Optimal. Leaf size=568 \[ \frac{b^{7/2} \sqrt [4]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{d^2 f \left (a^2+b^2\right )^{9/4} \sqrt{d \sec (e+f x)}}-\frac{b^{7/2} \sqrt [4]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{d^2 f \left (a^2+b^2\right )^{9/4} \sqrt{d \sec (e+f x)}}-\frac{2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 d^2 f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)}}+\frac{2 \left (a \left (3 a^2+8 b^2\right ) \tan (e+f x)+5 b^3\right )}{5 d^2 f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (a \tan (e+f x)+b)}{5 d^2 f \left (a^2+b^2\right ) \sqrt{d \sec (e+f x)}}+\frac{2 a \left (3 a^2+8 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{5 d^2 f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)}}-\frac{a b^3 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{d^2 f \left (a^2+b^2\right )^{5/2} \sqrt{d \sec (e+f x)}}+\frac{a b^3 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{d^2 f \left (a^2+b^2\right )^{5/2} \sqrt{d \sec (e+f x)}} \]
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Rubi [A] time = 0.6085, antiderivative size = 568, 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, 823, 844, 227, 196, 746, 399, 490, 1213, 537, 444, 63, 298, 205, 208} \[ \frac{b^{7/2} \sqrt [4]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{d^2 f \left (a^2+b^2\right )^{9/4} \sqrt{d \sec (e+f x)}}-\frac{b^{7/2} \sqrt [4]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{d^2 f \left (a^2+b^2\right )^{9/4} \sqrt{d \sec (e+f x)}}-\frac{2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 d^2 f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)}}+\frac{2 \left (a \left (3 a^2+8 b^2\right ) \tan (e+f x)+5 b^3\right )}{5 d^2 f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (a \tan (e+f x)+b)}{5 d^2 f \left (a^2+b^2\right ) \sqrt{d \sec (e+f x)}}+\frac{2 a \left (3 a^2+8 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{5 d^2 f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)}}-\frac{a b^3 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{d^2 f \left (a^2+b^2\right )^{5/2} \sqrt{d \sec (e+f x)}}+\frac{a b^3 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{d^2 f \left (a^2+b^2\right )^{5/2} \sqrt{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 741
Rule 823
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}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))} \, dx &=\frac{\sqrt [4]{\sec ^2(e+f x)} \operatorname{Subst}\left (\int \frac{1}{(a+x) \left (1+\frac{x^2}{b^2}\right )^{9/4}} \, dx,x,b \tan (e+f x)\right )}{b d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (2 b \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-5-\frac{3 a^2}{b^2}\right )-\frac{3 a x}{2 b^2}}{(a+x) \left (1+\frac{x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (4 b^5 \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{3 a^4+8 a^2 b^2-5 b^4}{4 b^6}-\frac{a \left (3 a^2+8 b^2\right ) x}{4 b^6}}{(a+x) \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (b^3 \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (a \left (3 a^2+8 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 )}{5 b \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}\\ &=-\frac{2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (b^3 \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (a b^3 \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (a \left (3 a^2+8 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 )}{5 b \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{2 a \left (3 a^2+8 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)}}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (b^3 \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 )}{2 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (2 a b^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 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{2 a \left (3 a^2+8 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)}}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (2 b^5 \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 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (a b^3 \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (a b^3 \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{2 a \left (3 a^2+8 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)}}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (b^4 \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (b^4 \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (a b^3 \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (a b^3 \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 )}{\left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{b^{7/2} \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)}}{\left (a^2+b^2\right )^{9/4} d^2 f \sqrt{d \sec (e+f x)}}-\frac{b^{7/2} \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)}}{\left (a^2+b^2\right )^{9/4} d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 a \left (3 a^2+8 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)}}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{2 a \left (3 a^2+8 b^2\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}-\frac{a b^3 \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)}}{\left (a^2+b^2\right )^{5/2} d^2 f \sqrt{d \sec (e+f x)}}+\frac{a b^3 \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)}}{\left (a^2+b^2\right )^{5/2} d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \left (5 b^3+a \left (3 a^2+8 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 32.6737, size = 17838, normalized size = 31.4 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.536, size = 14547, normalized size = 25.6 \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}{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{2}}{\left (b \tan \left (f x + e\right ) + a\right )}}\,{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}{\left (d \sec{\left (e + f x \right )}\right )^{\frac{5}{2}} \left (a + b \tan{\left (e + f x \right )}\right )}\, 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}{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{2}}{\left (b \tan \left (f x + e\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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