Optimal. Leaf size=396 \[ -\frac{d^2 \sqrt [4]{a^2+b^2} \sqrt{d \sec (e+f x)} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{b^{3/2} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{d^2 \sqrt [4]{a^2+b^2} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{b^{3/2} f \sqrt [4]{\sec ^2(e+f x)}}+\frac{a d^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac{a d^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac{2 a d^2 \sqrt{d \sec (e+f x)} F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f} \]
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Rubi [A] time = 0.384003, antiderivative size = 396, 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, 735, 844, 231, 747, 401, 108, 409, 1213, 537, 444, 63, 212, 208, 205} \[ -\frac{d^2 \sqrt [4]{a^2+b^2} \sqrt{d \sec (e+f x)} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{b^{3/2} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{d^2 \sqrt [4]{a^2+b^2} \sqrt{d \sec (e+f x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{b^{3/2} f \sqrt [4]{\sec ^2(e+f x)}}+\frac{a d^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac{a d^2 \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt{d \sec (e+f x)} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac{2 a d^2 \sqrt{d \sec (e+f x)} F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 735
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{(d \sec (e+f x))^{5/2}}{a+b \tan (e+f x)} \, dx &=\frac{\left (d^2 \sqrt{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [4]{1+\frac{x^2}{b^2}}}{a+x} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [4]{\sec ^2(e+f x)}}\\ &=\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f}+\frac{\left (d^2 \sqrt{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{a x}{b^2}}{(a+x) \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{2 d^2 \sqrt{d \sec (e+f x)}}{b f}-\frac{\left (a d^2 \sqrt{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{b^3 f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (\left (1+\frac{a^2}{b^2}\right ) d^2 \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 )}{b f \sqrt [4]{\sec ^2(e+f x)}}\\ &=\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f}-\frac{2 a d^2 F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac{\left (\left (1+\frac{a^2}{b^2}\right ) d^2 \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 )}{b f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (a \left (1+\frac{a^2}{b^2}\right ) d^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 )}{b f \sqrt [4]{\sec ^2(e+f x)}}\\ &=\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f}-\frac{2 a d^2 F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac{\left (\left (1+\frac{a^2}{b^2}\right ) d^2 \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 )}{2 b f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (a \left (1+\frac{a^2}{b^2}\right ) d^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 )}{2 b^2 f \sqrt [4]{\sec ^2(e+f x)}}\\ &=\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f}-\frac{2 a d^2 F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac{\left (2 \left (1+\frac{a^2}{b^2}\right ) b d^2 \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 )}{f \sqrt [4]{\sec ^2(e+f x)}}-\frac{\left (2 a \left (1+\frac{a^2}{b^2}\right ) d^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 f \sqrt [4]{\sec ^2(e+f x)}}\\ &=\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f}-\frac{2 a d^2 F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}-\frac{\left (\left (1+\frac{a^2}{b^2}\right ) b d^2 \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 )}{\sqrt{a^2+b^2} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{\left (\left (1+\frac{a^2}{b^2}\right ) b d^2 \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 )}{\sqrt{a^2+b^2} f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (a \left (1+\frac{a^2}{b^2}\right ) d^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 )}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (a \left (1+\frac{a^2}{b^2}\right ) d^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 )}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}\\ &=\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f}-\frac{\sqrt [4]{a^2+b^2} d^2 \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)}}{b^{3/2} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{\sqrt [4]{a^2+b^2} d^2 \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)}}{b^{3/2} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{2 a d^2 F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (a \left (1+\frac{a^2}{b^2}\right ) d^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 )}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}+\frac{\left (a \left (1+\frac{a^2}{b^2}\right ) d^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 )}{\left (a^2+b^2\right ) f \sqrt [4]{\sec ^2(e+f x)}}\\ &=\frac{2 d^2 \sqrt{d \sec (e+f x)}}{b f}-\frac{\sqrt [4]{a^2+b^2} d^2 \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)}}{b^{3/2} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{\sqrt [4]{a^2+b^2} d^2 \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)}}{b^{3/2} f \sqrt [4]{\sec ^2(e+f x)}}-\frac{2 a d^2 F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt{d \sec (e+f x)}}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac{a d^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)}}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}+\frac{a d^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)}}{b^2 f \sqrt [4]{\sec ^2(e+f x)}}\\ \end{align*}
Mathematica [B] time = 24.6276, size = 6631, normalized size = 16.74 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.497, size = 10704, normalized size = 27. \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{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{2}}}{b \tan \left (f x + e\right ) + a}\,{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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{2}}}{b \tan \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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