Optimal. Leaf size=552 \[ -\frac {\sqrt {3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{2 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}+\frac {\sqrt {3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{2 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}+\frac {\log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) (d \sec (e+f x))^{5/3}}{4 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}-\frac {\log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) (d \sec (e+f x))^{5/3}}{4 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}+\frac {F_1\left (\frac {1}{2};1,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a f \sec ^2(e+f x)^{5/6}} \]
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Rubi [A]
time = 0.62, antiderivative size = 552, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3593, 771,
440, 455, 65, 302, 648, 632, 210, 642, 214} \begin {gather*} \frac {\tan (e+f x) (d \sec (e+f x))^{5/3} F_1\left (\frac {1}{2};1,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a f \sec ^2(e+f x)^{5/6}}-\frac {\sqrt {3} (d \sec (e+f x))^{5/3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right )}{2 b^{2/3} f \sqrt [6]{a^2+b^2} \sec ^2(e+f x)^{5/6}}+\frac {\sqrt {3} (d \sec (e+f x))^{5/3} \text {ArcTan}\left (\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}+\frac {1}{\sqrt {3}}\right )}{2 b^{2/3} f \sqrt [6]{a^2+b^2} \sec ^2(e+f x)^{5/6}}+\frac {(d \sec (e+f x))^{5/3} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 b^{2/3} f \sqrt [6]{a^2+b^2} \sec ^2(e+f x)^{5/6}}-\frac {(d \sec (e+f x))^{5/3} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 b^{2/3} f \sqrt [6]{a^2+b^2} \sec ^2(e+f x)^{5/6}}-\frac {(d \sec (e+f x))^{5/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{b^{2/3} f \sqrt [6]{a^2+b^2} \sec ^2(e+f x)^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 210
Rule 214
Rule 302
Rule 440
Rule 455
Rule 632
Rule 642
Rule 648
Rule 771
Rule 3593
Rubi steps
\begin {align*} \int \frac {(d \sec (e+f x))^{5/3}}{a+b \tan (e+f x)} \, dx &=\frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \frac {1}{(a+x) \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}\\ &=\frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \left (\frac {a}{\left (a^2-x^2\right ) \sqrt [6]{1+\frac {x^2}{b^2}}}+\frac {x}{\left (-a^2+x^2\right ) \sqrt [6]{1+\frac {x^2}{b^2}}}\right ) \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}\\ &=\frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \frac {x}{\left (-a^2+x^2\right ) \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}+\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right ) \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}\\ &=\frac {F_1\left (\frac {1}{2};1,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a f \sec ^2(e+f x)^{5/6}}+\frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \frac {1}{\left (-a^2+x\right ) \sqrt [6]{1+\frac {x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{2 b f \sec ^2(e+f x)^{5/6}}\\ &=\frac {F_1\left (\frac {1}{2};1,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a f \sec ^2(e+f x)^{5/6}}+\frac {\left (3 b (d \sec (e+f x))^{5/3}\right ) \text {Subst}\left (\int \frac {x^4}{-a^2-b^2+b^2 x^6} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{f \sec ^2(e+f x)^{5/6}}\\ &=\frac {F_1\left (\frac {1}{2};1,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a f \sec ^2(e+f x)^{5/6}}-\frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{a^2+b^2}-b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{\sqrt [3]{b} f \sec ^2(e+f x)^{5/6}}-\frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \frac {-\frac {1}{2} \sqrt [6]{a^2+b^2}-\frac {\sqrt [3]{b} x}{2}}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{\sqrt [3]{b} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}-\frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \frac {-\frac {1}{2} \sqrt [6]{a^2+b^2}+\frac {\sqrt [3]{b} x}{2}}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{\sqrt [3]{b} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}+\frac {F_1\left (\frac {1}{2};1,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a f \sec ^2(e+f x)^{5/6}}+\frac {\left (3 (d \sec (e+f x))^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{4 \sqrt [3]{b} f \sec ^2(e+f x)^{5/6}}+\frac {\left (3 (d \sec (e+f x))^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{4 \sqrt [3]{b} f \sec ^2(e+f x)^{5/6}}+\frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \frac {-\sqrt [3]{b} \sqrt [6]{a^2+b^2}+2 b^{2/3} x}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{4 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}-\frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \frac {\sqrt [3]{b} \sqrt [6]{a^2+b^2}+2 b^{2/3} x}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{4 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}+\frac {\log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) (d \sec (e+f x))^{5/3}}{4 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}-\frac {\log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) (d \sec (e+f x))^{5/3}}{4 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}+\frac {F_1\left (\frac {1}{2};1,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a f \sec ^2(e+f x)^{5/6}}+\frac {\left (3 (d \sec (e+f x))^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{2 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}-\frac {\left (3 (d \sec (e+f x))^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{2 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}}{\sqrt {3}}\right ) (d \sec (e+f x))^{5/3}}{2 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}}{\sqrt {3}}\right ) (d \sec (e+f x))^{5/3}}{2 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}+\frac {\log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) (d \sec (e+f x))^{5/3}}{4 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}-\frac {\log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) (d \sec (e+f x))^{5/3}}{4 b^{2/3} \sqrt [6]{a^2+b^2} f \sec ^2(e+f x)^{5/6}}+\frac {F_1\left (\frac {1}{2};1,\frac {1}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a f \sec ^2(e+f x)^{5/6}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 25.22, size = 276, normalized size = 0.50 \begin {gather*} -\frac {24 d^2 F_1\left (\frac {1}{3};\frac {1}{6},\frac {1}{6};\frac {4}{3};\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right ) (a+b \tan (e+f x))}{b f \sqrt [3]{d \sec (e+f x)} \left ((a+i b) F_1\left (\frac {4}{3};\frac {1}{6},\frac {7}{6};\frac {7}{3};\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right )+(a-i b) F_1\left (\frac {4}{3};\frac {7}{6},\frac {1}{6};\frac {7}{3};\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right )+8 F_1\left (\frac {1}{3};\frac {1}{6},\frac {1}{6};\frac {4}{3};\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right ) (a+b \tan (e+f x))\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.98, size = 0, normalized size = 0.00 \[\int \frac {\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}}}{a +b \tan \left (f x +e \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 {\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}{a + b \tan {\left (e + f x \right )}}\, 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 {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}}{a+b\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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