Optimal. Leaf size=88 \[ -\frac {2 a d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 a d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3486, 3768, 3771, 2639} \[ -\frac {2 a d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 a d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3486
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int (d \sec (e+f x))^{3/2} (a+b \tan (e+f x)) \, dx &=\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}+a \int (d \sec (e+f x))^{3/2} \, dx\\ &=\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}+\frac {2 a d \sqrt {d \sec (e+f x)} \sin (e+f x)}{f}-\left (a d^2\right ) \int \frac {1}{\sqrt {d \sec (e+f x)}} \, dx\\ &=\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}+\frac {2 a d \sqrt {d \sec (e+f x)} \sin (e+f x)}{f}-\frac {\left (a d^2\right ) \int \sqrt {\cos (e+f x)} \, dx}{\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\\ &=-\frac {2 a d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 b (d \sec (e+f x))^{3/2}}{3 f}+\frac {2 a d \sqrt {d \sec (e+f x)} \sin (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 58, normalized size = 0.66 \[ \frac {(d \sec (e+f x))^{3/2} \left (3 a \sin (2 (e+f x))-6 a \cos ^{\frac {3}{2}}(e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+2 b\right )}{3 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b d \sec \left (f x + e\right ) \tan \left (f x + e\right ) + a d \sec \left (f x + e\right )\right )} \sqrt {d \sec \left (f x + e\right )}, x\right ) \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.80, size = 356, normalized size = 4.05 \[ -\frac {2 \left (1+\cos \left (f x +e \right )\right )^{2} \left (-1+\cos \left (f x +e \right )\right )^{2} \left (3 i \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) a -3 i \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) a +3 i \cos \left (f x +e \right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) a \sin \left (f x +e \right )-3 i \cos \left (f x +e \right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) a \sin \left (f x +e \right )+3 \left (\cos ^{2}\left (f x +e \right )\right ) a -3 a \cos \left (f x +e \right )-b \sin \left (f x +e \right )\right ) \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}{3 f \sin \left (f x +e \right )^{5}} \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}\,{d x} \]
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 \[ \int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right ) \,d x \]
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 \[ \int \left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (a + b \tan {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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