Optimal. Leaf size=213 \[ -\frac {152 a^4 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {32 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{7 d}+\frac {152 a^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {32 a^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {122 a^4 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {8 a^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \]
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Rubi [A]
time = 0.20, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3876, 3853,
3856, 2719, 2720} \begin {gather*} \frac {2 a^4 \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {8 a^4 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}+\frac {122 a^4 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{45 d}+\frac {32 a^4 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{7 d}+\frac {152 a^4 \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {32 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d}-\frac {152 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 3876
Rubi steps
\begin {align*} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \, dx &=\int \left (a^4 \sec ^{\frac {3}{2}}(c+d x)+4 a^4 \sec ^{\frac {5}{2}}(c+d x)+6 a^4 \sec ^{\frac {7}{2}}(c+d x)+4 a^4 \sec ^{\frac {9}{2}}(c+d x)+a^4 \sec ^{\frac {11}{2}}(c+d x)\right ) \, dx\\ &=a^4 \int \sec ^{\frac {3}{2}}(c+d x) \, dx+a^4 \int \sec ^{\frac {11}{2}}(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^{\frac {9}{2}}(c+d x) \, dx+\left (6 a^4\right ) \int \sec ^{\frac {7}{2}}(c+d x) \, dx\\ &=\frac {2 a^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {8 a^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {12 a^4 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {8 a^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{9} \left (7 a^4\right ) \int \sec ^{\frac {7}{2}}(c+d x) \, dx-a^4 \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (4 a^4\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{7} \left (20 a^4\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{5} \left (18 a^4\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {46 a^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {32 a^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {122 a^4 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {8 a^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{15} \left (7 a^4\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{21} \left (20 a^4\right ) \int \sqrt {\sec (c+d x)} \, dx-\frac {1}{5} \left (18 a^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx-\left (a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{3} \left (4 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 a^4 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{d}+\frac {8 a^4 \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 {152 a^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {32 a^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {122 a^4 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {8 a^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}-\frac {1}{15} \left (7 a^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (20 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{5} \left (18 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {46 a^4 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {32 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{7 d}+\frac {152 a^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {32 a^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {122 a^4 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {8 a^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}-\frac {1}{15} \left (7 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {152 a^4 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {32 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{7 d}+\frac {152 a^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {32 a^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {122 a^4 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {8 a^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^4 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 3.84, size = 287, normalized size = 1.35 \begin {gather*} \frac {a^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) (1+\sec (c+d x))^4 \left (-\frac {4 i \sqrt {2} e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^4(c+d x) \left (399 e^{i c} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )+e^{i d x} \left (180 \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 )+133 e^{i (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )\right )\right )}{-1+e^{2 i c}}+\frac {1596 \cos (d x) \csc (c)+\left (720+427 \sec (c+d x)+180 \sec ^2(c+d x)+35 \sec ^3(c+d x)\right ) \tan (c+d x)}{\sec ^{\frac {7}{2}}(c+d x)}\right )}{2520 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(491\) vs.
\(2(233)=466\).
time = 0.12, size = 492, normalized size = 2.31
method | result | size |
default | \(-\frac {a^{4} \sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{7 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{4}}-\frac {16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{7 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{2}}+\frac {1544 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{105 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{72 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{5}}-\frac {61 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{90 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{3}}-\frac {304 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 \sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {152 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (\EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{15 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}\right )}{\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}\) | \(492\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 0.70, size = 228, normalized size = 1.07 \begin {gather*} -\frac {2 \, {\left (360 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 360 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 798 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{4} {\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 ) - 798 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{4} {\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 (1596 \, a^{4} \cos \left (d x + c\right )^{4} + 720 \, a^{4} \cos \left (d x + c\right )^{3} + 427 \, a^{4} \cos \left (d x + c\right )^{2} + 180 \, a^{4} \cos \left (d x + c\right ) + 35 \, a^{4}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d \cos \left (d x + c\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^4\,{\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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