Optimal. Leaf size=67 \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 d}+\frac {2 b \sin (c+d x)}{3 d \sqrt {b \sec (c+d x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3854, 3856,
2720} \begin {gather*} \frac {2 b \sin (c+d x)}{3 d \sqrt {b \sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2720
Rule 3854
Rule 3856
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \sqrt {b \sec (c+d x)} \, dx &=b^2 \int \frac {1}{(b \sec (c+d x))^{3/2}} \, dx\\ &=\frac {2 b \sin (c+d x)}{3 d \sqrt {b \sec (c+d x)}}+\frac {1}{3} \int \sqrt {b \sec (c+d x)} \, dx\\ &=\frac {2 b \sin (c+d x)}{3 d \sqrt {b \sec (c+d x)}}+\frac {1}{3} \left (\sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 d}+\frac {2 b \sin (c+d x)}{3 d \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 51, normalized size = 0.76 \begin {gather*} \frac {\sqrt {b \sec (c+d x)} \left (2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sin (2 (c+d x))\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 31.47, size = 123, normalized size = 1.84
method | result | size |
default | \(\frac {2 \left (\cos \left (d x +c \right )-1\right ) \left (-i \EllipticF \left (\frac {i \left (\cos \left (d x +c \right )-1\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right )+\cos ^{2}\left (d x +c \right )-\cos \left (d x +c \right )\right ) \left (\cos \left (d x +c \right )+1\right )^{2} \sqrt {\frac {b}{\cos \left (d x +c \right )}}}{3 d \sin \left (d x +c \right )^{3}}\) | \(123\) |
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 = 0.85, size = 84, normalized size = 1.25 \begin {gather*} \frac {2 \, \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )}{3 \, d} \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 \sqrt {b \sec {\left (c + d x \right )}} \cos ^{2}{\left (c + d 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 {\cos \left (c+d\,x\right )}^2\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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