Optimal. Leaf size=139 \[ \frac {2 a^2 \tan ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{d e}+\frac {3 a^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {e \sin (c+d x)}} \]
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Rubi [A] time = 0.31, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3872, 2873, 2642, 2641, 2564, 329, 212, 206, 203, 2571} \[ \frac {2 a^2 \tan ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{d e}+\frac {3 a^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 2564
Rule 2571
Rule 2641
Rule 2642
Rule 2873
Rule 3872
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^2}{\sqrt {e \sin (c+d x)}} \, dx &=\int \frac {(-a-a \cos (c+d x))^2 \sec ^2(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\int \left (\frac {a^2}{\sqrt {e \sin (c+d x)}}+\frac {2 a^2 \sec (c+d x)}{\sqrt {e \sin (c+d x)}}+\frac {a^2 \sec ^2(c+d x)}{\sqrt {e \sin (c+d x)}}\right ) \, dx\\ &=a^2 \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx+a^2 \int \frac {\sec ^2(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx+\left (2 a^2\right ) \int \frac {\sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\frac {a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{d e}+\frac {1}{2} a^2 \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d e}+\frac {\left (a^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{\sqrt {e \sin (c+d x)}}\\ &=\frac {2 a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{d e}+\frac {\left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {x^4}{e^2}} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e}+\frac {\left (a^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{2 \sqrt {e \sin (c+d x)}}\\ &=\frac {3 a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{d e}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{e-x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{e+x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}\\ &=\frac {2 a^2 \tan ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {3 a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{d e}\\ \end {align*}
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Mathematica [C] time = 71.89, size = 164, normalized size = 1.18 \[ \frac {a^2 \sqrt {\sin (c+d x)} \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \sec ^4\left (\frac {1}{2} \sin ^{-1}(\sin (c+d x))\right ) \left (3 \sqrt {\sin (c+d x)} \sqrt {\cos ^2(c+d x)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(c+d x)\right )+\sqrt {\sin (c+d x)}+2 \sqrt {\cos ^2(c+d x)} \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right )+2 \sqrt {\cos ^2(c+d x)} \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )\right )}{d \sqrt {e \sin (c+d x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}\right )} \sqrt {e \sin \left (d x + c\right )}}{e \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\sqrt {e \sin \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 5.25, size = 163, normalized size = 1.17 \[ \frac {a^{2} \left (-3 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {e}+4 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, \arctanh \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )+4 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )+2 \sqrt {e}\, \sin \left (d x +c \right )\right )}{2 \sqrt {e}\, \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d} \]
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 \[ \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{\sqrt {e\,\sin \left (c+d\,x\right )}} \,d x \]
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 \[ a^{2} \left (\int \frac {1}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx + \int \frac {2 \sec {\left (c + d x \right )}}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx + \int \frac {\sec ^{2}{\left (c + d x \right )}}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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