Optimal. Leaf size=145 \[ \frac {2 \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 d^2 f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 \left (3 a^2+2 b^2\right ) \sin (e+f x)}{15 d f (d \sec (e+f x))^{3/2}}-\frac {2 a b}{15 f (d \sec (e+f x))^{5/2}}-\frac {2 b (a+b \tan (e+f x))}{3 f (d \sec (e+f x))^{5/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 145, 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, 3769, 3771, 2639} \[ \frac {2 \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 d^2 f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 \left (3 a^2+2 b^2\right ) \sin (e+f x)}{15 d f (d \sec (e+f x))^{3/2}}-\frac {2 a b}{15 f (d \sec (e+f x))^{5/2}}-\frac {2 b (a+b \tan (e+f x))}{3 f (d \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3486
Rule 3508
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{5/2}} \, dx &=-\frac {2 b (a+b \tan (e+f x))}{3 f (d \sec (e+f x))^{5/2}}-\frac {2}{3} \int \frac {-\frac {3 a^2}{2}-b^2-\frac {1}{2} a b \tan (e+f x)}{(d \sec (e+f x))^{5/2}} \, dx\\ &=-\frac {2 a b}{15 f (d \sec (e+f x))^{5/2}}-\frac {2 b (a+b \tan (e+f x))}{3 f (d \sec (e+f x))^{5/2}}-\frac {1}{3} \left (-3 a^2-2 b^2\right ) \int \frac {1}{(d \sec (e+f x))^{5/2}} \, dx\\ &=-\frac {2 a b}{15 f (d \sec (e+f x))^{5/2}}+\frac {2 \left (3 a^2+2 b^2\right ) \sin (e+f x)}{15 d f (d \sec (e+f x))^{3/2}}-\frac {2 b (a+b \tan (e+f x))}{3 f (d \sec (e+f x))^{5/2}}+\frac {\left (3 a^2+2 b^2\right ) \int \frac {1}{\sqrt {d \sec (e+f x)}} \, dx}{5 d^2}\\ &=-\frac {2 a b}{15 f (d \sec (e+f x))^{5/2}}+\frac {2 \left (3 a^2+2 b^2\right ) \sin (e+f x)}{15 d f (d \sec (e+f x))^{3/2}}-\frac {2 b (a+b \tan (e+f x))}{3 f (d \sec (e+f x))^{5/2}}+\frac {\left (3 a^2+2 b^2\right ) \int \sqrt {\cos (e+f x)} \, dx}{5 d^2 \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\\ &=-\frac {2 a b}{15 f (d \sec (e+f x))^{5/2}}+\frac {2 \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 d^2 f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 \left (3 a^2+2 b^2\right ) \sin (e+f x)}{15 d f (d \sec (e+f x))^{3/2}}-\frac {2 b (a+b \tan (e+f x))}{3 f (d \sec (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.92, size = 92, normalized size = 0.63 \[ \frac {\left (6 a^2+4 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+2 \cos ^{\frac {3}{2}}(e+f x) \left (\left (a^2-b^2\right ) \sin (e+f x)-2 a b \cos (e+f x)\right )}{5 f \cos ^{\frac {5}{2}}(e+f x) (d \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}\right )} \sqrt {d \sec \left (f x + e\right )}}{d^{3} \sec \left (f x + e\right )^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.88, size = 670, normalized size = 4.62 \[ -\frac {2 \left (-3 i \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 )}}\, \cos \left (f x +e \right ) \sin \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^{2}-2 i \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 )}}\, \cos \left (f x +e \right ) \sin \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 ) b^{2}+3 i \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 )}}\, \cos \left (f x +e \right ) \sin \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^{2}+2 i \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 )}}\, \cos \left (f x +e \right ) \sin \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 ) b^{2}-3 i \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 )}}\, \sin \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^{2}-2 i \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 )}}\, \sin \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 ) b^{2}+3 i \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 )}}\, \sin \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^{2}+2 i \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 )}}\, \sin \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 ) b^{2}+\left (\cos ^{4}\left (f x +e \right )\right ) a^{2}-\left (\cos ^{4}\left (f x +e \right )\right ) b^{2}+2 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) a b +2 a^{2} \left (\cos ^{2}\left (f x +e \right )\right )+3 b^{2} \left (\cos ^{2}\left (f x +e \right )\right )-3 \cos \left (f x +e \right ) a^{2}-2 \cos \left (f x +e \right ) b^{2}\right )}{5 f \cos \left (f x +e \right )^{3} \sin \left (f x +e \right ) \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{2}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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