Optimal. Leaf size=588 \[ -\frac{2 \left (116 a^2 A b^3+17 a^4 A b-80 a^3 b^2 B-5 a^5 B+80 a b^4 B-128 A b^5\right ) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (-71 a^2 A b^2+3 a^4 A+50 a^3 b B-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (12 a^2 A b-9 a^3 B+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{3 a^2 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left (-98 a^2 A b^3+14 a^4 A b+65 a^3 b^2 B-5 a^5 B-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^4 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)}}+\frac{2 \left (55 a^4 A b^2-212 a^2 A b^4+9 a^6 A+140 a^3 b^3 B-40 a^5 b B-80 a b^5 B+128 A b^6\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}} \]
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Rubi [A] time = 1.87864, antiderivative size = 588, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4030, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{2 \left (-71 a^2 A b^2+3 a^4 A+50 a^3 b B-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (12 a^2 A b-9 a^3 B+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{3 a^2 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left (-98 a^2 A b^3+14 a^4 A b+65 a^3 b^2 B-5 a^5 B-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^4 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)}}-\frac{2 \left (116 a^2 A b^3+17 a^4 A b-80 a^3 b^2 B-5 a^5 B+80 a b^4 B-128 A b^5\right ) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (55 a^4 A b^2-212 a^2 A b^4+9 a^6 A+140 a^3 b^3 B-40 a^5 b B-80 a b^5 B+128 A b^6\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 4030
Rule 4100
Rule 4104
Rule 4035
Rule 3856
Rule 2655
Rule 2653
Rule 3858
Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx &=\frac{2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (-3 a^2 A+8 A b^2-5 a b B\right )+\frac{3}{2} a (A b-a B) \sec (c+d x)-3 b (A b-a B) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac{2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{4 \int \frac{\frac{1}{4} \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right )-\frac{1}{4} a \left (6 a^2 A b-2 A b^3-3 a^3 B-a b^2 B\right ) \sec (c+d x)+b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{8 \int \frac{\frac{3}{8} \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right )-\frac{1}{8} a \left (9 a^4 A+27 a^2 A b^2-16 A b^4-30 a^3 b B+10 a b^3 B\right ) \sec (c+d x)-\frac{1}{4} b \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{15 a^3 \left (a^2-b^2\right )^2}\\ &=\frac{2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{16 \int \frac{\frac{3}{16} \left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right )-\frac{3}{16} a \left (8 a^4 A b+44 a^2 A b^3-32 A b^5-5 a^5 B-35 a^3 b^2 B+20 a b^4 B\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{45 a^4 \left (a^2-b^2\right )^2}\\ &=\frac{2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}-\frac{\left (17 a^4 A b+116 a^2 A b^3-128 A b^5-5 a^5 B-80 a^3 b^2 B+80 a b^4 B\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )}+\frac{\left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )^2}\\ &=\frac{2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}-\frac{\left (\left (17 a^4 A b+116 a^2 A b^3-128 A b^5-5 a^5 B-80 a^3 b^2 B+80 a b^4 B\right ) \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{b+a \cos (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{15 a^5 \left (a^2-b^2\right )^2 \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}-\frac{\left (\left (17 a^4 A b+116 a^2 A b^3-128 A b^5-5 a^5 B-80 a^3 b^2 B+80 a b^4 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{15 a^5 \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}} \, dx}{15 a^5 \left (a^2-b^2\right )^2 \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}\\ &=-\frac{2 \left (17 a^4 A b+116 a^2 A b^3-128 A b^5-5 a^5 B-80 a^3 b^2 B+80 a b^4 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{\sec (c+d x)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.74211, size = 392, normalized size = 0.67 \[ \frac{\sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) \left (a \left (\frac{10 b^4 (A b-a B) \sin (c+d x)}{b^2-a^2}-\frac{10 b^3 \left (-15 a^2 A b+12 a^3 B-8 a b^2 B+11 A b^3\right ) \sin (c+d x) (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^2}-2 (14 A b-5 a B) \sin (c+d x) (a \cos (c+d x)+b)^2+3 a A \sin (2 (c+d x)) (a \cos (c+d x)+b)^2\right )-\frac{2 \left (\frac{a \cos (c+d x)+b}{a+b}\right )^{3/2} \left (a^2 \left (44 a^2 A b^3+8 a^4 A b-35 a^3 b^2 B-5 a^5 B+20 a b^4 B-32 A b^5\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )-\left (55 a^4 A b^2-212 a^2 A b^4+9 a^6 A+140 a^3 b^3 B-40 a^5 b B-80 a b^5 B+128 A b^6\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )-b \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}\right )}{15 a^5 d (a+b \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.81, size = 8251, normalized size = 14. \begin{align*} \text{output 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{{\left (B \sec \left (d x + c\right ) + A\right )} \sqrt{b \sec \left (d x + c\right ) + a} \sqrt{\sec \left (d x + c\right )}}{b^{3} \sec \left (d x + c\right )^{6} + 3 \, a b^{2} \sec \left (d x + c\right )^{5} + 3 \, a^{2} b \sec \left (d x + c\right )^{4} + a^{3} \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{B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \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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