2} \int \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx\\ &=-\frac {\cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{10 b}-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x)}{14 b}+\frac {3}{10} \int \sqrt {\sin (2 a+2 b x)} \, dx\\ &=\frac {3 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{10 b}-\frac {\cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{10 b}-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x)}{14 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.24, size = 66, normalized size = 0.96 \begin {gather*} \frac {84 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\sqrt {\sin (2 (a+b x))} (-15 \sin (2 (a+b x))-14 \sin (4 (a+b x))+5 \sin (6 (a+b x)))}{280 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 130.06, size = 306311267, normalized size = 4439293.72
method | result | size |
default | \(\text {Expression too large to display}\) | \(306311267\) |
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 [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]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.01 \begin {gather*} \int {\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________