Optimal. Leaf size=240 \[ -\frac {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {2 a^2 e \text {ArcTan}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a^2 e \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}-\frac {5 a^2 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d}+\frac {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d} \]
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Rubi [A]
time = 0.23, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3963, 3957,
2952, 2716, 2719, 2644, 331, 335, 304, 209, 212, 2650, 2651} \begin {gather*} -\frac {2 a^2 e \sqrt {\sin (c+d x)} \text {ArcTan}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)}}{d}-\frac {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sec (c+d x) \sqrt {e \csc (c+d x)}}{d}+\frac {3 a^2 e \sin (c+d x) \tan (c+d x) \sqrt {e \csc (c+d x)}}{d}+\frac {2 a^2 e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d}-\frac {5 a^2 e \sqrt {\sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 304
Rule 331
Rule 335
Rule 2644
Rule 2650
Rule 2651
Rule 2716
Rule 2719
Rule 2952
Rule 3957
Rule 3963
Rubi steps
\begin {align*} \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx &=\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^2}{\sin ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {(-a-a \cos (c+d x))^2 \sec ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \left (\frac {a^2}{\sin ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 \sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \sec ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)}\right ) \, dx\\ &=\left (a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x)} \, dx+\left (a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\sec ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx+\left (2 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx\\ &=-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\left (a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx+\left (3 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sec ^2(c+d x) \sqrt {\sin (c+d x)} \, dx+\frac {\left (2 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{x^{3/2} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d}+\frac {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}-\frac {1}{2} \left (3 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx+\frac {\left (2 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {5 a^2 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d}+\frac {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}+\frac {\left (4 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d}\\ &=-\frac {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {5 a^2 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d}+\frac {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}+\frac {\left (2 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d}-\frac {\left (2 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d}\\ &=-\frac {4 a^2 e \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a^2 e \sqrt {e \csc (c+d x)} \sec (c+d x)}{d}-\frac {2 a^2 e \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a^2 e \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}-\frac {5 a^2 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d}+\frac {3 a^2 e \sqrt {e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 28.00, size = 307, normalized size = 1.28 \begin {gather*} \frac {\left (1+\cos \left (2 \left (\frac {c}{2}+\frac {d x}{2}\right )\right )\right )^2 \cos (c+d x) (e \csc (c+d x))^{3/2} \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 \left (2 \sqrt {\csc (c+d x)}+\frac {\csc ^{\frac {5}{2}}(c+d x) \sqrt {1-\sin ^2(c+d x)}}{2-2 \csc ^2(c+d x)}\right )}{d}-\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 {5 \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 {7}{2}}(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 = 0.20, size = 1559, normalized size = 6.50
method | result | size |
default | \(\text {Expression too large to display}\) | \(1559\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 1.27, size = 247, normalized size = 1.03 \begin {gather*} -\frac {5 \, \sqrt {2 i} a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} {\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 ) + 5 \, \sqrt {-2 i} a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} {\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 {\sin \left (d x + c\right ) - 1}{2 \, \sqrt {\sin \left (d x + c\right )}}\right ) \cos \left (d x + c\right ) e^{\frac {3}{2}} - a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} \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 (5 \, a^{2} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} + 4 \, a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} - a^{2} e^{\frac {3}{2}}\right )}}{\sqrt {\sin \left (d x + c\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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.00 \begin {gather*} \int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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