Optimal. Leaf size=153 \[ -\frac {2 a^2 \text {ArcTan}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a^2 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 \tan (c+d x)}{d \sqrt {e \csc (c+d x)}} \]
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Rubi [A]
time = 0.19, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3963, 3957,
2952, 2719, 2644, 335, 304, 209, 212, 2651} \begin {gather*} -\frac {2 a^2 \text {ArcTan}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}+\frac {a^2 \tan (c+d x)}{d \sqrt {e \csc (c+d x)}}+\frac {2 a^2 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}+\frac {a^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 2644
Rule 2651
Rule 2719
Rule 2952
Rule 3957
Rule 3963
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^2}{\sqrt {e \csc (c+d x)}} \, dx &=\frac {\int (a+a \sec (c+d x))^2 \sqrt {\sin (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int (-a-a \cos (c+d x))^2 \sec ^2(c+d x) \sqrt {\sin (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \left (a^2 \sqrt {\sin (c+d x)}+2 a^2 \sec (c+d x) \sqrt {\sin (c+d x)}+a^2 \sec ^2(c+d x) \sqrt {\sin (c+d x)}\right ) \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {a^2 \int \sqrt {\sin (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 \int \sec ^2(c+d x) \sqrt {\sin (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \int \sec (c+d x) \sqrt {\sin (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {2 a^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 \tan (c+d x)}{d \sqrt {e \csc (c+d x)}}-\frac {a^2 \int \sqrt {\sin (c+d x)} \, dx}{2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {a^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 \tan (c+d x)}{d \sqrt {e \csc (c+d x)}}+\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {a^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 \tan (c+d x)}{d \sqrt {e \csc (c+d x)}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {2 a^2 \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a^2 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 \tan (c+d x)}{d \sqrt {e \csc (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 27.70, size = 288, normalized size = 1.88 \begin {gather*} \frac {\left (1+\cos \left (2 \left (\frac {c}{2}+\frac {d x}{2}\right )\right )\right )^2 \cos (c+d x) \left (-1+\csc ^2(c+d x)\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \left (-\frac {-2+\csc ^2(c+d x)}{d \csc ^{\frac {3}{2}}(c+d x) \sqrt {1-\sin ^2(c+d x)}}-\frac {-6 \text {ArcTan}\left (\sqrt {\csc (c+d x)}\right )+3 \log \left (1-\sqrt {\csc (c+d x)}\right )-3 \log \left (1+\sqrt {\csc (c+d x)}\right )+\frac {\csc ^{\frac {5}{2}}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\csc ^2(c+d x)\right ) \sqrt {1-\sin ^2(c+d x)}}{\sqrt {1-\csc ^2(c+d x)}}}{3 d}\right )}{4 \left (1+\cos \left (2 \left (\frac {c}{2}+\frac {1}{2} \left (-c+\csc ^{-1}(\csc (c+d x))\right )\right )\right )\right )^2 \csc ^{\frac {3}{2}}(c+d x) \sqrt {e \csc (c+d x)} \sqrt {1-\sin ^2(c+d x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains complex when optimal does not.
time = 0.23, size = 1605, normalized size = 10.49
method | result | size |
default | \(\text {Expression too large to display}\) | \(1605\) |
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 = 1.04, size = 272, normalized size = 1.78 \begin {gather*} \frac {{\left (\sqrt {2 i} a^{2} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {-2 i} a^{2} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, a^{2} \arctan \left (\frac {76 \, \cos \left (d x + c\right )^{2} + \frac {425 \, {\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right ) - 1\right )}}{\sqrt {\sin \left (d x + c\right )}} - 152 \, \sin \left (d x + c\right ) - 152}{2 \, {\left (16 \, \cos \left (d x + c\right )^{2} + 393 \, \sin \left (d x + c\right ) - 32\right )}}\right ) \cos \left (d x + c\right ) + a^{2} \cos \left (d x + c\right ) \log \left (\frac {\cos \left (d x + c\right )^{2} + \frac {4 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right ) - 1\right )}}{\sqrt {\sin \left (d x + c\right )}} - 6 \, \sin \left (d x + c\right ) - 2}{\cos \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) - 2}\right ) - \frac {2 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )}}{\sqrt {\sin \left (d x + c\right )}}\right )} e^{\left (-\frac {1}{2}\right )}}{2 \, d \cos \left (d x + c\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*} a^{2} \left (\int \frac {1}{\sqrt {e \csc {\left (c + d x \right )}}}\, dx + \int \frac {2 \sec {\left (c + d x \right )}}{\sqrt {e \csc {\left (c + d x \right )}}}\, dx + \int \frac {\sec ^{2}{\left (c + d x \right )}}{\sqrt {e \csc {\left (c + d x \right )}}}\, dx\right ) \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+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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