Optimal. Leaf size=222 \[ -\frac {4 a^2}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}+\frac {2 a^2 \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}+\frac {2 a^2 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]
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Rubi [A] time = 0.31, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3878, 3872, 2873, 2635, 2641, 2564, 321, 329, 212, 206, 203, 2566} \[ -\frac {4 a^2}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}+\frac {2 a^2 \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}+\frac {2 a^2 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 321
Rule 329
Rule 2564
Rule 2566
Rule 2635
Rule 2641
Rule 2873
Rule 3872
Rule 3878
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx &=\frac {\int (a+a \sec (c+d x))^2 \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int (-a-a \cos (c+d x))^2 \sec ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \left (a^2 \sin ^{\frac {3}{2}}(c+d x)+2 a^2 \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x)+a^2 \sec ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x)\right ) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {a^2 \int \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 \int \sec ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \int \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {a^2 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {x^{3/2}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{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 {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{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 {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {\sin (c+d x)}\right )}{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 {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{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 {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}+\frac {2 a^2 \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a^2 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 7.82, size = 164, normalized size = 0.74 \[ \frac {2 a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \tan (c+d x) \sqrt {e \csc (c+d x)} \sec ^4\left (\frac {1}{2} \csc ^{-1}(\csc (c+d x))\right ) \left (-6 \sqrt {\cos ^2(c+d x)} \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};\csc ^2(c+d x)\right )+3 \sqrt {-\cot ^2(c+d x)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\csc ^2(c+d x)\right )+\sin ^2(c+d x) \sqrt {-\cot ^2(c+d x)} \, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};\csc ^2(c+d x)\right )+3\right )}{3 d e^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}\right )} \sqrt {e \csc \left (d x + c\right )}}{e^{2} \csc \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.33, size = 763, normalized size = 3.44 \[ -\frac {a^{2} \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 ) \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 )+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 ) \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 )-13 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 \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 ) \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 )+6 \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 ) \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 )+2 \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )+10 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-15 \cos \left (d x +c \right ) \sqrt {2}+3 \sqrt {2}\right ) \sqrt {2}}{6 d \left (-1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right ) \left (\frac {e}{\sin \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right )} \]
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 \[ \int \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ a^{2} \left (\int \frac {1}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {2 \sec {\left (c + d x \right )}}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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