Optimal. Leaf size=239 \[ \frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{d}+\frac {2 \left (a^3 A+9 a A b^2+9 a^2 b B+b^3 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 d}-\frac {2 b \left (2 a^2 A-3 A b^2-9 a b B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}-\frac {2 b^2 (a A-b B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}} \]
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Rubi [A]
time = 0.33, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4110, 4161,
4132, 3856, 2720, 4131, 2719} \begin {gather*} -\frac {2 b \left (2 a^2 A-9 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{3 d}+\frac {2 \left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \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 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3856
Rule 4110
Rule 4131
Rule 4132
Rule 4161
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}-\frac {2}{3} \int \frac {(a+b \sec (c+d x)) \left (-\frac {1}{2} a (7 A b+3 a B)-\frac {1}{2} \left (a^2 A+3 A b^2+6 a b B\right ) \sec (c+d x)+\frac {3}{2} b (a A-b B) \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\\ &=-\frac {2 b^2 (a A-b B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}-\frac {4}{9} \int \frac {-\frac {3}{4} a^2 (7 A b+3 a B)-\frac {3}{4} \left (a^3 A+9 a A b^2+9 a^2 b B+b^3 B\right ) \sec (c+d x)+\frac {3}{4} b \left (2 a^2 A-3 A b^2-9 a b B\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx\\ &=-\frac {2 b^2 (a A-b B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}-\frac {4}{9} \int \frac {-\frac {3}{4} a^2 (7 A b+3 a B)+\frac {3}{4} b \left (2 a^2 A-3 A b^2-9 a b B\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx-\frac {1}{3} \left (-a^3 A-9 a A b^2-9 a^2 b B-b^3 B\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=-\frac {2 b \left (2 a^2 A-3 A b^2-9 a b B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}-\frac {2 b^2 (a A-b B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}-\left (-3 a^2 A b+A b^3-a^3 B+3 a b^2 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx-\frac {1}{3} \left (\left (-a^3 A-9 a A b^2-9 a^2 b B-b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (a^3 A+9 a A b^2+9 a^2 b B+b^3 B\right ) \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 b \left (2 a^2 A-3 A b^2-9 a b B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}-\frac {2 b^2 (a A-b B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}-\left (\left (-3 a^2 A b+A b^3-a^3 B+3 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{d}+\frac {2 \left (a^3 A+9 a A b^2+9 a^2 b B+b^3 B\right ) \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 b \left (2 a^2 A-3 A b^2-9 a b B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}-\frac {2 b^2 (a A-b B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 1.98, size = 166, normalized size = 0.69 \begin {gather*} \frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (6 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \left (a^3 A+9 a A b^2+9 a^2 b B+b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {\left (a^3 A+2 b^3 B+6 b^2 (A b+3 a B) \cos (c+d x)+a^3 A \cos (2 (c+d x))\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}\right )}{3 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. \(1211\) vs.
\(2(269)=538\).
time = 3.90, size = 1212, normalized size = 5.07
method | result | size |
default | \(\text {Expression too large to display}\) | \(1212\) |
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 = 0.78, size = 298, normalized size = 1.25 \begin {gather*} \frac {\sqrt {2} {\left (-i \, A a^{3} - 9 i \, B a^{2} b - 9 i \, A a b^{2} - i \, B b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} {\left (i \, A a^{3} + 9 i \, B a^{2} b + 9 i \, A a b^{2} + i \, B b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 3 \, \sqrt {2} {\left (-i \, B a^{3} - 3 i \, A a^{2} b + 3 i \, B a b^{2} + i \, A b^{3}\right )} \cos \left (d x + c\right ) {\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 \, \sqrt {2} {\left (i \, B a^{3} + 3 i \, A a^{2} b - 3 i \, B a b^{2} - i \, A b^{3}\right )} \cos \left (d x + c\right ) {\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 {2 \, {\left (A a^{3} \cos \left (d x + c\right )^{2} + B b^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3 \, d \cos \left (d x + c\right )} \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*} \int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \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 {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3}{{\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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