Optimal. Leaf size=80 \[ -\frac {d^2 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sec (a+b x) \sqrt {\sin (2 a+2 b x)}}{3 b \sqrt {d \tan (a+b x)}}+\frac {2 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b} \]
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Rubi [A]
time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2691, 2694,
2653, 2720} \begin {gather*} \frac {2 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b}-\frac {d^2 \sqrt {\sin (2 a+2 b x)} \sec (a+b x) F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{3 b \sqrt {d \tan (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2653
Rule 2691
Rule 2694
Rule 2720
Rubi steps
\begin {align*} \int \sec (a+b x) (d \tan (a+b x))^{3/2} \, dx &=\frac {2 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b}-\frac {1}{3} d^2 \int \frac {\sec (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\\ &=\frac {2 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b}-\frac {\left (d^2 \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}} \, dx}{3 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\\ &=\frac {2 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b}-\frac {\left (d^2 \sec (a+b x) \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{3 \sqrt {d \tan (a+b x)}}\\ &=-\frac {d^2 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sec (a+b x) \sqrt {\sin (2 a+2 b x)}}{3 b \sqrt {d \tan (a+b x)}}+\frac {2 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{3 b}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.32, size = 69, normalized size = 0.86 \begin {gather*} \frac {2 d \cos (a+b x) \left (\sec ^2(a+b x)-\, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) \sqrt {d \tan (a+b x)}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 188, normalized size = 2.35
method | result | size |
default | \(\frac {\left (-1+\cos \left (b x +a \right )\right ) \left (\sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \cos \left (b x +a \right ) \sin \left (b x +a \right ) \EllipticF \left (\sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+\cos \left (b x +a \right ) \sqrt {2}-\sqrt {2}\right ) \left (\cos \left (b x +a \right )+1\right )^{2} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \sqrt {2}}{3 b \sin \left (b x +a \right )^{5}}\) | \(188\) |
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 complex when optimal does not.
time = 0.09, size = 95, normalized size = 1.19 \begin {gather*} \frac {\sqrt {i \, d} d \cos \left (b x + a\right ) {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + \sqrt {-i \, d} d \cos \left (b x + a\right ) {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) + 2 \, d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{3 \, b \cos \left (b x + a\right )} \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 \left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}} \sec {\left (a + b x \right )}\, 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 (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}}{\cos \left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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