Optimal. Leaf size=237 \[ -\frac{2 (43 A-91 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (31 A-7 B+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.758292, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4086, 4022, 4013, 3808, 206} \[ -\frac{2 (43 A-91 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (31 A-7 B+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 4022
Rule 4013
Rule 3808
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx &=\frac{2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 \int \frac{-\frac{1}{2} a (A-7 B)+\frac{1}{2} a (6 A+7 C) \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx}{7 a}\\ &=\frac{2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{4 \int \frac{\frac{1}{4} a^2 (31 A-7 B+35 C)-a^2 (A-7 B) \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx}{35 a^2}\\ &=\frac{2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 (31 A-7 B+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{8 \int \frac{-\frac{1}{8} a^3 (43 A-91 B+35 C)+\frac{1}{4} a^3 (31 A-7 B+35 C) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx}{105 a^3}\\ &=\frac{2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 (31 A-7 B+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{2 (43 A-91 B+35 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+(A-B+C) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx\\ &=\frac{2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 (31 A-7 B+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{2 (43 A-91 B+35 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}-\frac{(2 (A-B+C)) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sqrt{\sec (c+d x)} \sin (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{\sec (c+d x)} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 (31 A-7 B+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{2 (43 A-91 B+35 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.56284, size = 175, normalized size = 0.74 \[ \frac{4 \cos \left (\frac{1}{2} (c+d x)\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-140 (2 A+B+C) \sin ^3\left (\frac{1}{2} (c+d x)\right )+105 (A-B+C) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+168 (2 A+B) \sin ^5\left (\frac{1}{2} (c+d x)\right )-240 A \sin ^7\left (\frac{1}{2} (c+d x)\right )+210 B \sin \left (\frac{1}{2} (c+d x)\right )\right )}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.377, size = 296, normalized size = 1.3 \begin{align*} -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{105\,ad\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 30\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+105\,\arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}A\sin \left ( dx+c \right ) -36\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}-105\,\arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}B\sin \left ( dx+c \right ) +42\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+105\,C\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sin \left ( dx+c \right ) +68\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}-56\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+70\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}-148\,A\cos \left ( dx+c \right ) +196\,B\cos \left ( dx+c \right ) -140\,C\cos \left ( dx+c \right ) +86\,A-182\,B+70\,C \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.58547, size = 1465, normalized size = 6.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.566275, size = 1187, normalized size = 5.01 \begin{align*} \left [\frac{\frac{105 \, \sqrt{2}{\left ({\left (A - B + C\right )} a \cos \left (d x + c\right ) +{\left (A - B + C\right )} a\right )} \log \left (-\frac{\cos \left (d x + c\right )^{2} - \frac{2 \, \sqrt{2} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt{a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt{a}} + \frac{4 \,{\left (15 \, A \cos \left (d x + c\right )^{4} - 3 \,{\left (A - 7 \, B\right )} \cos \left (d x + c\right )^{3} +{\left (31 \, A - 7 \, B + 35 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (43 \, A - 91 \, B + 35 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{210 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}}, -\frac{105 \, \sqrt{2}{\left ({\left (A - B + C\right )} a \cos \left (d x + c\right ) +{\left (A - B + C\right )} a\right )} \sqrt{-\frac{1}{a}} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{-\frac{1}{a}} \sqrt{\cos \left (d x + c\right )}}{\sin \left (d x + c\right )}\right ) - \frac{2 \,{\left (15 \, A \cos \left (d x + c\right )^{4} - 3 \,{\left (A - 7 \, B\right )} \cos \left (d x + c\right )^{3} +{\left (31 \, A - 7 \, B + 35 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (43 \, A - 91 \, B + 35 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{105 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}}\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}{\sqrt{a \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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