Optimal. Leaf size=260 \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{b^4 d \sqrt{a^2+b^2}}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{\tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{2 b^2 d \sqrt{a^2+b^2}}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac{2 a}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{b \cos (c+d x)-a \sin (c+d x)}{2 b^2 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{\sec (c+d x)}{b^3 d} \]
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Rubi [A] time = 0.28615, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3106, 3076, 3074, 206, 3104, 3770, 3094} \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{b^4 d \sqrt{a^2+b^2}}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{\tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{2 b^2 d \sqrt{a^2+b^2}}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac{2 a}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{b \cos (c+d x)-a \sin (c+d x)}{2 b^2 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{\sec (c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
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Rule 3106
Rule 3076
Rule 3074
Rule 206
Rule 3104
Rule 3770
Rule 3094
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx &=\frac{\int \frac{\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^2}-\frac{(2 a) \int \frac{\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx}{b^2}+\frac{\left (a^2+b^2\right ) \int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx}{b^2}\\ &=\frac{\sec (c+d x)}{b^3 d}-\frac{b \cos (c+d x)-a \sin (c+d x)}{2 b^2 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{2 a}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{a \int \sec (c+d x) \, dx}{b^4}-\frac{(2 a) \int \sec (c+d x) \, dx}{b^4}+\frac{\left (2 a^2\right ) \int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^4}+\frac{\int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{2 b^2}+\frac{\left (a^2+b^2\right ) \int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^4}\\ &=-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac{\sec (c+d x)}{b^3 d}-\frac{b \cos (c+d x)-a \sin (c+d x)}{2 b^2 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{2 a}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{b^4 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{2 b^2 d}-\frac{\left (a^2+b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{b^4 d}\\ &=-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac{2 a^2 \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{b^4 \sqrt{a^2+b^2} d}-\frac{\tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{2 b^2 \sqrt{a^2+b^2} d}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{b^4 d}+\frac{\sec (c+d x)}{b^3 d}-\frac{b \cos (c+d x)-a \sin (c+d x)}{2 b^2 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{2 a}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 2.44481, size = 396, normalized size = 1.52 \[ \frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (\frac{b^2 \left (a^2+b^2\right ) \sin (c+d x)}{a}+\frac{6 \left (2 a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}}+\frac{2 b \sin \left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}-\frac{2 b \sin \left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+2 b (a \cos (c+d x)+b \sin (c+d x))^2+\frac{b (2 a-b) (2 a+b) (a \cos (c+d x)+b \sin (c+d x))}{a}+6 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2-6 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2\right )}{2 b^4 d (a+b \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.29, size = 611, normalized size = 2.4 \begin{align*} \text{result too large to display} \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 = 0.718939, size = 1175, normalized size = 4.52 \begin{align*} \frac{4 \, a^{2} b^{3} + 4 \, b^{5} + 6 \,{\left (2 \, a^{4} b + a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 18 \,{\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \,{\left ({\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) +{\left (2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt{a^{2} + b^{2}} \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 ) - 6 \,{\left ({\left (a^{5} - a b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) +{\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 6 \,{\left ({\left (a^{5} - a b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) +{\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{4 \,{\left ({\left (a^{4} b^{4} - b^{8}\right )} d \cos \left (d x + c\right )^{3} + 2 \,{\left (a^{3} b^{5} + a b^{7}\right )} d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) +{\left (a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.61047, size = 424, normalized size = 1.63 \begin{align*} -\frac{\frac{6 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} - \frac{6 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} + \frac{3 \,{\left (2 \, a^{2} + b^{2}\right )} \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 )}{\sqrt{a^{2} + b^{2}} b^{4}} + \frac{4}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} b^{3}} + \frac{2 \,{\left (3 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 2 \, b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 13 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 4 \, a^{4} + a^{2} b^{2}\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a\right )}^{2} a^{2} b^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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