Optimal. Leaf size=430 \[ -\frac{3 a \sqrt{b} \sqrt{d \sec (e+f x)} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{7/4} \sqrt [4]{\sec ^2(e+f x)}}-\frac{3 a \sqrt{b} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{7/4} \sqrt [4]{\sec ^2(e+f x)}}-\frac{b \sqrt{d \sec (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac{\sqrt{d \sec (e+f x)} F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{f \left (a^2+b^2\right ) \sqrt [4]{\sec ^2(e+f x)}}+\frac{3 a^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^2 \sqrt [4]{\sec ^2(e+f x)}}+\frac{3 a^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^2 \sqrt [4]{\sec ^2(e+f x)}} \]
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Rubi [A] time = 0.37999, antiderivative size = 430, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 15, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3512, 745, 844, 231, 747, 401, 108, 409, 1213, 537, 444, 63, 212, 208, 205} \[ -\frac{3 a \sqrt{b} \sqrt{d \sec (e+f x)} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{7/4} \sqrt [4]{\sec ^2(e+f x)}}-\frac{3 a \sqrt{b} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{7/4} \sqrt [4]{\sec ^2(e+f x)}}-\frac{b \sqrt{d \sec (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac{\sqrt{d \sec (e+f x)} F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{f \left (a^2+b^2\right ) \sqrt [4]{\sec ^2(e+f x)}}+\frac{3 a^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^2 \sqrt [4]{\sec ^2(e+f x)}}+\frac{3 a^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 f \left (a^2+b^2\right )^2 \sqrt [4]{\sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 745
Rule 844
Rule 231
Rule 747
Rule 401
Rule 108
Rule 409
Rule 1213
Rule 537
Rule 444
Rule 63
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx &=\frac{\sqrt{d \sec (e+f x)} \operatorname{Subst}\left (\int \frac{1}{(a+x)^2 \left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [4]{\sec ^2(e+f x)}}\\ &=-\frac{b \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac{\sqrt{d \sec (e+f x)} \operatorname{Subst}\left (\int \frac{-a+\frac{x}{2}}{(a+x) \left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{b \left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}\\ &=-\frac{b \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac{\sqrt{d \sec (e+f x)} \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (3 a \sqrt{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(a+x) \left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}\\ &=-\frac{F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}-\frac{b \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac{\left (3 a \sqrt{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a^2-x^2\right ) \left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (3 a^2 \sqrt{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x^2\right ) \left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}\\ &=-\frac{F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}-\frac{b \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac{\left (3 a \sqrt{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x\right ) \left (1+\frac{x}{b^2}\right )^{3/4}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{4 b \left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (3 a^2 \cot (e+f x) \sqrt{d \sec (e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x\right ) \sqrt{-\frac{x}{b^2}} \left (1+\frac{x}{b^2}\right )^{3/4}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{4 b^2 \left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}\\ &=-\frac{F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}-\frac{b \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac{\left (3 a b \sqrt{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-b^2 x^4} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}-\frac{\left (3 a^2 \cot (e+f x) \sqrt{d \sec (e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^4} \left (-1-\frac{a^2}{b^2}+x^4\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{b^2 \left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}\\ &=-\frac{F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}-\frac{b \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac{\left (3 a b \sqrt{d \sec (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 )^{3/2} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{\left (3 a b \sqrt{d \sec (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 )^{3/2} f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (3 a^2 \cot (e+f x) \sqrt{d \sec (e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{b x^2}{\sqrt{a^2+b^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 [4]{\sec ^2(e+f x)}}+\frac{\left (3 a^2 \cot (e+f x) \sqrt{d \sec (e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{b x^2}{\sqrt{a^2+b^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 [4]{\sec ^2(e+f x)}}\\ &=-\frac{3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt{d \sec (e+f x)}}{2 \left (a^2+b^2\right )^{7/4} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt{d \sec (e+f x)}}{2 \left (a^2+b^2\right )^{7/4} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}-\frac{b \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac{\left (3 a^2 \cot (e+f x) \sqrt{d \sec (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 (1-\frac{b x^2}{\sqrt{a^2+b^2}}\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (3 a^2 \cot (e+f x) \sqrt{d \sec (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 (1+\frac{b x^2}{\sqrt{a^2+b^2}}\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}\\ &=-\frac{3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt{d \sec (e+f x)}}{2 \left (a^2+b^2\right )^{7/4} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt{d \sec (e+f x)}}{2 \left (a^2+b^2\right )^{7/4} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}+\frac{3 a^2 \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{d \sec (e+f x)} \sqrt{-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac{3 a^2 \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{d \sec (e+f x)} \sqrt{-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac{b \sqrt{d \sec (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}\\ \end{align*}
Mathematica [C] time = 26.1265, size = 8876, normalized size = 20.64 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.877, size = 14318, normalized size = 33.3 \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(-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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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{\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{\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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