Optimal. Leaf size=207 \[ \frac{8 a b \left (a^2+3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}-\frac{2 b^2 \left (a^2-5 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left (30 a^2 b^2+3 a^4-5 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{28 a^3 b \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.36943, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3841, 4074, 4047, 3771, 2641, 4046, 2639} \[ -\frac{2 b^2 \left (a^2-5 b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a b \left (a^2+3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 \left (30 a^2 b^2+3 a^4-5 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{28 a^3 b \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3841
Rule 4074
Rule 4047
Rule 3771
Rule 2641
Rule 4046
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2}{5} \int \frac{(a+b \sec (c+d x)) \left (7 a^2 b+\frac{3}{2} a \left (a^2+5 b^2\right ) \sec (c+d x)-\frac{1}{2} b \left (a^2-5 b^2\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{28 a^3 b \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{4}{15} \int \frac{-\frac{3}{4} a^2 \left (3 a^2+29 b^2\right )-5 a b \left (a^2+3 b^2\right ) \sec (c+d x)+\frac{3}{4} b^2 \left (a^2-5 b^2\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{28 a^3 b \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{4}{15} \int \frac{-\frac{3}{4} a^2 \left (3 a^2+29 b^2\right )+\frac{3}{4} b^2 \left (a^2-5 b^2\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (4 a b \left (a^2+3 b^2\right )\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{28 a^3 b \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 \left (a^2-5 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{1}{5} \left (-3 a^4-30 a^2 b^2+5 b^4\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{3} \left (4 a b \left (a^2+3 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 (a^2+3 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)}}{3 d}+\frac{28 a^3 b \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 \left (a^2-5 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{1}{5} \left (\left (-3 a^4-30 a^2 b^2+5 b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (3 a^4+30 a^2 b^2-5 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)}}{5 d}+\frac{8 a b \left (a^2+3 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)}}{3 d}+\frac{28 a^3 b \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 \left (a^2-5 b^2\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.58226, size = 138, normalized size = 0.67 \[ \frac{\sqrt{\sec (c+d x)} \left (80 a b \left (a^2+3 b^2\right ) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+2 \sin (c+d x) \left (40 a^3 b \cos (c+d x)+3 a^4 \cos (2 (c+d x))+3 a^4+30 b^4\right )+12 \left (30 a^2 b^2+3 a^4-5 b^4\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{30 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.752, size = 619, normalized size = 3. \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{5}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \sec{\left (c + d x \right )}\right )^{4}}{\sec ^{\frac{5}{2}}{\left (c + d x \right )}}\, dx \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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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