Optimal. Leaf size=53 \[ \frac {\tan ^3(a+b x)}{3 b}+\frac {3 \tan (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}-\frac {3 \cot (a+b x)}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2620, 270} \[ \frac {\tan ^3(a+b x)}{3 b}+\frac {3 \tan (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}-\frac {3 \cot (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2620
Rubi steps
\begin {align*} \int \csc ^4(a+b x) \sec ^4(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^4} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (3+\frac {1}{x^4}+\frac {3}{x^2}+x^2\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac {3 \cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}+\frac {3 \tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 0.81 \[ 16 \left (-\frac {\cot (2 (a+b x))}{3 b}-\frac {\cot (2 (a+b x)) \csc ^2(2 (a+b x))}{6 b}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 66, normalized size = 1.25 \[ -\frac {16 \, \cos \left (b x + a\right )^{6} - 24 \, \cos \left (b x + a\right )^{4} + 6 \, \cos \left (b x + a\right )^{2} + 1}{3 \, {\left (b \cos \left (b x + a\right )^{5} - b \cos \left (b x + a\right )^{3}\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 31, normalized size = 0.58 \[ -\frac {8 \, {\left (3 \, \tan \left (2 \, b x + 2 \, a\right )^{2} + 1\right )}}{3 \, b \tan \left (2 \, b x + 2 \, a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 68, normalized size = 1.28 \[ \frac {\frac {1}{3 \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{3}}-\frac {2}{3 \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )}+\frac {8}{3 \sin \left (b x +a \right ) \cos \left (b x +a \right )}-\frac {16 \cot \left (b x +a \right )}{3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 44, normalized size = 0.83 \[ \frac {\tan \left (b x + a\right )^{3} - \frac {9 \, \tan \left (b x + a\right )^{2} + 1}{\tan \left (b x + a\right )^{3}} + 9 \, \tan \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 45, normalized size = 0.85 \[ -\frac {-{\mathrm {tan}\left (a+b\,x\right )}^6-9\,{\mathrm {tan}\left (a+b\,x\right )}^4+9\,{\mathrm {tan}\left (a+b\,x\right )}^2+1}{3\,b\,{\mathrm {tan}\left (a+b\,x\right )}^3} \]
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 \[ \int \frac {\sec ^{4}{\left (a + b x \right )}}{\sin ^{4}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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