Optimal. Leaf size=143 \[ -\frac {2 d^2 \left (5 a^2-2 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 d \left (5 a^2-2 b^2\right ) \sin (e+f x) \sqrt {d \sec (e+f x)}}{5 f}+\frac {14 a b (d \sec (e+f x))^{3/2}}{15 f}+\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}{5 f} \]
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Rubi [A] time = 0.16, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3508, 3486, 3768, 3771, 2639} \[ -\frac {2 d^2 \left (5 a^2-2 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 d \left (5 a^2-2 b^2\right ) \sin (e+f x) \sqrt {d \sec (e+f x)}}{5 f}+\frac {14 a b (d \sec (e+f x))^{3/2}}{15 f}+\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}{5 f} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3486
Rule 3508
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2 \, dx &=\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}{5 f}+\frac {2}{5} \int (d \sec (e+f x))^{3/2} \left (\frac {5 a^2}{2}-b^2+\frac {7}{2} a b \tan (e+f x)\right ) \, dx\\ &=\frac {14 a b (d \sec (e+f x))^{3/2}}{15 f}+\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}{5 f}+\frac {1}{5} \left (5 a^2-2 b^2\right ) \int (d \sec (e+f x))^{3/2} \, dx\\ &=\frac {14 a b (d \sec (e+f x))^{3/2}}{15 f}+\frac {2 \left (5 a^2-2 b^2\right ) d \sqrt {d \sec (e+f x)} \sin (e+f x)}{5 f}+\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}{5 f}-\frac {1}{5} \left (\left (5 a^2-2 b^2\right ) d^2\right ) \int \frac {1}{\sqrt {d \sec (e+f x)}} \, dx\\ &=\frac {14 a b (d \sec (e+f x))^{3/2}}{15 f}+\frac {2 \left (5 a^2-2 b^2\right ) d \sqrt {d \sec (e+f x)} \sin (e+f x)}{5 f}+\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}{5 f}-\frac {\left (\left (5 a^2-2 b^2\right ) d^2\right ) \int \sqrt {\cos (e+f x)} \, dx}{5 \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\\ &=-\frac {2 \left (5 a^2-2 b^2\right ) d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {14 a b (d \sec (e+f x))^{3/2}}{15 f}+\frac {2 \left (5 a^2-2 b^2\right ) d \sqrt {d \sec (e+f x)} \sin (e+f x)}{5 f}+\frac {2 b (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}{5 f}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 126, normalized size = 0.88 \[ -\frac {2 d^2 (a+b \tan (e+f x))^2 \left (\left (3 b^2-\frac {15 a^2}{2}\right ) \sin (2 (e+f x))+3 \left (5 a^2-2 b^2\right ) \cos ^{\frac {3}{2}}(e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )-b (10 a+3 b \tan (e+f x))\right )}{15 f \sqrt {d \sec (e+f x)} (a \cos (e+f x)+b \sin (e+f x))^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} d \sec \left (f x + e\right ) \tan \left (f x + e\right )^{2} + 2 \, a b d \sec \left (f x + e\right ) \tan \left (f x + e\right ) + a^{2} d \sec \left (f x + e\right )\right )} \sqrt {d \sec \left (f x + e\right )}, x\right ) \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.91, size = 712, normalized size = 4.98 \[ -\frac {2 \left (1+\cos \left (f x +e \right )\right )^{2} \left (-1+\cos \left (f x +e \right )\right )^{2} \left (15 i \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) a^{2}-6 i \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) b^{2}-15 i \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) a^{2}+6 i \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) b^{2}+15 i \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) a^{2}-6 i \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) b^{2}-15 i \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) a^{2}+6 i \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) b^{2}+15 \left (\cos ^{3}\left (f x +e \right )\right ) a^{2}-6 \left (\cos ^{3}\left (f x +e \right )\right ) b^{2}-15 a^{2} \left (\cos ^{2}\left (f x +e \right )\right )+9 b^{2} \left (\cos ^{2}\left (f x +e \right )\right )-10 a \cos \left (f x +e \right ) b \sin \left (f x +e \right )-3 b^{2}\right ) \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}{15 f \sin \left (f x +e \right )^{5} \cos \left (f x +e \right )} \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
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 \[ \int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2 \,d x \]
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 \[ \int \left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (a + b \tan {\left (e + f x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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