Optimal. Leaf size=16 \[ \frac {\tanh ^{-1}\left (\sin ^2(c+d x)\right )}{2 d} \]
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Rubi [A] time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {275, 206} \[ \frac {\tanh ^{-1}\left (\sin ^2(c+d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 275
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{1-x^4} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {\tanh ^{-1}\left (\sin ^2(c+d x)\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 30, normalized size = 1.88 \[ \frac {\log \left (2-\cos ^2(c+d x)\right )-2 \log (\cos (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 30, normalized size = 1.88 \[ \frac {\log \left (-\cos \left (d x + c\right )^{2} + 2\right ) - 2 \, \log \left (-\cos \left (d x + c\right )\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 79, normalized size = 4.94 \[ -\frac {2 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \log \left ({\left | -\frac {6 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1 \right |}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 19, normalized size = 1.19 \[ \frac {\ln \left (2 \left (\sec ^{2}\left (d x +c \right )\right )-1\right )}{4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 39, normalized size = 2.44 \[ \frac {\log \left (\sin \left (d x + c\right )^{2} + 1\right ) - \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 14, normalized size = 0.88 \[ \frac {\mathrm {atanh}\left ({\sin \left (c+d\,x\right )}^2\right )}{2\,d} \]
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 \[ \int \frac {\sec {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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