Optimal. Leaf size=201 \[ \frac {4 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {4 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a^2 d}-\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {16 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a^2 d}+\frac {20 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{21 a^2 d} \]
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Rubi [A] time = 0.45, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3878, 3872, 2875, 2873, 2567, 2636, 2641, 2564, 14} \[ \frac {4 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {4 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a^2 d}-\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {16 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a^2 d}+\frac {20 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{21 a^2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rule 2567
Rule 2636
Rule 2641
Rule 2873
Rule 2875
Rule 3872
Rule 3878
Rubi steps
\begin {align*} \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx &=\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {\sin (c+d x)}} \, dx\\ &=\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{(-a-a \cos (c+d x))^2 \sqrt {\sin (c+d x)}} \, dx\\ &=\frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{\sin ^{\frac {9}{2}}(c+d x)} \, dx}{a^4}\\ &=\frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \left (\frac {a^2 \cos ^2(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}-\frac {2 a^2 \cos ^3(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}+\frac {a^2 \cos ^4(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}\right ) \, dx}{a^4}\\ &=\frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)} \, dx}{a^2}+\frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^4(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)} \, dx}{a^2}-\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^3(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)} \, dx}{a^2}\\ &=-\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{7 a^2}-\frac {\left (6 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{7 a^2}-\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{x^{9/2}} \, dx,x,\sin (c+d x)\right )}{a^2 d}\\ &=\frac {16 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a^2 d}-\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 a^2}+\frac {\left (4 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{7 a^2}-\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{9/2}}-\frac {1}{x^{5/2}}\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d}\\ &=\frac {16 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a^2 d}-\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {4 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {4 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {20 \sqrt {e \csc (c+d x)} F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 82, normalized size = 0.41 \[ -\frac {4 \csc ^3(c+d x) \sqrt {e \csc (c+d x)} \left (2 \sin ^4\left (\frac {1}{2} (c+d x)\right ) (11 \cos (c+d x)+8)+5 \sin ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{21 a^2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )}}{a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {e \csc \left (d x + c\right )}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.56, size = 474, normalized size = 2.36 \[ -\frac {\sqrt {\frac {e}{\sin \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{3} \left (10 i \sin \left (d x +c \right ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+20 i \cos \left (d x +c \right ) \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sin \left (d x +c \right ) \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+10 i \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+11 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-3 \cos \left (d x +c \right ) \sqrt {2}-8 \sqrt {2}\right ) \sqrt {2}}{21 a^{2} d \sin \left (d x +c \right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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 \[ \frac {\int \frac {\sqrt {e \csc {\left (c + d x \right )}}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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