Optimal. Leaf size=158 \[ \frac {4 a^2 A \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{d}+\frac {4 a^2 (2 A+3 B) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 d}-\frac {2 a^2 (A-3 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}} \]
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Rubi [A]
time = 0.18, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4102, 4082,
3872, 3856, 2719, 2720} \begin {gather*} -\frac {2 a^2 (A-3 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{3 d}+\frac {4 a^2 (2 A+3 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 A \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{3 d \sqrt {\sec (c+d x)}}+\frac {4 a^2 A \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3856
Rule 3872
Rule 4082
Rule 4102
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\frac {2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2}{3} \int \frac {(a+a \sec (c+d x)) \left (\frac {1}{2} a (5 A+3 B)-\frac {1}{2} a (A-3 B) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\\ &=-\frac {2 a^2 (A-3 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {4}{3} \int \frac {\frac {3 a^2 A}{2}+\frac {1}{2} a^2 (2 A+3 B) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx\\ &=-\frac {2 a^2 (A-3 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\left (2 a^2 A\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (2 a^2 (2 A+3 B)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=-\frac {2 a^2 (A-3 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\left (2 a^2 A \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{3} \left (2 a^2 (2 A+3 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {4 a^2 A \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{d}+\frac {4 a^2 (2 A+3 B) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 d}-\frac {2 a^2 (A-3 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 3.09, size = 299, normalized size = 1.89 \begin {gather*} \frac {a^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) (1+\sec (c+d x))^2 (A+B \sec (c+d x)) \left (\frac {i \sqrt {2} \cos ^3(c+d x) \left (3 A \sqrt {1+e^{2 i (c+d x)}}+3 A \left (-1+e^{2 i c}\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )-(2 A+3 B) e^{i (c+d x)} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right )\right )}{d \left (-1+e^{2 i c}\right ) \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}}}+\frac {-3 (2 A-B) \cos (d x) \csc (c)-3 (2 A+B) \cos (2 c+d x) \csc (c)+A \sin (2 (c+d x))}{4 d \sec ^{\frac {5}{2}}(c+d x)}\right )}{3 (B+A \cos (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.34, size = 245, normalized size = 1.55
method | result | size |
default | \(-\frac {4 a^{2} \left (2 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) A +2 A \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-3 A \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-3 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 B \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\right )}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(245\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.81, size = 166, normalized size = 1.05 \begin {gather*} -\frac {2 \, {\left (i \, \sqrt {2} {\left (2 \, A + 3 \, B\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - i \, \sqrt {2} {\left (2 \, A + 3 \, B\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 3 i \, \sqrt {2} A a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 i \, \sqrt {2} A a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (A a^{2} \cos \left (d x + c\right ) + 3 \, B a^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{3 \, d} \end {gather*}
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 \begin {gather*} a^{2} \left (\int \frac {A}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {2 A}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int A \sqrt {\sec {\left (c + d x \right )}}\, dx + \int \frac {B}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int 2 B \sqrt {\sec {\left (c + d x \right )}}\, dx + \int B \sec ^{\frac {3}{2}}{\left (c + d x \right )}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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