Optimal. Leaf size=104 \[ -\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {4 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{b d^2 \sqrt {\sin (2 a+2 b x)}}+\frac {4 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d^3} \]
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Rubi [A]
time = 0.10, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2688, 2693,
2695, 2652, 2719} \begin {gather*} \frac {4 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d^3}-\frac {4 \cos (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2652
Rule 2688
Rule 2693
Rule 2695
Rule 2719
Rubi steps
\begin {align*} \int \frac {\sec ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {2 \int \sec (a+b x) \sqrt {d \tan (a+b x)} \, dx}{d^2}\\ &=-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {4 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d^3}-\frac {4 \int \cos (a+b x) \sqrt {d \tan (a+b x)} \, dx}{d^2}\\ &=-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {4 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d^3}-\frac {\left (4 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)} \, dx}{d^2 \sqrt {\sin (a+b x)}}\\ &=-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {4 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d^3}-\frac {\left (4 \cos (a+b x) \sqrt {d \tan (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{d^2 \sqrt {\sin (2 a+2 b x)}}\\ &=-\frac {2 \sec (a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {4 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{b d^2 \sqrt {\sin (2 a+2 b x)}}+\frac {4 \cos (a+b x) (d \tan (a+b x))^{3/2}}{b d^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.47, size = 93, normalized size = 0.89 \begin {gather*} -\frac {2 \csc (a+b x) \sqrt {d \tan (a+b x)} \left (3 \cos (2 (a+b x)) \sqrt {\sec ^2(a+b x)}+4 \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(a+b x)\right ) \tan ^2(a+b x)\right )}{3 b d^2 \sqrt {\sec ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs.
\(2(121)=242\).
time = 0.34, size = 502, normalized size = 4.83
method | result | size |
default | \(-\frac {\left (-4 \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticE \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 ) \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 )+2 \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \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 )}}\, \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 )-4 \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticE \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 ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}+2 \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\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 ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}+2 \cos \left (b x +a \right ) \sqrt {2}-\sqrt {2}\right ) \sin \left (b x +a \right ) \sqrt {2}}{b \cos \left (b x +a \right )^{2} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}}\) | \(502\) |
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\sec ^{3}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{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 {1}{{\cos \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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