Optimal. Leaf size=285 \[ -\frac{(5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 a^{3/2} d}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(A+2 C) \sin (c+d x)}{2 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.910585, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216, Rules used = {4265, 4085, 4021, 4023, 3808, 206, 3801, 215} \[ -\frac{(5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 a^{3/2} d}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(A+2 C) \sin (c+d x)}{2 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4265
Rule 4085
Rule 4021
Rule 4023
Rule 3808
Rule 206
Rule 3801
Rule 215
Rubi steps
\begin{align*} \int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{5}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{5}{2}}(c+d x) \left (\frac{1}{2} a (A+5 C)-2 a (A+2 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{(A+2 C) \sin (c+d x)}{2 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (-3 a^2 (A+2 C)+a^2 (2 A+7 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{4 a^3}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{(A+2 C) \sin (c+d x)}{2 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\sec (c+d x)} \left (\frac{1}{2} a^3 (2 A+7 C)-\frac{1}{2} a^3 (8 A+19 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{4 a^4}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{(A+2 C) \sin (c+d x)}{2 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{\left ((5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx}{4 a}+\frac{\left ((8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx}{8 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{(A+2 C) \sin (c+d x)}{2 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{\left ((5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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 )}{2 a d}-\frac{\left ((8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{4 a^2 d}\\ &=\frac{(8 A+19 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{4 a^{3/2} d}-\frac{(5 A+13 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{\cos (c+d x)} \sqrt{\sec (c+d x)}}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac{(A+2 C) \sin (c+d x)}{2 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.76652, size = 213, normalized size = 0.75 \[ -\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (A \cos ^2(c+d x)+C\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right ) ((2 A+7 C) \cos (2 (c+d x))+2 A+6 C \cos (c+d x)+3 C)+(5 A+13 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{3}{2} (c+d x)\right )\right )^2 \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\frac{(8 A+19 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{3}{2} (c+d x)\right )\right )^2 \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{2}}\right )}{4 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.3, size = 508, normalized size = 1.8 \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 [A] time = 0.869465, size = 1956, normalized size = 6.86 \begin{align*} \text{result too large to display} \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} + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \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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