Optimal. Leaf size=119 \[ \frac {3 a b}{10 f (d \sec (e+f x))^{5/3}}-\frac {3 \left (2 a^2+3 b^2\right ) d \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};\cos ^2(e+f x)\right ) \sin (e+f x)}{16 f (d \sec (e+f x))^{8/3} \sqrt {\sin ^2(e+f x)}}-\frac {3 b (a+b \tan (e+f x))}{2 f (d \sec (e+f x))^{5/3}} \]
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Rubi [A]
time = 0.11, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3589, 3567,
3857, 2722} \begin {gather*} -\frac {3 d \left (2 a^2+3 b^2\right ) \sin (e+f x) \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};\cos ^2(e+f x)\right )}{16 f \sqrt {\sin ^2(e+f x)} (d \sec (e+f x))^{8/3}}+\frac {3 a b}{10 f (d \sec (e+f x))^{5/3}}-\frac {3 b (a+b \tan (e+f x))}{2 f (d \sec (e+f x))^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3567
Rule 3589
Rule 3857
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx &=-\frac {3 b (a+b \tan (e+f x))}{2 f (d \sec (e+f x))^{5/3}}-\frac {3}{2} \int \frac {-\frac {2 a^2}{3}-b^2+\frac {1}{3} a b \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx\\ &=\frac {3 a b}{10 f (d \sec (e+f x))^{5/3}}-\frac {3 b (a+b \tan (e+f x))}{2 f (d \sec (e+f x))^{5/3}}-\frac {1}{2} \left (-2 a^2-3 b^2\right ) \int \frac {1}{(d \sec (e+f x))^{5/3}} \, dx\\ &=\frac {3 a b}{10 f (d \sec (e+f x))^{5/3}}-\frac {3 b (a+b \tan (e+f x))}{2 f (d \sec (e+f x))^{5/3}}-\frac {1}{2} \left (\left (-2 a^2-3 b^2\right ) \sqrt [3]{\frac {\cos (e+f x)}{d}} \sqrt [3]{d \sec (e+f x)}\right ) \int \left (\frac {\cos (e+f x)}{d}\right )^{5/3} \, dx\\ &=\frac {3 a b}{10 f (d \sec (e+f x))^{5/3}}-\frac {3 \left (2 a^2+3 b^2\right ) \cos ^3(e+f x) \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};\cos ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \sin (e+f x)}{16 d^2 f \sqrt {\sin ^2(e+f x)}}-\frac {3 b (a+b \tan (e+f x))}{2 f (d \sec (e+f x))^{5/3}}\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 119, normalized size = 1.00 \begin {gather*} \frac {\sec ^2(e+f x) \left (-6 a b-6 a b \cos (2 (e+f x))+3 a^2 \sin (2 (e+f x))-3 b^2 \sin (2 (e+f x))+2 \left (2 a^2+3 b^2\right ) \cos ^2(e+f x)^{2/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\sin ^2(e+f x)\right ) \tan (e+f x)\right )}{10 f (d \sec (e+f x))^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{2}}{\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}}}\, dx\]
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{2}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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