Optimal. Leaf size=38 \[ -\frac {\csc ^3(a+b x)}{3 b}-\frac {\csc (a+b x)}{b}+\frac {\tanh ^{-1}(\sin (a+b x))}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2621, 302, 207} \[ -\frac {\csc ^3(a+b x)}{3 b}-\frac {\csc (a+b x)}{b}+\frac {\tanh ^{-1}(\sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 207
Rule 302
Rule 2621
Rubi steps
\begin {align*} \int \csc ^4(a+b x) \sec (a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac {\tanh ^{-1}(\sin (a+b x))}{b}-\frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 31, normalized size = 0.82 \[ -\frac {\csc ^3(a+b x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\sin ^2(a+b x)\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 94, normalized size = 2.47 \[ \frac {3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 6 \, \cos \left (b x + a\right )^{2} + 8}{6 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 52, normalized size = 1.37 \[ -\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 3 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 3 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 46, normalized size = 1.21 \[ -\frac {1}{3 \sin \left (b x +a \right )^{3} b}-\frac {1}{b \sin \left (b x +a \right )}+\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 50, normalized size = 1.32 \[ -\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 3 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\sin \left (b x + a\right ) - 1\right )}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 32, normalized size = 0.84 \[ \frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )-\frac {{\sin \left (a+b\,x\right )}^2+\frac {1}{3}}{{\sin \left (a+b\,x\right )}^3}}{b} \]
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 \[ \int \frac {\sec {\left (a + b x \right )}}{\sin ^{4}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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