Optimal. Leaf size=40 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d \sqrt{a-b}} \]
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Rubi [A] time = 0.0461854, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3676, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d \sqrt{a-b}} \]
Antiderivative was successfully verified.
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Rule 3676
Rule 208
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{a+b \tan ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a-(a-b) x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{a-b} d}\\ \end{align*}
Mathematica [A] time = 0.0472236, size = 40, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d \sqrt{a-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 36, normalized size = 0.9 \begin{align*}{\frac{1}{d}{\it Artanh} \left ({ \left ( a-b \right ) \sin \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32869, size = 279, normalized size = 6.98 \begin{align*} \left [\frac{\log \left (-\frac{{\left (a - b\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt{a^{2} - a b} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right )}{2 \, \sqrt{a^{2} - a b} d}, -\frac{\sqrt{-a^{2} + a b} \arctan \left (\frac{\sqrt{-a^{2} + a b} \sin \left (d x + c\right )}{a}\right )}{{\left (a^{2} - a b\right )} d}\right ] \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{\sec{\left (c + d x \right )}}{a + b \tan ^{2}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.72895, size = 63, normalized size = 1.58 \begin{align*} -\frac{\arctan \left (\frac{a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt{-a^{2} + a b}}\right )}{\sqrt{-a^{2} + a b} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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