Optimal. Leaf size=162 \[ -\frac{2 b \left (-2 a^2 b^2+4 a^4+b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{b^2 \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{x}{a^2} \]
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Rubi [A] time = 0.339561, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {4042, 3923, 4060, 3919, 3831, 2659, 208} \[ -\frac{2 b \left (-2 a^2 b^2+4 a^4+b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{b^2 \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{x}{a^2} \]
Antiderivative was successfully verified.
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Rule 4042
Rule 3923
Rule 4060
Rule 3919
Rule 3831
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx &=-\int \frac{-a+b \sec (c+d x)}{(a+b \sec (c+d x))^3} \, dx\\ &=\frac{b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\int \frac{2 a \left (a^2-b^2\right )-4 a^2 b \sec (c+d x)+2 a b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac{b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\int \frac{-2 a \left (a^2-b^2\right )^2+6 a^4 b \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{x}{a^2}+\frac{b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (b \left (4 a^4-2 a^2 b^2+b^4\right )\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )^2}\\ &=\frac{x}{a^2}+\frac{b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (4 a^4-2 a^2 b^2+b^4\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{a^2 \left (a^2-b^2\right )^2}\\ &=\frac{x}{a^2}+\frac{b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (2 \left (4 a^4-2 a^2 b^2+b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^2 \left (a^2-b^2\right )^2 d}\\ &=\frac{x}{a^2}-\frac{2 b \left (4 a^4-2 a^2 b^2+b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^2 (a-b)^{5/2} (a+b)^{5/2} d}+\frac{b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.775569, size = 223, normalized size = 1.38 \[ \frac{\sec ^2(c+d x) (a \cos (c+d x)+b) (a-b \sec (c+d x)) \left (\frac{a b^2 \left (5 a^2-2 b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}+\frac{2 b \left (-2 a^2 b^2+4 a^4+b^4\right ) (a \cos (c+d x)+b)^2 \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac{a b^3 \sin (c+d x)}{(b-a) (a+b)}+(c+d x) (a \cos (c+d x)+b)^2\right )}{a^2 d (a \cos (c+d x)-b) (a+b \sec (c+d x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.102, size = 659, normalized size = 4.1 \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 = 0.684094, size = 1925, normalized size = 11.88 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a - b \sec{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30622, size = 428, normalized size = 2.64 \begin{align*} \frac{\frac{2 \,{\left (4 \, a^{4} b - 2 \, a^{2} b^{3} + b^{5}\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 )}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt{-a^{2} + b^{2}}} + \frac{d x + c}{a^{2}} - \frac{2 \,{\left (5 \, a^{3} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, a b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 5 \, a^{3} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 4 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, a b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\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 )}^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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