Optimal. Leaf size=195 \[ -\frac{2 b^2 \left (3 a^2-4 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d \sqrt{a-b} \sqrt{a+b}}-\frac{\left (a^2-12 b^2\right ) \tan (c+d x)}{3 a^4 d}+\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{2 b \tan (c+d x) \sec (c+d x)}{a^3 d}+\frac{4 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{a d (a+b \cos (c+d x))} \]
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Rubi [A] time = 0.916285, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3056, 3055, 3001, 3770, 2659, 205} \[ -\frac{2 b^2 \left (3 a^2-4 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d \sqrt{a-b} \sqrt{a+b}}-\frac{\left (a^2-12 b^2\right ) \tan (c+d x)}{3 a^4 d}+\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{2 b \tan (c+d x) \sec (c+d x)}{a^3 d}+\frac{4 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{a d (a+b \cos (c+d x))} \]
Antiderivative was successfully verified.
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Rule 3056
Rule 3055
Rule 3001
Rule 3770
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (1-\cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx &=-\frac{\sec ^2(c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}+\frac{\int \frac{\left (4 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac{4 \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}-\frac{\sec ^2(c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}+\frac{\int \frac{\left (-12 b \left (a^2-b^2\right )-a \left (a^2-b^2\right ) \cos (c+d x)+8 b \left (a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{3 a^2 \left (a^2-b^2\right )}\\ &=-\frac{2 b \sec (c+d x) \tan (c+d x)}{a^3 d}+\frac{4 \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}-\frac{\sec ^2(c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}+\frac{\int \frac{\left (-2 \left (a^4-13 a^2 b^2+12 b^4\right )+4 a b \left (a^2-b^2\right ) \cos (c+d x)-12 b^2 \left (a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )}\\ &=-\frac{\left (a^2-12 b^2\right ) \tan (c+d x)}{3 a^4 d}-\frac{2 b \sec (c+d x) \tan (c+d x)}{a^3 d}+\frac{4 \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}-\frac{\sec ^2(c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}+\frac{\int \frac{\left (6 b \left (a^4-5 a^2 b^2+4 b^4\right )-12 a b^2 \left (a^2-b^2\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )}\\ &=-\frac{\left (a^2-12 b^2\right ) \tan (c+d x)}{3 a^4 d}-\frac{2 b \sec (c+d x) \tan (c+d x)}{a^3 d}+\frac{4 \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}-\frac{\sec ^2(c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}+\frac{\left (b \left (a^2-4 b^2\right )\right ) \int \sec (c+d x) \, dx}{a^5}-\frac{\left (b^2 \left (3 a^2-4 b^2\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{a^5}\\ &=\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{\left (a^2-12 b^2\right ) \tan (c+d x)}{3 a^4 d}-\frac{2 b \sec (c+d x) \tan (c+d x)}{a^3 d}+\frac{4 \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}-\frac{\sec ^2(c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}-\frac{\left (2 b^2 \left (3 a^2-4 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 d}\\ &=-\frac{2 b^2 \left (3 a^2-4 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 \sqrt{a-b} \sqrt{a+b} d}+\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{\left (a^2-12 b^2\right ) \tan (c+d x)}{3 a^4 d}-\frac{2 b \sec (c+d x) \tan (c+d x)}{a^3 d}+\frac{4 \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}-\frac{\sec ^2(c+d x) \tan (c+d x)}{a d (a+b \cos (c+d x))}\\ \end{align*}
Mathematica [B] time = 6.21069, size = 475, normalized size = 2.44 \[ \frac{b^3 \sin (c+d x)}{a^4 d (a+b \cos (c+d x))}+\frac{9 b^2 \sin \left (\frac{1}{2} (c+d x)\right )-a^2 \sin \left (\frac{1}{2} (c+d x)\right )}{3 a^4 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{9 b^2 \sin \left (\frac{1}{2} (c+d x)\right )-a^2 \sin \left (\frac{1}{2} (c+d x)\right )}{3 a^4 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{2 b^2 \left (3 a^2-4 b^2\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{a^5 d \sqrt{b^2-a^2}}+\frac{\left (4 b^3-a^2 b\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 d}+\frac{\left (a^2 b-4 b^3\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 d}+\frac{a-6 b}{12 a^3 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{6 b-a}{12 a^3 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{6 a^2 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{6 a^2 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 458, normalized size = 2.4 \begin{align*} 2\,{\frac{{b}^{3}\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}- \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) }}-6\,{\frac{{b}^{2}}{d{a}^{3}\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+8\,{\frac{{b}^{4}}{d{a}^{5}\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-{\frac{1}{3\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{2\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-{\frac{b}{d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-{\frac{b}{d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-3\,{\frac{{b}^{2}}{d{a}^{4} \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }}-{\frac{b}{d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+4\,{\frac{{b}^{3}\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }{d{a}^{5}}}-{\frac{1}{3\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{2\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{b}{d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{b}{d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-3\,{\frac{{b}^{2}}{d{a}^{4} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}+{\frac{b}{d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-4\,{\frac{{b}^{3}\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }{d{a}^{5}}} \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 = 2.78843, size = 1886, normalized size = 9.67 \begin{align*} \text{result too large to display} \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 = 1.50607, size = 427, normalized size = 2.19 \begin{align*} \frac{\frac{6 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a + b\right )} a^{4}} + \frac{3 \,{\left (a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac{3 \,{\left (a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} + \frac{6 \,{\left (3 \, a^{2} b^{2} - 4 \, b^{4}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{\sqrt{a^{2} - b^{2}} a^{5}} - \frac{2 \,{\left (3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 18 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 9 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3} a^{4}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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