Optimal. Leaf size=214 \[ -\frac{(5 A-5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 a d}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{(7 A-5 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(5 A-5 B+3 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{3 (7 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]
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Rubi [A] time = 0.253706, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {4084, 3787, 3769, 3771, 2639, 2641} \[ -\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{(7 A-5 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(5 A-5 B+3 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(5 A-5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{3 (7 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d} \]
Antiderivative was successfully verified.
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Rule 4084
Rule 3787
Rule 3769
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx &=-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}+\frac{\int \frac{\frac{1}{2} a (7 A-5 B+5 C)-\frac{1}{2} a (5 A-5 B+3 C) \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{a^2}\\ &=-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{(5 A-5 B+3 C) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{2 a}+\frac{(7 A-5 B+5 C) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{2 a}\\ &=\frac{(7 A-5 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(5 A-5 B+3 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{(5 A-5 B+3 C) \int \sqrt{\sec (c+d x)} \, dx}{6 a}+\frac{(3 (7 A-5 B+5 C)) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{10 a}\\ &=\frac{(7 A-5 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(5 A-5 B+3 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}-\frac{\left ((5 A-5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a}+\frac{\left (3 (7 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{10 a}\\ &=\frac{3 (7 A-5 B+5 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(5 A-5 B+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 a d}+\frac{(7 A-5 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(5 A-5 B+3 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))}\\ \end{align*}
Mathematica [C] time = 6.77207, size = 1350, normalized size = 6.31 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 2.247, size = 320, normalized size = 1.5 \begin{align*} -{\frac{1}{15\,ad}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1} \left ( 63\,A{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +25\,A{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -45\,B{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -25\,B{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +45\,C{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +15\,C{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) +48\,A \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+ \left ( -56\,A-40\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( -30\,A+90\,B-30\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( 23\,A-35\,B+15\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \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{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt{\sec \left (d x + c\right )}}{a \sec \left (d x + c\right )^{4} + a \sec \left (d x + c\right )^{3}}, 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{C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )} \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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