Optimal. Leaf size=13 \[ \frac {\tan (a+b x)}{4 b} \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4288, 3767, 8} \[ \frac {\tan (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 4288
Rubi steps
\begin {align*} \int \csc ^2(2 a+2 b x) \sin ^2(a+b x) \, dx &=\frac {1}{4} \int \sec ^2(a+b x) \, dx\\ &=-\frac {\operatorname {Subst}(\int 1 \, dx,x,-\tan (a+b x))}{4 b}\\ &=\frac {\tan (a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \frac {\tan (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 19, normalized size = 1.46 \[ \frac {\sin \left (b x + a\right )}{4 \, b \cos \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.52, size = 152, normalized size = 11.69 \[ -\frac {\tan \left (\frac {1}{2} \, a\right )^{12} + 6 \, \tan \left (\frac {1}{2} \, a\right )^{10} + 15 \, \tan \left (\frac {1}{2} \, a\right )^{8} + 20 \, \tan \left (\frac {1}{2} \, a\right )^{6} + 15 \, \tan \left (\frac {1}{2} \, a\right )^{4} + 6 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 1}{8 \, {\left (6 \, \tan \left (b x + 4 \, a\right ) \tan \left (\frac {1}{2} \, a\right )^{5} - \tan \left (\frac {1}{2} \, a\right )^{6} - 20 \, \tan \left (b x + 4 \, a\right ) \tan \left (\frac {1}{2} \, a\right )^{3} + 15 \, \tan \left (\frac {1}{2} \, a\right )^{4} + 6 \, \tan \left (b x + 4 \, a\right ) \tan \left (\frac {1}{2} \, a\right ) - 15 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 1\right )} {\left (3 \, \tan \left (\frac {1}{2} \, a\right )^{5} - 10 \, \tan \left (\frac {1}{2} \, a\right )^{3} + 3 \, \tan \left (\frac {1}{2} \, a\right )\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.07, size = 12, normalized size = 0.92 \[ \frac {\tan \left (b x +a \right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 53, normalized size = 4.08 \[ \frac {\sin \left (2 \, b x + 2 \, a\right )}{2 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 11, normalized size = 0.85 \[ \frac {\mathrm {tan}\left (a+b\,x\right )}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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