Optimal. Leaf size=190 \[ -\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {a e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {a e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {2 a e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{3 d} \]
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Rubi [A] time = 0.17, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3878, 3872, 2838, 2564, 325, 329, 212, 206, 203, 2636, 2641} \[ -\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {a e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {a e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d}+\frac {2 a e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 325
Rule 329
Rule 2564
Rule 2636
Rule 2641
Rule 2838
Rule 3872
Rule 3878
Rubi steps
\begin {align*} \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx &=\left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {a+a \sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx\\ &=-\left (\left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {(-a-a \cos (c+d x)) \sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx\right )\\ &=\left (a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {5}{2}}(c+d x)} \, dx+\left (a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx\\ &=-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {1}{3} \left (a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx+\frac {\left (a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{5/2} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {2 a e^2 \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)}}{3 d}+\frac {\left (a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {2 a e^2 \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)}}{3 d}+\frac {\left (2 a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d}\\ &=-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {2 a e^2 \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)}}{3 d}+\frac {\left (a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d}+\frac {\left (a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d}\\ &=-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {a e^2 \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {a e^2 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a e^2 \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)}}{3 d}\\ \end {align*}
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Mathematica [A] time = 1.52, size = 135, normalized size = 0.71 \[ -\frac {a (e \csc (c+d x))^{5/2} \left (4 \cot \left (\frac {1}{2} (c+d x)\right ) \sqrt {\csc (c+d x)}+3 \log \left (1-\sqrt {\csc (c+d x)}\right )-3 \log \left (\sqrt {\csc (c+d x)}+1\right )+6 \tan ^{-1}\left (\sqrt {\csc (c+d x)}\right )+4 \sqrt {\sin (c+d x)} \sqrt {\csc (c+d x)} F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{6 d \csc ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e^{2} \csc \left (d x + c\right )^{2} \sec \left (d x + c\right ) + a e^{2} \csc \left (d x + c\right )^{2}\right )} \sqrt {e \csc \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.10, size = 679, normalized size = 3.57 \[ \frac {a \left (-1+\cos \left (d x +c \right )\right ) \left (4 i \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 ) \sin \left (d x +c \right ) \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\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 \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}-3 i \EllipticPi \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c \right ) \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\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 \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}-3 i \EllipticPi \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c \right ) \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\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 \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}-3 \EllipticPi \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c \right ) \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\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 \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}+3 \EllipticPi \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c \right ) \sqrt {\frac {-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i}{\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 \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}+2 \sqrt {2}\right ) \left (1+\cos \left (d x +c \right )\right )^{2} \left (\frac {e}{\sin \left (d x +c \right )}\right )^{\frac {5}{2}} \sqrt {2}}{6 d \sin \left (d x +c \right )} \]
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 \[ \int \left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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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