Optimal. Leaf size=49 \[ \frac {3 \tan (c+d x)}{2 a d}-\frac {\sin ^2(c+d x) \tan (c+d x)}{2 a d}-\frac {3 x}{2 a} \]
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Rubi [A] time = 0.08, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3175, 2591, 288, 321, 203} \[ \frac {3 \tan (c+d x)}{2 a d}-\frac {\sin ^2(c+d x) \tan (c+d x)}{2 a d}-\frac {3 x}{2 a} \]
Antiderivative was successfully verified.
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Rule 203
Rule 288
Rule 321
Rule 2591
Rule 3175
Rubi steps
\begin {align*} \int \frac {\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx &=\frac {\int \sin ^2(c+d x) \tan ^2(c+d x) \, dx}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{a d}\\ &=-\frac {\sin ^2(c+d x) \tan (c+d x)}{2 a d}+\frac {3 \operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 a d}\\ &=\frac {3 \tan (c+d x)}{2 a d}-\frac {\sin ^2(c+d x) \tan (c+d x)}{2 a d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 a d}\\ &=-\frac {3 x}{2 a}+\frac {3 \tan (c+d x)}{2 a d}-\frac {\sin ^2(c+d x) \tan (c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 34, normalized size = 0.69 \[ \frac {-6 (c+d x)+\sin (2 (c+d x))+4 \tan (c+d x)}{4 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 45, normalized size = 0.92 \[ -\frac {3 \, d x \cos \left (d x + c\right ) - {\left (\cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right )}{2 \, a d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 50, normalized size = 1.02 \[ -\frac {\frac {3 \, {\left (d x + c\right )}}{a} - \frac {2 \, \tan \left (d x + c\right )}{a} - \frac {\tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )} a}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 56, normalized size = 1.14 \[ \frac {\tan \left (d x +c \right )}{a d}+\frac {\tan \left (d x +c \right )}{2 a d \left (\tan ^{2}\left (d x +c \right )+1\right )}-\frac {3 \arctan \left (\tan \left (d x +c \right )\right )}{2 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 49, normalized size = 1.00 \[ -\frac {\frac {3 \, {\left (d x + c\right )}}{a} - \frac {\tan \left (d x + c\right )}{a \tan \left (d x + c\right )^{2} + a} - \frac {2 \, \tan \left (d x + c\right )}{a}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.53, size = 45, normalized size = 0.92 \[ \frac {\mathrm {tan}\left (c+d\,x\right )}{2\,d\,\left (a\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}-\frac {3\,x}{2\,a}+\frac {\mathrm {tan}\left (c+d\,x\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.75, size = 502, normalized size = 10.24 \[ \begin {cases} - \frac {3 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} - \frac {3 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} + \frac {3 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} + \frac {3 d x}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} - \frac {6 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} - \frac {4 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} - \frac {6 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\relax (c )}}{- a \sin ^{2}{\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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