Optimal. Leaf size=234 \[ \frac {2 a \left (5 a^2+21 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 b \left (9 a^2+5 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a \left (5 a^2+21 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 b \left (9 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}+\frac {32 a^2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{35 d} \]
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Rubi [A] time = 0.27, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3238, 3842, 4047, 3768, 3771, 2641, 4046, 2639} \[ \frac {2 a \left (5 a^2+21 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 b \left (9 a^2+5 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a \left (5 a^2+21 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 b \left (9 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}+\frac {32 a^2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{35 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3238
Rule 3768
Rule 3771
Rule 3842
Rule 4046
Rule 4047
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \, dx &=\int \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^3 \, dx\\ &=\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) (b+a \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} b \left (3 a^2+7 b^2\right )+\frac {1}{2} a \left (5 a^2+21 b^2\right ) \sec (c+d x)+8 a^2 b \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) (b+a \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} b \left (3 a^2+7 b^2\right )+8 a^2 b \sec ^2(c+d x)\right ) \, dx+\frac {1}{7} \left (a \left (5 a^2+21 b^2\right )\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {2 a \left (5 a^2+21 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {32 a^2 b \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{5} \left (b \left (9 a^2+5 b^2\right )\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{21} \left (a \left (5 a^2+21 b^2\right )\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {2 b \left (9 a^2+5 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a \left (5 a^2+21 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {32 a^2 b \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d}-\frac {1}{5} \left (b \left (9 a^2+5 b^2\right )\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (a \left (5 a^2+21 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 a \left (5 a^2+21 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 b \left (9 a^2+5 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a \left (5 a^2+21 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {32 a^2 b \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d}-\frac {1}{5} \left (b \left (9 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 b \left (9 a^2+5 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 a \left (5 a^2+21 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 b \left (9 a^2+5 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a \left (5 a^2+21 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {32 a^2 b \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 1.20, size = 191, normalized size = 0.82 \[ \frac {\sec ^{\frac {7}{2}}(c+d x) \left (30 a^3 \sin (c+d x)+50 a^3 \sin (c+d x) \cos ^2(c+d x)+10 a \left (5 a^2+21 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-42 b \left (9 a^2+5 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+63 a^2 b \sin (2 (c+d x))+378 a^2 b \sin (c+d x) \cos ^3(c+d x)+210 a b^2 \sin (c+d x) \cos ^2(c+d x)+210 b^3 \sin (c+d x) \cos ^3(c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sec \left (d x + c\right )^{\frac {9}{2}}, x\right ) \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.34, size = 847, normalized size = 3.62 \[ \text {result too large to display} \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
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 \[ \int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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