Optimal. Leaf size=61 \[ -\frac {a \cos (c+d x)}{d}+\frac {2 b E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d}-\frac {c \sin (c+d x) \cos (c+d x)}{2 d}+\frac {c x}{2} \]
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Rubi [A] time = 0.29, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4395, 4401, 2639, 2638, 2635, 8} \[ -\frac {a \cos (c+d x)}{d}+\frac {2 b E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d}-\frac {c \sin (c+d x) \cos (c+d x)}{2 d}+\frac {c x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 2639
Rule 4395
Rule 4401
Rubi steps
\begin {align*} \int \sin (c+d x) \left (a+\frac {b}{\sqrt {\sin (c+d x)}}+c \sin (c+d x)\right ) \, dx &=\int \sqrt {\sin (c+d x)} \left (b+a \sqrt {\sin (c+d x)}+c \sin ^{\frac {3}{2}}(c+d x)\right ) \, dx\\ &=\int \left (b \sqrt {\sin (c+d x)}+a \sin (c+d x)+c \sin ^2(c+d x)\right ) \, dx\\ &=a \int \sin (c+d x) \, dx+b \int \sqrt {\sin (c+d x)} \, dx+c \int \sin ^2(c+d x) \, dx\\ &=-\frac {a \cos (c+d x)}{d}+\frac {2 b E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d}-\frac {c \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} c \int 1 \, dx\\ &=\frac {c x}{2}-\frac {a \cos (c+d x)}{d}+\frac {2 b E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d}-\frac {c \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 55, normalized size = 0.90 \[ \frac {-4 a \cos (c+d x)-8 b E\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )+c (-\sin (2 (c+d x))+2 c+2 d x)}{4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-c \cos \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + b \sqrt {\sin \left (d x + c\right )} + c, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c \sin \left (d x + c\right ) + a + \frac {b}{\sqrt {\sin \left (d x + c\right )}}\right )} \sin \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 136, normalized size = 2.23 \[ c x -\frac {a \cos \left (d x +c \right )}{d}-\frac {c \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {b \sqrt {1+\sin \left (d x +c \right )}\, \sqrt {-2 \sin \left (d x +c \right )+2}\, \sqrt {-\sin \left (d x +c \right )}\, \left (2 \EllipticE \left (\sqrt {1+\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1+\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (d x +c \right ) \sqrt {\sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, c d x - 4 \, a \cos \left (d x + c\right ) - 2 \, d \int \frac {{\left ({\left (b \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - b \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - b \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), -\cos \left (d x + c\right ) + 1\right )\right ) - {\left (b \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - b \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + b \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), -\cos \left (d x + c\right ) + 1\right )\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right )\right ) + {\left ({\left (b \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - b \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + b \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), -\cos \left (d x + c\right ) + 1\right )\right ) + {\left (b \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - b \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - b \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), -\cos \left (d x + c\right ) + 1\right )\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right )\right )}{{\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )}^{\frac {1}{4}} {\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )}^{\frac {1}{4}}}\,{d x} - c \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.25, size = 51, normalized size = 0.84 \[ \frac {c\,x}{2}-\frac {c\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}-\frac {a\,\cos \left (c+d\,x\right )}{d}+\frac {2\,b\,\mathrm {E}\left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\middle |2\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sqrt {\sin {\left (c + d x \right )}} + b + c \sin ^{\frac {3}{2}}{\left (c + d x \right )}\right ) \sqrt {\sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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