Optimal. Leaf size=79 \[ -\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{a \tanh ^{-1}(\sin (c+d x))}{b^2 d}+\frac{\sec (c+d x)}{b d} \]
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Rubi [A] time = 0.0944538, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3510, 3486, 3770, 3509, 206} \[ -\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right )}{b^2 d}-\frac{a \tanh ^{-1}(\sin (c+d x))}{b^2 d}+\frac{\sec (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 3510
Rule 3486
Rule 3770
Rule 3509
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{a+b \tan (c+d x)} \, dx &=-\frac{\int \sec (c+d x) (a-b \tan (c+d x)) \, dx}{b^2}+\frac{\left (a^2+b^2\right ) \int \frac{\sec (c+d x)}{a+b \tan (c+d x)} \, dx}{b^2}\\ &=\frac{\sec (c+d x)}{b d}-\frac{a \int \sec (c+d x) \, dx}{b^2}-\frac{\left (a^2+b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,\cos (c+d x) (b-a \tan (c+d x))\right )}{b^2 d}\\ &=-\frac{a \tanh ^{-1}(\sin (c+d x))}{b^2 d}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right )}{b^2 d}+\frac{\sec (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.141189, size = 109, normalized size = 1.38 \[ \frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{a^2+b^2}}\right )+a \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+b \sec (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 174, normalized size = 2.2 \begin{align*}{\frac{1}{bd} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{a}{{b}^{2}d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+2\,{\frac{{a}^{2}}{{b}^{2}d\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }+2\,{\frac{1}{d\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }-{\frac{1}{bd} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{a}{{b}^{2}d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \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 [B] time = 2.10969, size = 471, normalized size = 5.96 \begin{align*} -\frac{a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) - \sqrt{a^{2} + b^{2}} \cos \left (d x + c\right ) \log \left (-\frac{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 2 \, b}{2 \, b^{2} d \cos \left (d x + c\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 ^{3}{\left (c + d x \right )}}{a + b \tan{\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 = 1.65507, size = 184, normalized size = 2.33 \begin{align*} -\frac{\frac{a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{b^{2}} - \frac{a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{b^{2}} + \frac{\sqrt{a^{2} + b^{2}} \log \left (\frac{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{b^{2}} + \frac{2}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} b}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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