Optimal. Leaf size=140 \[ \frac{\left (a^2+b^2\right ) \sec (c+d x)}{b^3 d}-\frac{a \left (2 a^2+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{\left (a^2+b^2\right )^{3/2} \tanh ^{-1}\left (\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{\sec ^3(c+d x)}{3 b d} \]
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Rubi [A] time = 0.198018, antiderivative size = 152, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3510, 3486, 3768, 3770, 3509, 206} \[ \frac{\left (a^2+b^2\right ) \sec (c+d x)}{b^3 d}-\frac{a \left (a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac{\left (a^2+b^2\right )^{3/2} \tanh ^{-1}\left (\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right )}{b^4 d}-\frac{a \tanh ^{-1}(\sin (c+d x))}{2 b^2 d}-\frac{a \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{\sec ^3(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
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Rule 3510
Rule 3486
Rule 3768
Rule 3770
Rule 3509
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x)}{a+b \tan (c+d x)} \, dx &=-\frac{\int \sec ^3(c+d x) (a-b \tan (c+d x)) \, dx}{b^2}+\frac{\left (a^2+b^2\right ) \int \frac{\sec ^3(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2}\\ &=\frac{\sec ^3(c+d x)}{3 b d}-\frac{a \int \sec ^3(c+d x) \, dx}{b^2}-\frac{\left (a^2+b^2\right ) \int \sec (c+d x) (a-b \tan (c+d x)) \, dx}{b^4}+\frac{\left (a^2+b^2\right )^2 \int \frac{\sec (c+d x)}{a+b \tan (c+d x)} \, dx}{b^4}\\ &=\frac{\left (a^2+b^2\right ) \sec (c+d x)}{b^3 d}+\frac{\sec ^3(c+d x)}{3 b d}-\frac{a \sec (c+d x) \tan (c+d x)}{2 b^2 d}-\frac{a \int \sec (c+d x) \, dx}{2 b^2}-\frac{\left (a \left (a^2+b^2\right )\right ) \int \sec (c+d x) \, dx}{b^4}-\frac{\left (a^2+b^2\right )^2 \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^4 d}\\ &=-\frac{a \tanh ^{-1}(\sin (c+d x))}{2 b^2 d}-\frac{a \left (a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac{\left (a^2+b^2\right )^{3/2} \tanh ^{-1}\left (\frac{\cos (c+d x) (b-a \tan (c+d x))}{\sqrt{a^2+b^2}}\right )}{b^4 d}+\frac{\left (a^2+b^2\right ) \sec (c+d x)}{b^3 d}+\frac{\sec ^3(c+d x)}{3 b d}-\frac{a \sec (c+d x) \tan (c+d x)}{2 b^2 d}\\ \end{align*}
Mathematica [B] time = 1.90942, size = 321, normalized size = 2.29 \[ \frac{48 \left (a^2+b^2\right )^{3/2} \tanh ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{a^2+b^2}}\right )+\sec ^3(c+d x) \left (12 b \left (a^2+b^2\right ) \cos (2 (c+d x))+9 a \left (2 a^2+3 b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+12 a^2 b+6 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-6 a^3 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-6 a b^2 \sin (2 (c+d x))+9 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-9 a b^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+20 b^3\right )}{24 b^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.088, size = 488, normalized size = 3.5 \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 [A] time = 2.95833, size = 633, normalized size = 4.52 \begin{align*} \frac{6 \,{\left (a^{2} + b^{2}\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{3} \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 ) - 3 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 6 \, a b^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, b^{3} + 12 \,{\left (a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}}{12 \, b^{4} d \cos \left (d x + c\right )^{3}} \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 ^{5}{\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 [B] time = 1.60799, size = 375, normalized size = 2.68 \begin{align*} -\frac{\frac{3 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \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^{3} + 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} + \frac{6 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\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{2 \,{\left (3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 12 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 12 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, a^{2} + 8 \, b^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3} b^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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