Optimal. Leaf size=43 \[ \frac{\log \left (a+b \sin ^2(c+d x)\right )}{2 d (a+b)}-\frac{\log (\cos (c+d x))}{d (a+b)} \]
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Rubi [A] time = 0.0385448, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3194, 36, 31} \[ \frac{\log \left (a+b \sin ^2(c+d x)\right )}{2 d (a+b)}-\frac{\log (\cos (c+d x))}{d (a+b)} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{\tan (c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) (a+b x)} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sin ^2(c+d x)\right )}{2 (a+b) d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,\sin ^2(c+d x)\right )}{2 (a+b) d}\\ &=-\frac{\log (\cos (c+d x))}{(a+b) d}+\frac{\log \left (a+b \sin ^2(c+d x)\right )}{2 (a+b) d}\\ \end{align*}
Mathematica [A] time = 0.0323504, size = 37, normalized size = 0.86 \[ \frac{\log \left (a-b \cos ^2(c+d x)+b\right )-2 \log (\cos (c+d x))}{2 a d+2 b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 47, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( \cos \left ( dx+c \right ) \right ) }{ \left ( a+b \right ) d}}+{\frac{\ln \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b \right ) }{2\, \left ( a+b \right ) d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983951, size = 58, normalized size = 1.35 \begin{align*} \frac{\frac{\log \left (b \sin \left (d x + c\right )^{2} + a\right )}{a + b} - \frac{\log \left (\sin \left (d x + c\right )^{2} - 1\right )}{a + b}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87446, size = 99, normalized size = 2.3 \begin{align*} \frac{\log \left (-b \cos \left (d x + c\right )^{2} + a + b\right ) - 2 \, \log \left (-\cos \left (d x + c\right )\right )}{2 \,{\left (a + b\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1399, size = 149, normalized size = 3.47 \begin{align*} \frac{\frac{\log \left (a - \frac{2 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{4 \, b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a + b} - \frac{2 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a + b}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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