Optimal. Leaf size=287 \[ \frac{8 a b \left (7 a^2+5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{14 b^2 \left (7 a^2+b^2\right ) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{8 a b \left (7 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left (54 a^2 b^2+15 a^4+7 b^4\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 \left (54 a^2 b^2+15 a^4+7 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{44 a b^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d} \]
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Rubi [A] time = 0.409396, antiderivative size = 287, normalized size of antiderivative = 1., 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 = {3842, 4076, 4047, 3768, 3771, 2641, 4046, 2639} \[ \frac{14 b^2 \left (7 a^2+b^2\right ) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{8 a b \left (7 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left (54 a^2 b^2+15 a^4+7 b^4\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{8 a b \left (7 a^2+5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 \left (54 a^2 b^2+15 a^4+7 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{44 a b^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 3842
Rule 4076
Rule 4047
Rule 3768
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx &=\frac{2 b^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2}{9} \int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) \left (\frac{3}{2} a \left (3 a^2+b^2\right )+\frac{1}{2} b \left (27 a^2+7 b^2\right ) \sec (c+d x)+11 a b^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac{44 a b^3 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{4}{63} \int \sec ^{\frac{3}{2}}(c+d x) \left (\frac{21}{4} a^2 \left (3 a^2+b^2\right )+9 a b \left (7 a^2+5 b^2\right ) \sec (c+d x)+\frac{49}{4} b^2 \left (7 a^2+b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{44 a b^3 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{4}{63} \int \sec ^{\frac{3}{2}}(c+d x) \left (\frac{21}{4} a^2 \left (3 a^2+b^2\right )+\frac{49}{4} b^2 \left (7 a^2+b^2\right ) \sec ^2(c+d x)\right ) \, dx+\frac{1}{7} \left (4 a b \left (7 a^2+5 b^2\right )\right ) \int \sec ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{44 a b^3 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{21} \left (4 a b \left (7 a^2+5 b^2\right )\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{15} \left (15 a^4+54 a^2 b^2+7 b^4\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{44 a b^3 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{15} \left (-15 a^4-54 a^2 b^2-7 b^4\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (4 a b \left (7 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{8 a b \left (7 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 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{44 a b^3 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{15} \left (\left (-15 a^4-54 a^2 b^2-7 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (15 a^4+54 a^2 b^2+7 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 (7 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 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{44 a b^3 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 2.1412, size = 256, normalized size = 0.89 \[ -\frac{2 (a+b \sec (c+d x))^4 \left (-60 a b \left (7 a^2+5 b^2\right ) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-1134 a^2 b^2 \sin (c+d x)+21 \left (54 a^2 b^2+15 a^4+7 b^4\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )-378 a^2 b^2 \tan (c+d x) \sec (c+d x)-420 a^3 b \tan (c+d x)-315 a^4 \sin (c+d x)-300 a b^3 \tan (c+d x)-180 a b^3 \tan (c+d x) \sec ^2(c+d x)-147 b^4 \sin (c+d x)-35 b^4 \tan (c+d x) \sec ^3(c+d x)-49 b^4 \tan (c+d x) \sec (c+d x)\right )}{315 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 6.618, size = 1174, normalized size = 4.1 \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 ({\left (b^{4} \sec \left (d x + c\right )^{5} + 4 \, a b^{3} \sec \left (d x + c\right )^{4} + 6 \, a^{2} b^{2} \sec \left (d x + c\right )^{3} + 4 \, a^{3} b \sec \left (d x + c\right )^{2} + a^{4} \sec \left (d x + c\right )\right )} \sqrt{\sec \left (d x + c\right )}, 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{\left (b \sec \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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