Optimal. Leaf size=581 \[ \frac {3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}+\frac {\sqrt {3} b^{8/3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{2 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {\sqrt {3} b^{8/3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{2 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {b^{8/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac {b^{8/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sec ^2(e+f x)^{5/6}}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {b^{8/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sec ^2(e+f x)^{5/6}}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac {F_1\left (\frac {1}{2};1,\frac {11}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}} \]
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Rubi [A]
time = 0.58, antiderivative size = 581, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3593, 771,
440, 455, 53, 65, 216, 648, 632, 210, 642, 214} \begin {gather*} \frac {\tan (e+f x) \sec ^2(e+f x)^{5/6} F_1\left (\frac {1}{2};1,\frac {11}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a f (d \sec (e+f x))^{5/3}}+\frac {\sqrt {3} b^{8/3} \sec ^2(e+f x)^{5/6} \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 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}-\frac {\sqrt {3} b^{8/3} \sec ^2(e+f x)^{5/6} \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 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}+\frac {3 b}{5 f \left (a^2+b^2\right ) (d \sec (e+f x))^{5/3}}+\frac {b^{8/3} \sec ^2(e+f x)^{5/6} \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 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}-\frac {b^{8/3} \sec ^2(e+f x)^{5/6} \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 f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}}-\frac {b^{8/3} \sec ^2(e+f x)^{5/6} \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{f \left (a^2+b^2\right )^{11/6} (d \sec (e+f x))^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 210
Rule 214
Rule 216
Rule 440
Rule 455
Rule 632
Rule 642
Rule 648
Rule 771
Rule 3593
Rubi steps
\begin {align*} \int \frac {1}{(d \sec (e+f x))^{5/3} (a+b \tan (e+f x))} \, dx &=\frac {\sec ^2(e+f x)^{5/6} \text {Subst}\left (\int \frac {1}{(a+x) \left (1+\frac {x^2}{b^2}\right )^{11/6}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}}\\ &=\frac {\sec ^2(e+f x)^{5/6} \text {Subst}\left (\int \left (\frac {a}{\left (a^2-x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{11/6}}+\frac {x}{\left (-a^2+x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{11/6}}\right ) \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}}\\ &=\frac {\sec ^2(e+f x)^{5/6} \text {Subst}\left (\int \frac {x}{\left (-a^2+x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{11/6}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}}+\frac {\left (a \sec ^2(e+f x)^{5/6}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{11/6}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{5/3}}\\ &=\frac {F_1\left (\frac {1}{2};1,\frac {11}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}+\frac {\sec ^2(e+f x)^{5/6} \text {Subst}\left (\int \frac {1}{\left (-a^2+x\right ) \left (1+\frac {x}{b^2}\right )^{11/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{2 b f (d \sec (e+f x))^{5/3}}\\ &=\frac {3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}+\frac {F_1\left (\frac {1}{2};1,\frac {11}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}+\frac {\left (b \sec ^2(e+f x)^{5/6}\right ) \text {Subst}\left (\int \frac {1}{\left (-a^2+x\right ) \left (1+\frac {x}{b^2}\right )^{5/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{2 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}\\ &=\frac {3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}+\frac {F_1\left (\frac {1}{2};1,\frac {11}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}+\frac {\left (3 b^3 \sec ^2(e+f x)^{5/6}\right ) \text {Subst}\left (\int \frac {1}{-a^2-b^2+b^2 x^6} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}\\ &=\frac {3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}+\frac {F_1\left (\frac {1}{2};1,\frac {11}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}-\frac {\left (b^3 \sec ^2(e+f x)^{5/6}\right ) \text {Subst}\left (\int \frac {\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 )}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {\left (b^3 \sec ^2(e+f x)^{5/6}\right ) \text {Subst}\left (\int \frac {\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 )}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {\left (b^3 \sec ^2(e+f x)^{5/6}\right ) \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 )}{\left (a^2+b^2\right )^{5/3} f (d \sec (e+f x))^{5/3}}\\ &=\frac {3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}-\frac {b^{8/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac {F_1\left (\frac {1}{2};1,\frac {11}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}+\frac {\left (b^{8/3} \sec ^2(e+f x)^{5/6}\right ) \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 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {\left (b^{8/3} \sec ^2(e+f x)^{5/6}\right ) \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 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {\left (3 b^3 \sec ^2(e+f x)^{5/6}\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 \left (a^2+b^2\right )^{5/3} f (d \sec (e+f x))^{5/3}}-\frac {\left (3 b^3 \sec ^2(e+f x)^{5/6}\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 \left (a^2+b^2\right )^{5/3} f (d \sec (e+f x))^{5/3}}\\ &=\frac {3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}-\frac {b^{8/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac {b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac {F_1\left (\frac {1}{2};1,\frac {11}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}-\frac {\left (3 b^{8/3} \sec ^2(e+f x)^{5/6}\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 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac {\left (3 b^{8/3} \sec ^2(e+f x)^{5/6}\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 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}\\ &=\frac {3 b}{5 \left (a^2+b^2\right ) f (d \sec (e+f x))^{5/3}}+\frac {\sqrt {3} b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{2 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {\sqrt {3} b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{2 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {b^{8/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sec ^2(e+f x)^{5/6}}{\left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac {b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}-\frac {b^{8/3} \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 ) \sec ^2(e+f x)^{5/6}}{4 \left (a^2+b^2\right )^{11/6} f (d \sec (e+f x))^{5/3}}+\frac {F_1\left (\frac {1}{2};1,\frac {11}{6};\frac {3}{2};\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{5/6} \tan (e+f x)}{a f (d \sec (e+f x))^{5/3}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(6862\) vs. \(2(581)=1162\).
time = 129.53, size = 6862, normalized size = 11.81 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.90, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}} \left (a +b \tan \left (f x +e \right )\right )}\, dx\]
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 {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}} \left (a + b \tan {\left (e + f x \right )}\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 {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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