Optimal. Leaf size=234 \[ -\frac {2 a \left (5 a^2+9 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 b \left (21 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a \left (5 a^2+9 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b \left (21 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {32 a b^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d} \]
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Rubi [A]
time = 0.17, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3927, 4132,
3853, 3856, 2720, 4131, 2719} \begin {gather*} \frac {2 b \left (21 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 a \left (5 a^2+9 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 b \left (21 a^2+5 b^2\right ) \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 {2 a \left (5 a^2+9 b^2\right ) \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 {2 b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {32 a b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{35 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 3927
Rule 4131
Rule 4132
Rubi steps
\begin {align*} \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx &=\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {2}{7} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a \left (7 a^2+3 b^2\right )+\frac {1}{2} b \left (21 a^2+5 b^2\right ) \sec (c+d x)+8 a b^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {2}{7} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a \left (7 a^2+3 b^2\right )+8 a b^2 \sec ^2(c+d x)\right ) \, dx+\frac {1}{7} \left (b \left (21 a^2+5 b^2\right )\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {2 b \left (21 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {32 a b^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{21} \left (b \left (21 a^2+5 b^2\right )\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{5} \left (a \left (5 a^2+9 b^2\right )\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 a \left (5 a^2+9 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b \left (21 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {32 a b^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}-\frac {1}{5} \left (a \left (5 a^2+9 b^2\right )\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (b \left (21 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b \left (21 a^2+5 b^2\right ) \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 {2 a \left (5 a^2+9 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b \left (21 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {32 a b^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}-\frac {1}{5} \left (a \left (5 a^2+9 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 a \left (5 a^2+9 b^2\right ) \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 {2 b \left (21 a^2+5 b^2\right ) \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 {2 a \left (5 a^2+9 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b \left (21 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {32 a b^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A]
time = 2.24, size = 177, normalized size = 0.76 \begin {gather*} \frac {\sec ^{\frac {7}{2}}(c+d x) \left (-168 a \left (5 a^2+9 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 b \left (21 a^2+5 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \left (210 a^2 b+110 b^3+63 a \left (5 a^2+13 b^2\right ) \cos (c+d x)+10 \left (21 a^2 b+5 b^3\right ) \cos (2 (c+d x))+105 a^3 \cos (3 (c+d x))+189 a b^2 \cos (3 (c+d x))\right ) \sin (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. \(819\) vs.
\(2(258)=516\).
time = 0.33, size = 820, normalized size = 3.50
method | result | size |
default | \(\text {Expression too large to display}\) | \(820\) |
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 = 1.19, size = 270, normalized size = 1.15 \begin {gather*} -\frac {5 \, \sqrt {2} {\left (21 i \, a^{2} b + 5 i \, b^{3}\right )} \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 \, \sqrt {2} {\left (-21 i \, a^{2} b - 5 i \, b^{3}\right )} \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 \, \sqrt {2} {\left (5 i \, a^{3} + 9 i \, a b^{2}\right )} \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 \, \sqrt {2} {\left (-5 i \, a^{3} - 9 i \, a b^{2}\right )} \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 {2 \, {\left (63 \, a b^{2} \cos \left (d x + c\right ) + 21 \, {\left (5 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, b^{3} + 5 \, {\left (21 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\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 {\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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