Optimal. Leaf size=170 \[ \frac {2 a \left (5 a^2+6 b^2\right ) F\left (\left .\frac {1}{2} \text {ArcTan}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{21 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \left (2 b \left (3 a^2+2 b^2\right )-a \left (5 a^2+3 b^2\right ) \tan (e+f x)\right )}{21 d^2 f (d \sec (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 170, 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 = {3593, 753, 792,
237} \begin {gather*} \frac {2 a \left (5 a^2+6 b^2\right ) \sec ^2(e+f x)^{3/4} F\left (\left .\frac {1}{2} \text {ArcTan}(\tan (e+f x))\right |2\right )}{21 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \left (2 b \left (3 a^2+2 b^2\right )-a \left (5 a^2+3 b^2\right ) \tan (e+f x)\right )}{21 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 753
Rule 792
Rule 3593
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{7/2}} \, dx &=\frac {\sec ^2(e+f x)^{3/4} \text {Subst}\left (\int \frac {(a+x)^3}{\left (1+\frac {x^2}{b^2}\right )^{11/4}} \, dx,x,b \tan (e+f x)\right )}{b d^2 f (d \sec (e+f x))^{3/2}}\\ &=-\frac {2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}}+\frac {\left (2 b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {(a+x) \left (\frac {1}{2} \left (4+\frac {5 a^2}{b^2}\right )+\frac {a x}{2 b^2}\right )}{\left (1+\frac {x^2}{b^2}\right )^{7/4}} \, dx,x,b \tan (e+f x)\right )}{7 d^2 f (d \sec (e+f x))^{3/2}}\\ &=-\frac {2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \left (2 b \left (3 a^2+2 b^2\right )-a \left (5 a^2+3 b^2\right ) \tan (e+f x)\right )}{21 d^2 f (d \sec (e+f x))^{3/2}}+\frac {\left (a \left (6+\frac {5 a^2}{b^2}\right ) b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{21 d^2 f (d \sec (e+f x))^{3/2}}\\ &=\frac {2 a \left (5 a^2+6 b^2\right ) F\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{21 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \cos ^2(e+f x) (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{7 d^2 f (d \sec (e+f x))^{3/2}}-\frac {2 \left (2 b \left (3 a^2+2 b^2\right )-a \left (5 a^2+3 b^2\right ) \tan (e+f x)\right )}{21 d^2 f (d \sec (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 2.82, size = 150, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)} \left (4 \left (5 a^3+6 a b^2\right ) F\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} \left (-b \left (27 a^2+19 b^2\right ) \cos (e+f x)+\left (-9 a^2 b+3 b^3\right ) \cos (3 (e+f x))+2 a \left (13 a^2+3 b^2+3 \left (a^2-3 b^2\right ) \cos (2 (e+f x))\right ) \sin (e+f x)\right )\right )}{42 d^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.56, size = 391, normalized size = 2.30
method | result | size |
default | \(-\frac {2 \left (-5 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \cos \left (f x +e \right ) a^{3}-6 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \cos \left (f x +e \right ) a \,b^{2}+9 \left (\cos ^{4}\left (f x +e \right )\right ) a^{2} b -3 \left (\cos ^{4}\left (f x +e \right )\right ) b^{3}-3 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) a^{3}+9 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) a \,b^{2}-5 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) a^{3}-6 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) a \,b^{2}+7 b^{3} \left (\cos ^{2}\left (f x +e \right )\right )-5 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a^{3}-6 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a \,b^{2}\right )}{21 f \cos \left (f x +e \right )^{4} \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}}}\) | \(391\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.15, size = 191, normalized size = 1.12 \begin {gather*} \frac {\sqrt {2} {\left (-5 i \, a^{3} - 6 i \, a b^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2} {\left (5 i \, a^{3} + 6 i \, a b^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, {\left (7 \, b^{3} \cos \left (f x + e\right )^{2} + 3 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (f x + e\right )^{4} - {\left (3 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (5 \, a^{3} + 6 \, a b^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{21 \, d^{4} f} \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 )^{3}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, 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 )}^3}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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