Optimal. Leaf size=566 \[ \frac{\left (3 a^2-2 b^2\right ) (d \sec (e+f x))^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 \sqrt{b} f \left (a^2+b^2\right )^{9/4} \sec ^2(e+f x)^{3/4}}-\frac{\left (3 a^2-2 b^2\right ) (d \sec (e+f x))^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 \sqrt{b} f \left (a^2+b^2\right )^{9/4} \sec ^2(e+f x)^{3/4}}-\frac{5 a b (d \sec (e+f x))^{3/2}}{4 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac{b (d \sec (e+f x))^{3/2}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac{5 a \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{4 f \left (a^2+b^2\right )^2}-\frac{5 a (d \sec (e+f x))^{3/2} E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{4 f \left (a^2+b^2\right )^2 \sec ^2(e+f x)^{3/4}}-\frac{a \left (3 a^2-2 b^2\right ) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 b f \left (a^2+b^2\right )^{5/2} \sec ^2(e+f x)^{3/4}}+\frac{a \left (3 a^2-2 b^2\right ) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 b f \left (a^2+b^2\right )^{5/2} \sec ^2(e+f x)^{3/4}} \]
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Rubi [A] time = 0.539162, antiderivative size = 566, 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, 745, 835, 844, 227, 196, 746, 399, 490, 1213, 537, 444, 63, 298, 205, 208} \[ \frac{\left (3 a^2-2 b^2\right ) (d \sec (e+f x))^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 \sqrt{b} f \left (a^2+b^2\right )^{9/4} \sec ^2(e+f x)^{3/4}}-\frac{\left (3 a^2-2 b^2\right ) (d \sec (e+f x))^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 \sqrt{b} f \left (a^2+b^2\right )^{9/4} \sec ^2(e+f x)^{3/4}}-\frac{5 a b (d \sec (e+f x))^{3/2}}{4 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac{b (d \sec (e+f x))^{3/2}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac{5 a \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{4 f \left (a^2+b^2\right )^2}-\frac{5 a (d \sec (e+f x))^{3/2} E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{4 f \left (a^2+b^2\right )^2 \sec ^2(e+f x)^{3/4}}-\frac{a \left (3 a^2-2 b^2\right ) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 b f \left (a^2+b^2\right )^{5/2} \sec ^2(e+f x)^{3/4}}+\frac{a \left (3 a^2-2 b^2\right ) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 b f \left (a^2+b^2\right )^{5/2} \sec ^2(e+f x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 745
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{(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx &=\frac{(d \sec (e+f x))^{3/2} \operatorname{Subst}\left (\int \frac{1}{(a+x)^3 \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{3/4}}\\ &=-\frac{b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{(d \sec (e+f x))^{3/2} \operatorname{Subst}\left (\int \frac{-2 a+\frac{x}{2}}{(a+x)^2 \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}\\ &=-\frac{b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac{\left (b (d \sec (e+f x))^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-1+\frac{4 a^2}{b^2}\right )+\frac{5 a x}{4 b^2}}{(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 \sec ^2(e+f x)^{3/4}}\\ &=-\frac{b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac{\left (5 a (d \sec (e+f x))^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 b \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac{\left (\left (-2+\frac{3 a^2}{b^2}\right ) b (d \sec (e+f x))^{3/2}\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 )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}\\ &=\frac{5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}-\frac{b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac{\left (5 a (d \sec (e+f x))^{3/2}\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 )}{8 b \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}-\frac{\left (\left (-2+\frac{3 a^2}{b^2}\right ) b (d \sec (e+f x))^{3/2}\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 )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac{\left (a \left (-2+\frac{3 a^2}{b^2}\right ) b (d \sec (e+f x))^{3/2}\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 )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}\\ &=-\frac{5 a E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac{5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}-\frac{b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac{\left (\left (-2+\frac{3 a^2}{b^2}\right ) b (d \sec (e+f x))^{3/2}\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 )}{16 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac{\left (a \left (-2+\frac{3 a^2}{b^2}\right ) \cot (e+f x) (d \sec (e+f x))^{3/2} \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 )}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}\\ &=-\frac{5 a E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac{5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}-\frac{b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac{\left (\left (-2+\frac{3 a^2}{b^2}\right ) b^3 (d \sec (e+f x))^{3/2}\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 )}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac{\left (a \left (-2+\frac{3 a^2}{b^2}\right ) b \cot (e+f x) (d \sec (e+f x))^{3/2} \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 )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}-\frac{\left (a \left (-2+\frac{3 a^2}{b^2}\right ) b \cot (e+f x) (d \sec (e+f x))^{3/2} \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 )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}\\ &=-\frac{5 a E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac{5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}-\frac{b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac{\left (\left (-2+\frac{3 a^2}{b^2}\right ) b^2 (d \sec (e+f x))^{3/2}\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 )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac{\left (\left (-2+\frac{3 a^2}{b^2}\right ) b^2 (d \sec (e+f x))^{3/2}\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 )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac{\left (a \left (-2+\frac{3 a^2}{b^2}\right ) b \cot (e+f x) (d \sec (e+f x))^{3/2} \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 )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}-\frac{\left (a \left (-2+\frac{3 a^2}{b^2}\right ) b \cot (e+f x) (d \sec (e+f x))^{3/2} \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 )}{8 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}\\ &=-\frac{\left (2-\frac{3 a^2}{b^2}\right ) b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{8 \left (a^2+b^2\right )^{9/4} f \sec ^2(e+f x)^{3/4}}+\frac{\left (2-\frac{3 a^2}{b^2}\right ) b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{8 \left (a^2+b^2\right )^{9/4} f \sec ^2(e+f x)^{3/4}}-\frac{5 a E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f \sec ^2(e+f x)^{3/4}}+\frac{5 a \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 \left (a^2+b^2\right )^2 f}+\frac{a \left (2-\frac{3 a^2}{b^2}\right ) 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 ) (d \sec (e+f x))^{3/2} \sqrt{-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac{a \left (2-\frac{3 a^2}{b^2}\right ) 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 ) (d \sec (e+f x))^{3/2} \sqrt{-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^{5/2} f \sec ^2(e+f x)^{3/4}}-\frac{b (d \sec (e+f x))^{3/2}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{5 a b (d \sec (e+f x))^{3/2}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}\\ \end{align*}
Mathematica [C] time = 29.1131, size = 14396, normalized size = 25.43 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 4.376, size = 80250, normalized size = 141.8 \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{\left (d \sec{\left (e + f x \right )}\right )^{\frac{3}{2}}}{\left (a + b \tan{\left (e + f x \right )}\right )^{3}}\, 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{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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