Optimal. Leaf size=110 \[ -\frac {5 \text {ArcSin}(\cos (a+b x)-\sin (a+b x))}{32 b}+\frac {5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{32 b}-\frac {5 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 110, 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, 4387,
4390} \begin {gather*} -\frac {5 \text {ArcSin}(\cos (a+b x)-\sin (a+b x))}{32 b}-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}-\frac {5 \sqrt {\sin (2 a+2 b x)} \cos (a+b x)}{16 b}+\frac {5 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{32 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 4383
Rule 4387
Rule 4390
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx &=-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}+\frac {5}{8} \int \sin (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx\\ &=-\frac {5 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}+\frac {5}{16} \int \frac {\cos (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=-\frac {5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{32 b}+\frac {5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{32 b}-\frac {5 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.24, size = 86, normalized size = 0.78 \begin {gather*} \frac {5 \left (-\text {ArcSin}(\cos (a+b x)-\sin (a+b x))+\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right )\right )+2 (-6 \cos (a+b x)+\cos (3 (a+b x))) \sqrt {\sin (2 (a+b x))}}{32 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 17.74, size = 57690707, normalized size = 524460.97
method | result | size |
default | \(\text {Expression too large to display}\) | \(57690707\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 281 vs.
\(2 (96) = 192\).
time = 4.01, size = 281, normalized size = 2.55 \begin {gather*} \frac {8 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - 9 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 10 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) - 10 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) - 5 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{128 \, b} \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 )}^3\,\sqrt {\sin \left (2\,a+2\,b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________