Optimal. Leaf size=172 \[ -\frac {4}{a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \cos ^3(c+d x)}{7 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {26 \cos (c+d x)}{21 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {4 \sin ^2(c+d x)}{5 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {52 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 a^2 d e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]
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Rubi [A] time = 0.46, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3878, 3872, 2875, 2873, 2569, 2641, 2564, 14} \[ -\frac {4}{a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \cos ^3(c+d x)}{7 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {26 \cos (c+d x)}{21 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {4 \sin ^2(c+d x)}{5 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {52 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 a^2 d e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rule 2569
Rule 2641
Rule 2873
Rule 2875
Rule 3872
Rule 3878
Rubi steps
\begin {align*} \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx &=\frac {\int \frac {\sin ^{\frac {7}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx}{e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \frac {\cos ^2(c+d x) \sin ^{\frac {7}{2}}(c+d x)}{(-a-a \cos (c+d x))^2} \, dx}{e^3 \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}{\sqrt {\sin (c+d x)}} \, dx}{a^4 e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \left (\frac {a^2 \cos ^2(c+d x)}{\sqrt {\sin (c+d x)}}-\frac {2 a^2 \cos ^3(c+d x)}{\sqrt {\sin (c+d x)}}+\frac {a^2 \cos ^4(c+d x)}{\sqrt {\sin (c+d x)}}\right ) \, dx}{a^4 e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \frac {\cos ^2(c+d x)}{\sqrt {\sin (c+d x)}} \, dx}{a^2 e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\int \frac {\cos ^4(c+d x)}{\sqrt {\sin (c+d x)}} \, dx}{a^2 e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \int \frac {\cos ^3(c+d x)}{\sqrt {\sin (c+d x)}} \, dx}{a^2 e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {2 \cos (c+d x)}{3 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \cos ^3(c+d x)}{7 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^2 e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {6 \int \frac {\cos ^2(c+d x)}{\sqrt {\sin (c+d x)}} \, dx}{7 a^2 e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {x}} \, dx,x,\sin (c+d x)\right )}{a^2 d e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {26 \cos (c+d x)}{21 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \cos ^3(c+d x)}{7 a^2 d e^3 \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^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {4 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{7 a^2 e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {x}}-x^{3/2}\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {4}{a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {26 \cos (c+d x)}{21 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \cos ^3(c+d x)}{7 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {52 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{21 a^2 d e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {4 \sin ^2(c+d x)}{5 a^2 d e^3 \sqrt {e \csc (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 2.20, size = 94, normalized size = 0.55 \[ \frac {\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (\sqrt {\sin (c+d x)} (305 \cos (c+d x)-84 \cos (2 (c+d x))+15 \cos (3 (c+d x))-756)-520 F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{210 a^2 d e^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )}}{a^{2} e^{4} \csc \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + 2 \, a^{2} e^{4} \csc \left (d x + c\right )^{4} \sec \left (d x + c\right ) + a^{2} e^{4} \csc \left (d x + c\right )^{4}}, x\right ) \]
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 \[ \int \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.24, size = 224, normalized size = 1.30 \[ \frac {\left (-130 i \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+15 \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}-57 \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )+107 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-233 \cos \left (d x +c \right ) \sqrt {2}+168 \sqrt {2}\right ) \sqrt {2}}{105 a^{2} d \left (-1+\cos \left (d x +c \right )\right ) \left (\frac {e}{\sin \left (d x +c \right )}\right )^{\frac {7}{2}} \sin \left (d x +c \right )^{3}} \]
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 = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{7/2}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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