Optimal. Leaf size=253 \[ -\frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (35 A+13 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {8 a^3 (70 A+53 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {12 C \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d} \]
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Rubi [A]
time = 0.37, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4174, 4103,
4082, 3872, 3856, 2719, 2720} \begin {gather*} \frac {8 a^3 (70 A+53 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{105 d}+\frac {2 (5 A+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac {4 a^3 (35 A+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {12 C \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{35 a d}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3856
Rule 3872
Rule 4082
Rule 4103
Rule 4174
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx &=\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {2 \int \frac {(a+a \sec (c+d x))^3 \left (\frac {1}{2} a (7 A-C)+3 a C \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{7 a}\\ &=\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {12 C \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {4 \int \frac {(a+a \sec (c+d x))^2 \left (\frac {1}{4} a^2 (35 A-11 C)+\frac {7}{4} a^2 (5 A+7 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{35 a}\\ &=\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {12 C \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {8 \int \frac {(a+a \sec (c+d x)) \left (\frac {1}{4} a^3 (35 A-41 C)+\frac {1}{2} a^3 (70 A+53 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{105 a}\\ &=\frac {8 a^3 (70 A+53 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {12 C \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {16 \int \frac {-\frac {21}{8} a^4 (5 A+7 C)+\frac {5}{8} a^4 (35 A+13 C) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{105 a}\\ &=\frac {8 a^3 (70 A+53 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {12 C \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {1}{5} \left (2 a^3 (5 A+7 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^3 (35 A+13 C)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {8 a^3 (70 A+53 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {12 C \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {1}{5} \left (2 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (2 a^3 (35 A+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {4 a^3 (5 A+7 C) \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 {4 a^3 (35 A+13 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {8 a^3 (70 A+53 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {12 C \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 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 = 3.74, size = 280, normalized size = 1.11 \begin {gather*} \frac {a^3 e^{-i d x} \sec ^{\frac {7}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left (-630 i A-882 i C-840 i A \cos (2 (c+d x))-1176 i C \cos (2 (c+d x))-210 i A \cos (4 (c+d x))-294 i C \cos (4 (c+d x))+80 (35 A+13 C) \cos ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+14 i (5 A+7 C) e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{7/2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+70 A \sin (c+d x)+380 C \sin (c+d x)+630 A \sin (2 (c+d x))+840 C \sin (2 (c+d x))+70 A \sin (3 (c+d x))+260 C \sin (3 (c+d x))+315 A \sin (4 (c+d x))+294 C \sin (4 (c+d x))\right )}{420 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. \(986\) vs.
\(2(277)=554\).
time = 11.06, size = 987, normalized size = 3.90
method | result | size |
default | \(\text {Expression too large to display}\) | \(987\) |
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 = 0.85, size = 259, normalized size = 1.02 \begin {gather*} -\frac {2 \, {\left (5 i \, \sqrt {2} {\left (35 \, A + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (35 \, A + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\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 ) - 21 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\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 (21 \, {\left (15 \, A + 14 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 5 \, {\left (7 \, A + 26 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 63 \, C a^{3} \cos \left (d x + c\right ) + 15 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d \cos \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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