Optimal. Leaf size=213 \[ \frac {4 \csc ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}+\frac {4}{a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 \cos (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{a^2 d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]
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Rubi [A] time = 0.48, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 13, 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, 2641, 2564, 14, 2569} \[ \frac {4 \csc ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}+\frac {4}{a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 \cos (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{a^2 d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rule 2567
Rule 2569
Rule 2641
Rule 2873
Rule 2875
Rule 3872
Rule 3878
Rubi steps
\begin {align*} \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx &=\frac {\int \frac {\sin ^{\frac {3}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \frac {\cos ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x)}{(-a-a \cos (c+d x))^2} \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a^4 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \left (\frac {a^2 \cos ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}-\frac {2 a^2 \cos ^3(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}+\frac {a^2 \cos ^4(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}\right ) \, dx}{a^4 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \frac {\cos ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\int \frac {\cos ^4(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \int \frac {\cos ^3(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \int \frac {\cos ^2(c+d x)}{\sqrt {\sin (c+d x)}} \, dx}{a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \operatorname {Subst}\left (\int \frac {1-x^2}{x^{5/2}} \, dx,x,\sin (c+d x)\right )}{a^2 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {4 \cos (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 a^2 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {4 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \operatorname {Subst}\left (\int \left (\frac {1}{x^{5/2}}-\frac {1}{\sqrt {x}}\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {4}{a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 \cos (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}+\frac {4 \csc ^2(c+d x)}{3 a^2 d e \sqrt {e \csc (c+d x)}}-\frac {4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{a^2 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 101, normalized size = 0.47 \[ \frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (\sqrt {\sin (c+d x)} (10 \cos (c+d x)-\cos (2 (c+d x))+15)+12 (\cos (c+d x)+1) F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{6 a^2 d \sin ^{\frac {3}{2}}(c+d x) (e \csc (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )}}{a^{2} e^{2} \csc \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} e^{2} \csc \left (d x + c\right )^{2} \sec \left (d x + c\right ) + a^{2} e^{2} \csc \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.23, size = 332, normalized size = 1.56 \[ \frac {\left (-6 i \cos \left (d x +c \right ) \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sin \left (d x +c \right ) \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )-6 i \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )-6 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-3 \cos \left (d x +c \right ) \sqrt {2}+8 \sqrt {2}\right ) \sqrt {2}}{3 a^{2} d \left (\frac {e}{\sin \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right )^{3}} \]
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 \[ \int \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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 \[ \frac {\int \frac {1}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec ^{2}{\left (c + d x \right )} + 2 \left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec {\left (c + d x \right )} + \left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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