Optimal. Leaf size=814 \[ \frac {9 b^{7/2} \left (11 a^2-2 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{8 \left (a^2+b^2\right )^{17/4} d^2 f \sqrt {d \sec (e+f x)}}-\frac {9 b^{7/2} \left (11 a^2-2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{8 \left (a^2+b^2\right )^{17/4} d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) E\left (\left .\frac {1}{2} \text {ArcTan}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt {d \sec (e+f x)}}-\frac {3 a \left (8 a^4+64 a^2 b^2-139 b^4\right ) \tan (e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt {d \sec (e+f x)}}-\frac {9 a b^3 \left (11 a^2-2 b^2\right ) \cot (e+f x) \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^{9/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {9 a b^3 \left (11 a^2-2 b^2\right ) \cot (e+f x) \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^{9/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 b \left (4 a^4+28 a^2 b^2-15 b^4\right ) \sec ^2(e+f x)}{10 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac {3 a b \left (8 a^4+64 a^2 b^2-139 b^4\right ) \sec ^2(e+f x)}{20 \left (a^2+b^2\right )^4 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {2 \left (b \left (4 a^2-9 b^2\right )-a \left (3 a^2+16 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.69, antiderivative size = 814, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 17, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used =
{3593, 755, 837, 849, 858, 233, 202, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214}
\begin {gather*} \frac {9 \left (11 a^2-2 b^2\right ) \text {ArcTan}\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 .\text {ArcSin}\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 .\text {ArcSin}\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} \text {ArcTan}(\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 202
Rule 211
Rule 214
Rule 233
Rule 304
Rule 408
Rule 455
Rule 504
Rule 551
Rule 755
Rule 760
Rule 837
Rule 849
Rule 858
Rule 1227
Rule 3593
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)} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 92.92, size = 9297, normalized size = 11.42 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 114406 vs. \(2 (755 ) = 1510\).
time = 3.84, size = 114407, normalized size = 140.55
method | result | size |
default | \(\text {Expression too large to display}\) | \(114407\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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