Optimal. Leaf size=91 \[ \frac {f x}{4 a d}+\frac {f \cos (c+d x)}{a d^2}+\frac {(e+f x) \sin (c+d x)}{a d}-\frac {f \cos (c+d x) \sin (c+d x)}{4 a d^2}-\frac {(e+f x) \sin ^2(c+d x)}{2 a d} \]
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Rubi [A]
time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4619, 3377,
2718, 4489, 2715, 8} \begin {gather*} \frac {f \cos (c+d x)}{a d^2}-\frac {f \sin (c+d x) \cos (c+d x)}{4 a d^2}-\frac {(e+f x) \sin ^2(c+d x)}{2 a d}+\frac {(e+f x) \sin (c+d x)}{a d}+\frac {f x}{4 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2718
Rule 3377
Rule 4489
Rule 4619
Rubi steps
\begin {align*} \int \frac {(e+f x) \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x) \cos (c+d x) \, dx}{a}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x) \, dx}{a}\\ &=\frac {(e+f x) \sin (c+d x)}{a d}-\frac {(e+f x) \sin ^2(c+d x)}{2 a d}+\frac {f \int \sin ^2(c+d x) \, dx}{2 a d}-\frac {f \int \sin (c+d x) \, dx}{a d}\\ &=\frac {f \cos (c+d x)}{a d^2}+\frac {(e+f x) \sin (c+d x)}{a d}-\frac {f \cos (c+d x) \sin (c+d x)}{4 a d^2}-\frac {(e+f x) \sin ^2(c+d x)}{2 a d}+\frac {f \int 1 \, dx}{4 a d}\\ &=\frac {f x}{4 a d}+\frac {f \cos (c+d x)}{a d^2}+\frac {(e+f x) \sin (c+d x)}{a d}-\frac {f \cos (c+d x) \sin (c+d x)}{4 a d^2}-\frac {(e+f x) \sin ^2(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 52, normalized size = 0.57 \begin {gather*} \frac {-f \cos (c+d x) (-4+\sin (c+d x))+d (e+f x) (\cos (2 (c+d x))+4 \sin (c+d x))}{4 a d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 113, normalized size = 1.24
method | result | size |
risch | \(\frac {f \cos \left (d x +c \right )}{a \,d^{2}}+\frac {\left (f x +e \right ) \sin \left (d x +c \right )}{a d}+\frac {\left (f x +e \right ) \cos \left (2 d x +2 c \right )}{4 a d}-\frac {f \sin \left (2 d x +2 c \right )}{8 a \,d^{2}}\) | \(74\) |
derivativedivides | \(\frac {-\frac {f c \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\left (\cos ^{2}\left (d x +c \right )\right ) d e}{2}-f \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-\sin \left (d x +c \right ) c f +\sin \left (d x +c \right ) d e +f \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2} a}\) | \(113\) |
default | \(\frac {-\frac {f c \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\left (\cos ^{2}\left (d x +c \right )\right ) d e}{2}-f \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-\sin \left (d x +c \right ) c f +\sin \left (d x +c \right ) d e +f \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2} a}\) | \(113\) |
norman | \(\frac {\frac {2 f}{a \,d^{2}}+\frac {\left (2 d e +2 f \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {\left (2 d e +4 f \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {f x}{4 a d}+\frac {5 f \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \,d^{2}}+\frac {7 f \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \,d^{2}}+\frac {\left (4 d e +f \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \,d^{2}}+\frac {\left (4 d e +3 f \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a \,d^{2}}+\frac {9 f x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {3 f x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}+\frac {11 f x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}+\frac {11 f x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}+\frac {3 f x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}+\frac {9 f x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}+\frac {f x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(337\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 114, normalized size = 1.25 \begin {gather*} -\frac {\frac {4 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} e}{a} - \frac {4 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} c f}{a d} - \frac {{\left (2 \, {\left (d x + c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 8 \, \cos \left (d x + c\right ) - \sin \left (2 \, d x + 2 \, c\right )\right )} f}{a d}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 69, normalized size = 0.76 \begin {gather*} -\frac {d f x - 2 \, {\left (d f x + d e\right )} \cos \left (d x + c\right )^{2} - 4 \, f \cos \left (d x + c\right ) - {\left (4 \, d f x - f \cos \left (d x + c\right ) + 4 \, d e\right )} \sin \left (d x + c\right )}{4 \, a d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 724 vs.
\(2 (78) = 156\).
time = 2.94, size = 724, normalized size = 7.96 \begin {gather*} \begin {cases} \frac {8 d e \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} - \frac {8 d e \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {8 d e \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {d f x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {8 d f x \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} - \frac {6 d f x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {8 d f x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {d f x}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {2 f \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {8 f \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} - \frac {2 f \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} + \frac {8 f}{4 a d^{2} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8 a d^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4 a d^{2}} & \text {for}\: d \neq 0 \\\frac {\left (e x + \frac {f x^{2}}{2}\right ) \cos ^{3}{\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 26480 vs.
\(2 (87) = 174\).
time = 10.74, size = 26480, normalized size = 290.99 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.04, size = 84, normalized size = 0.92 \begin {gather*} -\frac {\frac {f\,\sin \left (2\,c+2\,d\,x\right )}{2}+8\,f\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,d\,e\,\sin \left (c+d\,x\right )+2\,d\,e\,{\sin \left (c+d\,x\right )}^2-4\,d\,f\,x\,\sin \left (c+d\,x\right )+d\,f\,x\,\left (2\,{\sin \left (c+d\,x\right )}^2-1\right )}{4\,a\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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