Optimal. Leaf size=176 \[ -\frac {2 \cos ^2(e+f x) \left (2 b \left (5 a^2+2 b^2\right )-a \left (7 a^2+b^2\right ) \tan (e+f x)\right )}{45 d^4 f \sqrt {d \sec (e+f x)}}+\frac {2 a \left (7 a^2+6 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{15 d^4 f \sqrt {d \sec (e+f x)}}-\frac {2 \cos ^4(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{9 d^4 f \sqrt {d \sec (e+f x)}} \]
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Rubi [A] time = 0.14, antiderivative size = 176, 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 = {3512, 739, 778, 196} \[ -\frac {2 \cos ^2(e+f x) \left (2 b \left (5 a^2+2 b^2\right )-a \left (7 a^2+b^2\right ) \tan (e+f x)\right )}{45 d^4 f \sqrt {d \sec (e+f x)}}+\frac {2 a \left (7 a^2+6 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{15 d^4 f \sqrt {d \sec (e+f x)}}-\frac {2 \cos ^4(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{9 d^4 f \sqrt {d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 739
Rule 778
Rule 3512
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{9/2}} \, dx &=\frac {\sqrt [4]{\sec ^2(e+f x)} \operatorname {Subst}\left (\int \frac {(a+x)^3}{\left (1+\frac {x^2}{b^2}\right )^{13/4}} \, dx,x,b \tan (e+f x)\right )}{b d^4 f \sqrt {d \sec (e+f x)}}\\ &=-\frac {2 \cos ^4(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{9 d^4 f \sqrt {d \sec (e+f x)}}+\frac {\left (2 b \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {(a+x) \left (\frac {1}{2} \left (4+\frac {7 a^2}{b^2}\right )+\frac {3 a x}{2 b^2}\right )}{\left (1+\frac {x^2}{b^2}\right )^{9/4}} \, dx,x,b \tan (e+f x)\right )}{9 d^4 f \sqrt {d \sec (e+f x)}}\\ &=-\frac {2 \cos ^4(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{9 d^4 f \sqrt {d \sec (e+f x)}}-\frac {2 \cos ^2(e+f x) \left (2 b \left (5 a^2+2 b^2\right )-a \left (7 a^2+b^2\right ) \tan (e+f x)\right )}{45 d^4 f \sqrt {d \sec (e+f x)}}+\frac {\left (a \left (6+\frac {7 a^2}{b^2}\right ) b \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{15 d^4 f \sqrt {d \sec (e+f x)}}\\ &=\frac {2 a \left (7 a^2+6 b^2\right ) E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{15 d^4 f \sqrt {d \sec (e+f x)}}-\frac {2 \cos ^4(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{9 d^4 f \sqrt {d \sec (e+f x)}}-\frac {2 \cos ^2(e+f x) \left (2 b \left (5 a^2+2 b^2\right )-a \left (7 a^2+b^2\right ) \tan (e+f x)\right )}{45 d^4 f \sqrt {d \sec (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 6.40, size = 372, normalized size = 2.11 \[ \frac {\sec ^2(e+f x) (a+b \tan (e+f x))^3 \left (\frac {1}{180} a \left (19 a^2-3 b^2\right ) \sin (e+f x)+\frac {1}{360} a \left (43 a^2-21 b^2\right ) \sin (3 (e+f x))+\frac {1}{72} a \left (a^2-3 b^2\right ) \sin (5 (e+f x))-\frac {1}{90} b \left (15 a^2+4 b^2\right ) \cos (e+f x)-\frac {1}{360} b \left (75 a^2+11 b^2\right ) \cos (3 (e+f x))-\frac {1}{72} b \left (3 a^2-b^2\right ) \cos (5 (e+f x))\right )}{f (d \sec (e+f x))^{9/2} (a \cos (e+f x)+b \sin (e+f x))^3}+\frac {\sec ^{\frac {3}{2}}(e+f x) (a+b \tan (e+f x))^3 \left (\frac {2 \left (56 a^3+48 a b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\sqrt {\cos (e+f x)} \sqrt {\sec (e+f x)}}-\frac {2 \left (15 a^2 b+7 b^3\right ) \sin ^2(e+f x)}{\sqrt {1-\cos ^2(e+f x)} \sqrt {\sec (e+f x)} \sqrt {\cos ^2(e+f x) \left (\sec ^2(e+f x)-1\right )}}\right )}{120 f (d \sec (e+f x))^{9/2} (a \cos (e+f x)+b \sin (e+f x))^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \tan \left (f x + e\right )^{3} + 3 \, a b^{2} \tan \left (f x + e\right )^{2} + 3 \, a^{2} b \tan \left (f x + e\right ) + a^{3}\right )} \sqrt {d \sec \left (f x + e\right )}}{d^{5} \sec \left (f x + e\right )^{5}}, x\right ) \]
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 \[ \int \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{\left (d \sec \left (f x + e\right )\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.11, size = 745, normalized size = 4.23 \[ -\frac {2 \left (21 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^{3}+21 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^{3}-18 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 \,b^{2}+18 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 \,b^{2}+5 \left (\cos ^{6}\left (f x +e \right )\right ) a^{3}-15 \left (\cos ^{6}\left (f x +e \right )\right ) a \,b^{2}+15 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right ) a^{2} b -5 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right ) b^{3}-18 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 \,b^{2}-21 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^{3}-21 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^{3}+18 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 \,b^{2}+2 \left (\cos ^{4}\left (f x +e \right )\right ) a^{3}+21 \left (\cos ^{4}\left (f x +e \right )\right ) a \,b^{2}+9 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) b^{3}+14 \left (\cos ^{2}\left (f x +e \right )\right ) a^{3}+12 \left (\cos ^{2}\left (f x +e \right )\right ) a \,b^{2}-21 a^{3} \cos \left (f x +e \right )-18 a \cos \left (f x +e \right ) b^{2}\right )}{45 f \cos \left (f x +e \right )^{5} \sin \left (f x +e \right ) \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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