Optimal. Leaf size=814 \[ \frac{9 \left (11 a^2-2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)} b^{7/2}}{8 \left (a^2+b^2\right )^{17/4} d^2 f \sqrt{d \sec (e+f x)}}-\frac{9 \left (11 a^2-2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)} b^{7/2}}{8 \left (a^2+b^2\right )^{17/4} d^2 f \sqrt{d \sec (e+f x)}}-\frac{9 a \left (11 a^2-2 b^2\right ) \cot (e+f x) \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)} b^3}{8 \left (a^2+b^2\right )^{9/2} d^2 f \sqrt{d \sec (e+f x)}}+\frac{9 a \left (11 a^2-2 b^2\right ) \cot (e+f x) \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)} b^3}{8 \left (a^2+b^2\right )^{9/2} d^2 f \sqrt{d \sec (e+f x)}}+\frac{3 a \left (8 a^4+64 b^2 a^2-139 b^4\right ) \sec ^2(e+f x) b}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{3 \left (4 a^4+28 b^2 a^2-15 b^4\right ) \sec ^2(e+f x) b}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}-\frac{3 a \left (8 a^4+64 b^2 a^2-139 b^4\right ) \tan (e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{3 a \left (8 a^4+64 b^2 a^2-139 b^4\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2} \]
[Out]
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Rubi [A] time = 0.915303, antiderivative size = 814, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 17, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.68, Rules used = {3512, 741, 823, 835, 844, 227, 196, 746, 399, 490, 1213, 537, 444, 63, 298, 205, 208} \[ \frac{9 \left (11 a^2-2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)} b^{7/2}}{8 \left (a^2+b^2\right )^{17/4} d^2 f \sqrt{d \sec (e+f x)}}-\frac{9 \left (11 a^2-2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)} b^{7/2}}{8 \left (a^2+b^2\right )^{17/4} d^2 f \sqrt{d \sec (e+f x)}}-\frac{9 a \left (11 a^2-2 b^2\right ) \cot (e+f x) \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)} b^3}{8 \left (a^2+b^2\right )^{9/2} d^2 f \sqrt{d \sec (e+f x)}}+\frac{9 a \left (11 a^2-2 b^2\right ) \cot (e+f x) \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)} b^3}{8 \left (a^2+b^2\right )^{9/2} d^2 f \sqrt{d \sec (e+f x)}}+\frac{3 a \left (8 a^4+64 b^2 a^2-139 b^4\right ) \sec ^2(e+f x) b}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{3 \left (4 a^4+28 b^2 a^2-15 b^4\right ) \sec ^2(e+f x) b}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}-\frac{3 a \left (8 a^4+64 b^2 a^2-139 b^4\right ) \tan (e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{3 a \left (8 a^4+64 b^2 a^2-139 b^4\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3512
Rule 741
Rule 823
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}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3} \, dx &=\frac{\sqrt [4]{\sec ^2(e+f x)} \operatorname{Subst}\left (\int \frac{1}{(a+x)^3 \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)} (a+b \tan (e+f x))^2}-\frac{\left (2 b \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{3}{2} \left (3+\frac{a^2}{b^2}\right )-\frac{7 a x}{2 b^2}}{(a+x)^3 \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)} (a+b \tan (e+f x))^2}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 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)} (a+b \tan (e+f x))^2}+\frac{\left (4 b^5 \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{3 \left (a^4+12 a^2 b^2-15 b^4\right )}{4 b^6}+\frac{3 a \left (3 a^2+16 b^2\right ) x}{4 b^6}}{(a+x)^3 \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{3 b \left (4 a^4+28 a^2 b^2-15 b^4\right ) \sec ^2(e+f x)}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\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)} (a+b \tan (e+f x))^2}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 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)} (a+b \tan (e+f x))^2}-\frac{\left (2 b^7 \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\frac{3 a \left (a^4+9 a^2 b^2-31 b^4\right )}{2 b^8}-\frac{3 \left (4 a^4+28 a^2 b^2-15 b^4\right ) x}{8 b^8}}{(a+x)^2 \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{3 b \left (4 a^4+28 a^2 b^2-15 b^4\right ) \sec ^2(e+f x)}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\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)} (a+b \tan (e+f x))^2}+\frac{3 a b \left (8 a^4+64 a^2 b^2-139 b^4\right ) \sec ^2(e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 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)} (a+b \tan (e+f x))^2}+\frac{\left (2 b^9 \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{3 \left (4 a^6+32 a^4 b^2-152 a^2 b^4+15 b^6\right )}{8 b^{10}}-\frac{3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) x}{16 b^{10}}}{(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 )^4 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{3 b \left (4 a^4+28 a^2 b^2-15 b^4\right ) \sec ^2(e+f x)}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\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)} (a+b \tan (e+f x))^2}+\frac{3 a b \left (8 a^4+64 a^2 b^2-139 b^4\right ) \sec ^2(e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 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)} (a+b \tan (e+f x))^2}+\frac{\left (9 b^3 \left (11 a^2-2 b^2\right ) \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 )}{8 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (3 a \left (8 a^4+64 a^2 b^2-139 b^4\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 )}{40 b \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}\\ &=-\frac{3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \tan (e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{3 b \left (4 a^4+28 a^2 b^2-15 b^4\right ) \sec ^2(e+f x)}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\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)} (a+b \tan (e+f x))^2}+\frac{3 a b \left (8 a^4+64 a^2 b^2-139 b^4\right ) \sec ^2(e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 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)} (a+b \tan (e+f x))^2}-\frac{\left (9 b^3 \left (11 a^2-2 b^2\right ) \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 )}{8 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (9 a b^3 \left (11 a^2-2 b^2\right ) \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 )}{8 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (3 a \left (8 a^4+64 a^2 b^2-139 b^4\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 )}{40 b \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \tan (e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{3 b \left (4 a^4+28 a^2 b^2-15 b^4\right ) \sec ^2(e+f x)}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\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)} (a+b \tan (e+f x))^2}+\frac{3 a b \left (8 a^4+64 a^2 b^2-139 b^4\right ) \sec ^2(e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 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)} (a+b \tan (e+f x))^2}-\frac{\left (9 b^3 \left (11 a^2-2 b^2\right ) \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 )}{16 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (9 a b^2 \left (11 a^2-2 b^2\right ) \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 )}{4 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \tan (e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{3 b \left (4 a^4+28 a^2 b^2-15 b^4\right ) \sec ^2(e+f x)}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\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)} (a+b \tan (e+f x))^2}+\frac{3 a b \left (8 a^4+64 a^2 b^2-139 b^4\right ) \sec ^2(e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 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)} (a+b \tan (e+f x))^2}-\frac{\left (9 b^5 \left (11 a^2-2 b^2\right ) \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 )}{4 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (9 a b^3 \left (11 a^2-2 b^2\right ) \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 )}{8 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (9 a b^3 \left (11 a^2-2 b^2\right ) \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 )}{8 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \tan (e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{3 b \left (4 a^4+28 a^2 b^2-15 b^4\right ) \sec ^2(e+f x)}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\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)} (a+b \tan (e+f x))^2}+\frac{3 a b \left (8 a^4+64 a^2 b^2-139 b^4\right ) \sec ^2(e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 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)} (a+b \tan (e+f x))^2}-\frac{\left (9 b^4 \left (11 a^2-2 b^2\right ) \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 )}{8 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (9 b^4 \left (11 a^2-2 b^2\right ) \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 )}{8 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}+\frac{\left (9 a b^3 \left (11 a^2-2 b^2\right ) \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 )}{8 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{\left (9 a b^3 \left (11 a^2-2 b^2\right ) \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 )}{8 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{9 b^{7/2} \left (11 a^2-2 b^2\right ) \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)}}{8 \left (a^2+b^2\right )^{17/4} d^2 f \sqrt{d \sec (e+f x)}}-\frac{9 b^{7/2} \left (11 a^2-2 b^2\right ) \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)}}{8 \left (a^2+b^2\right )^{17/4} d^2 f \sqrt{d \sec (e+f x)}}+\frac{3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \tan (e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)}}-\frac{9 a b^3 \left (11 a^2-2 b^2\right ) \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)}}{8 \left (a^2+b^2\right )^{9/2} d^2 f \sqrt{d \sec (e+f x)}}+\frac{9 a b^3 \left (11 a^2-2 b^2\right ) \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)}}{8 \left (a^2+b^2\right )^{9/2} d^2 f \sqrt{d \sec (e+f x)}}+\frac{3 b \left (4 a^4+28 a^2 b^2-15 b^4\right ) \sec ^2(e+f x)}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\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)} (a+b \tan (e+f x))^2}+\frac{3 a b \left (8 a^4+64 a^2 b^2-139 b^4\right ) \sec ^2(e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 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)} (a+b \tan (e+f x))^2}\\ \end{align*}
Mathematica [C] time = 28.7014, size = 15513, normalized size = 19.06 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 4.79, size = 114407, normalized size = 140.6 \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(-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(-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{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{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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