Optimal. Leaf size=49 \[ \frac {3 \tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {3 \csc (a+b x)}{16 b}+\frac {\csc (a+b x) \sec ^2(a+b x)}{16 b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {4373, 2701,
294, 327, 213} \begin {gather*} -\frac {3 \csc (a+b x)}{16 b}+\frac {3 \tanh ^{-1}(\sin (a+b x))}{16 b}+\frac {\csc (a+b x) \sec ^2(a+b x)}{16 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 213
Rule 294
Rule 327
Rule 2701
Rule 4373
Rubi steps
\begin {align*} \int \csc ^3(2 a+2 b x) \sin (a+b x) \, dx &=\frac {1}{8} \int \csc ^2(a+b x) \sec ^3(a+b x) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\csc (a+b x)\right )}{8 b}\\ &=\frac {\csc (a+b x) \sec ^2(a+b x)}{16 b}-\frac {3 \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=-\frac {3 \csc (a+b x)}{16 b}+\frac {\csc (a+b x) \sec ^2(a+b x)}{16 b}-\frac {3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=\frac {3 \tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {3 \csc (a+b x)}{16 b}+\frac {\csc (a+b x) \sec ^2(a+b x)}{16 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 29, normalized size = 0.59 \begin {gather*} -\frac {\csc (a+b x) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\sin ^2(a+b x)\right )}{8 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 51, normalized size = 1.04
method | result | size |
default | \(\frac {\frac {1}{2 \sin \left (x b +a \right ) \cos \left (x b +a \right )^{2}}-\frac {3}{2 \sin \left (x b +a \right )}+\frac {3 \ln \left (\sec \left (x b +a \right )+\tan \left (x b +a \right )\right )}{2}}{8 b}\) | \(51\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{5 i \left (x b +a \right )}+2 \,{\mathrm e}^{3 i \left (x b +a \right )}+3 \,{\mathrm e}^{i \left (x b +a \right )}\right )}{8 b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}+\frac {3 \ln \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}{16 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-i\right )}{16 b}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 808 vs.
\(2 (43) = 86\).
time = 0.52, size = 808, normalized size = 16.49 \begin {gather*} \frac {4 \, {\left (3 \, \sin \left (5 \, b x + 5 \, a\right ) + 2 \, \sin \left (3 \, b x + 3 \, a\right ) + 3 \, \sin \left (b x + a\right )\right )} \cos \left (6 \, b x + 6 \, a\right ) - 12 \, {\left (\sin \left (4 \, b x + 4 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} \cos \left (5 \, b x + 5 \, a\right ) + 4 \, {\left (2 \, \sin \left (3 \, b x + 3 \, a\right ) + 3 \, \sin \left (b x + a\right )\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \, {\left (2 \, {\left (\cos \left (4 \, b x + 4 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \cos \left (6 \, b x + 6 \, a\right ) + \cos \left (6 \, b x + 6 \, a\right )^{2} - 2 \, {\left (\cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + \cos \left (4 \, b x + 4 \, a\right )^{2} + \cos \left (2 \, b x + 2 \, a\right )^{2} + 2 \, {\left (\sin \left (4 \, b x + 4 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + \sin \left (6 \, b x + 6 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} - 2 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\frac {\cos \left (b x + 2 \, a\right )^{2} + \cos \left (a\right )^{2} - 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} + 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}}{\cos \left (b x + 2 \, a\right )^{2} + \cos \left (a\right )^{2} + 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} - 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) - 4 \, {\left (3 \, \cos \left (5 \, b x + 5 \, a\right ) + 2 \, \cos \left (3 \, b x + 3 \, a\right ) + 3 \, \cos \left (b x + a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + 12 \, {\left (\cos \left (4 \, b x + 4 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \sin \left (5 \, b x + 5 \, a\right ) - 4 \, {\left (2 \, \cos \left (3 \, b x + 3 \, a\right ) + 3 \, \cos \left (b x + a\right )\right )} \sin \left (4 \, b x + 4 \, a\right ) - 8 \, {\left (\cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \sin \left (3 \, b x + 3 \, a\right ) + 8 \, \cos \left (3 \, b x + 3 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 12 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 12 \, \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) - 12 \, \sin \left (b x + a\right )}{32 \, {\left (b \cos \left (6 \, b x + 6 \, a\right )^{2} + b \cos \left (4 \, b x + 4 \, a\right )^{2} + b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (6 \, b x + 6 \, a\right )^{2} + b \sin \left (4 \, b x + 4 \, a\right )^{2} - 2 \, b \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + b \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, {\left (b \cos \left (4 \, b x + 4 \, a\right ) - b \cos \left (2 \, b x + 2 \, a\right ) - b\right )} \cos \left (6 \, b x + 6 \, a\right ) - 2 \, {\left (b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \cos \left (4 \, b x + 4 \, a\right ) + 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + 2 \, {\left (b \sin \left (4 \, b x + 4 \, a\right ) - b \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.56, size = 85, normalized size = 1.73 \begin {gather*} \frac {3 \, \cos \left (b x + a\right )^{2} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, \cos \left (b x + a\right )^{2} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 6 \, \cos \left (b x + a\right )^{2} + 2}{32 \, b \cos \left (b x + a\right )^{2} \sin \left (b x + a\right )} \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 [A]
time = 0.48, size = 63, normalized size = 1.29 \begin {gather*} -\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{2} - 2\right )}}{\sin \left (b x + a\right )^{3} - \sin \left (b x + a\right )} - 3 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 3 \, \log \left (-\sin \left (b x + a\right ) + 1\right )}{32 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.12, size = 48, normalized size = 0.98 \begin {gather*} \frac {3\,\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{16\,b}+\frac {\frac {3\,{\sin \left (a+b\,x\right )}^2}{16}-\frac {1}{8}}{b\,\left (\sin \left (a+b\,x\right )-{\sin \left (a+b\,x\right )}^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________