Optimal. Leaf size=555 \[ \frac{5 a b^{3/2} \sqrt [4]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{9/4} \sqrt{d \sec (e+f x)}}+\frac{b \left (2 a^2-3 b^2\right ) \sec ^2(e+f x)}{f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{5 a b^{3/2} \sqrt [4]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{9/4} \sqrt{d \sec (e+f x)}}-\frac{\left (2 a^2-3 b^2\right ) \tan (e+f x)}{f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)}}+\frac{2 (a \tan (e+f x)+b)}{f \left (a^2+b^2\right ) \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{\left (2 a^2-3 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)}}-\frac{5 a^2 b \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^{5/2} \sqrt{d \sec (e+f x)}}+\frac{5 a^2 b \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^{5/2} \sqrt{d \sec (e+f x)}} \]
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Rubi [A] time = 0.537244, antiderivative size = 555, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {3512, 741, 835, 844, 227, 196, 746, 399, 490, 1213, 537, 444, 63, 298, 205, 208} \[ \frac{5 a b^{3/2} \sqrt [4]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{9/4} \sqrt{d \sec (e+f x)}}+\frac{b \left (2 a^2-3 b^2\right ) \sec ^2(e+f x)}{f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{5 a b^{3/2} \sqrt [4]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{9/4} \sqrt{d \sec (e+f x)}}-\frac{\left (2 a^2-3 b^2\right ) \tan (e+f x)}{f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)}}+\frac{2 (a \tan (e+f x)+b)}{f \left (a^2+b^2\right ) \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{\left (2 a^2-3 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)}}-\frac{5 a^2 b \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^{5/2} \sqrt{d \sec (e+f x)}}+\frac{5 a^2 b \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^{5/2} \sqrt{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 741
Rule 835
Rule 844
Rule 227
Rule 196
Rule 746
Rule 399
Rule 490
Rule 1213
Rule 537
Rule 444
Rule 63
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2} \, dx &=\frac{\sqrt [4]{\sec ^2(e+f x)} \operatorname{Subst}\left (\int \frac{1}{(a+x)^2 \left (1+\frac{x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt{d \sec (e+f x)}}\\ &=\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left (2 b \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-3+\frac{a^2}{b^2}\right )-\frac{a x}{2 b^2}}{(a+x)^2 \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)}}\\ &=\frac{b \left (2 a^2-3 b^2\right ) \sec ^2(e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{\left (2 b^3 \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{a \left (a^2-4 b^2\right )}{2 b^4}-\frac{\left (2 a^2-3 b^2\right ) x}{4 b^4}}{(a+x) \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{b \left (2 a^2-3 b^2\right ) \sec ^2(e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{\left (5 a b \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (\left (2 a^2-3 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}\\ &=-\frac{\left (2 a^2-3 b^2\right ) \tan (e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}+\frac{b \left (2 a^2-3 b^2\right ) \sec ^2(e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left (5 a b \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a^2-x^2\right ) \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (5 a^2 b \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x^2\right ) \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (\left (2 a^2-3 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{\left (2 a^2-3 b^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (2 a^2-3 b^2\right ) \tan (e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}+\frac{b \left (2 a^2-3 b^2\right ) \sec ^2(e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left (5 a b \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x\right ) \sqrt [4]{1+\frac{x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{4 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (5 a^2 \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^4} \left (1+\frac{a^2}{b^2}-x^4\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{\left (2 a^2-3 b^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (2 a^2-3 b^2\right ) \tan (e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}+\frac{b \left (2 a^2-3 b^2\right ) \sec ^2(e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left (5 a b^3 \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{a^2+b^2-b^2 x^4} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (5 a^2 b \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{a^2+b^2}-b x^2\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (5 a^2 b \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{a^2+b^2}+b x^2\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{\left (2 a^2-3 b^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (2 a^2-3 b^2\right ) \tan (e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}+\frac{b \left (2 a^2-3 b^2\right ) \sec ^2(e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left (5 a b^2 \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+b^2}-b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (5 a b^2 \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+b^2}+b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (5 a^2 b \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (\sqrt{a^2+b^2}-b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (5 a^2 b \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (\sqrt{a^2+b^2}+b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{5 a b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{9/4} f \sqrt{d \sec (e+f x)}}-\frac{5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{9/4} f \sqrt{d \sec (e+f x)}}+\frac{\left (2 a^2-3 b^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (2 a^2-3 b^2\right ) \tan (e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}-\frac{5 a^2 b \cot (e+f x) \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^{5/2} f \sqrt{d \sec (e+f x)}}+\frac{5 a^2 b \cot (e+f x) \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^{5/2} f \sqrt{d \sec (e+f x)}}+\frac{b \left (2 a^2-3 b^2\right ) \sec ^2(e+f x)}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}\\ \end{align*}
Mathematica [C] time = 32.1259, size = 17812, normalized size = 32.09 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 2.362, size = 38627, normalized size = 69.6 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \sec \left (f x + e\right )}{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \sec{\left (e + f x \right )}} \left (a + b \tan{\left (e + f x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \sec \left (f x + e\right )}{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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