Optimal. Leaf size=245 \[ \frac{8 a b \left (5 a^2+7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{14 a^2 \left (a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left (5 a^2+7 b^2\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left (54 a^2 b^2+7 a^4+15 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{44 a^3 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.402599, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {3841, 4074, 4047, 3769, 3771, 2641, 4045, 2639} \[ \frac{14 a^2 \left (a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left (5 a^2+7 b^2\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{8 a b \left (5 a^2+7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (54 a^2 b^2+7 a^4+15 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{44 a^3 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3841
Rule 4074
Rule 4047
Rule 3769
Rule 3771
Rule 2641
Rule 4045
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2}{9} \int \frac{(a+b \sec (c+d x)) \left (11 a^2 b+\frac{1}{2} a \left (7 a^2+27 b^2\right ) \sec (c+d x)+\frac{3}{2} b \left (a^2+3 b^2\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{44 a^3 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{4}{63} \int \frac{-\frac{49}{4} a^2 \left (a^2+7 b^2\right )-9 a b \left (5 a^2+7 b^2\right ) \sec (c+d x)-\frac{21}{4} b^2 \left (a^2+3 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{44 a^3 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{4}{63} \int \frac{-\frac{49}{4} a^2 \left (a^2+7 b^2\right )-\frac{21}{4} b^2 \left (a^2+3 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx+\frac{1}{7} \left (4 a b \left (5 a^2+7 b^2\right )\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{44 a^3 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{14 a^2 \left (a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left (5 a^2+7 b^2\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{21} \left (4 a b \left (5 a^2+7 b^2\right )\right ) \int \sqrt{\sec (c+d x)} \, dx-\frac{1}{15} \left (-7 a^4-54 a^2 b^2-15 b^4\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{44 a^3 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{14 a^2 \left (a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left (5 a^2+7 b^2\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{21} \left (4 a b \left (5 a^2+7 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{15} \left (\left (-7 a^4-54 a^2 b^2-15 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (7 a^4+54 a^2 b^2+15 b^4\right ) \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{8 a b \left (5 a^2+7 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{44 a^3 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{14 a^2 \left (a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left (5 a^2+7 b^2\right ) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 1.47051, size = 168, normalized size = 0.69 \[ \frac{\sqrt{\sec (c+d x)} \left (480 a b \left (5 a^2+7 b^2\right ) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+a \sin (2 (c+d x)) \left (7 a \left (43 a^2+216 b^2\right ) \cos (c+d x)+5 \left (72 a^2 b \cos (2 (c+d x))+312 a^2 b+7 a^3 \cos (3 (c+d x))+336 b^3\right )\right )+168 \left (54 a^2 b^2+7 a^4+15 b^4\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{1260 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.666, size = 529, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{4} \sec \left (d x + c\right )^{4} + 4 \, a b^{3} \sec \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \sec \left (d x + c\right )^{2} + 4 \, a^{3} b \sec \left (d x + c\right ) + a^{4}}{\sec \left (d x + c\right )^{\frac{9}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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