Optimal. Leaf size=650 \[ -\frac{\sqrt{a+b} \cot (c+d x) \left (-4 a^2 b (39 A+28 B+60 C)-8 a^3 (9 A+16 B+12 C)-6 a b^2 (A+4 B)+9 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{192 a^2 d}-\frac{\sin (c+d x) \left (-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{192 a^2 d}+\frac{\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{96 a d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a^2 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left (24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+96 a^3 b B-8 a b^3 B+3 A b^4\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{64 a^3 d}+\frac{(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d} \]
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Rubi [A] time = 1.90629, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4094, 4104, 4058, 3921, 3784, 3832, 4004} \[ -\frac{\sin (c+d x) \left (-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{192 a^2 d}+\frac{\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{96 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left (-4 a^2 b (39 A+28 B+60 C)-8 a^3 (9 A+16 B+12 C)-6 a b^2 (A+4 B)+9 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a^2 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left (24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+96 a^3 b B-8 a b^3 B+3 A b^4\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{64 a^3 d}+\frac{(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 4094
Rule 4104
Rule 4058
Rule 3921
Rule 3784
Rule 3832
Rule 4004
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{1}{4} \int \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left (\frac{1}{2} (3 A b+8 a B)+(3 a A+4 b B+4 a C) \sec (c+d x)+\frac{1}{2} b (3 A+8 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{1}{12} \int \frac{\cos ^2(c+d x) \left (\frac{1}{4} \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right )+\frac{1}{2} \left (33 a A b+16 a^2 B+24 b^2 B+48 a b C\right ) \sec (c+d x)+\frac{3}{4} b (9 A b+8 a B+16 b C) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}-\frac{\int \frac{\cos (c+d x) \left (\frac{1}{8} \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right )-\frac{1}{4} a \left (104 a b B+12 a^2 (3 A+4 C)+3 b^2 (19 A+32 C)\right ) \sec (c+d x)-\frac{1}{8} b \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a}\\ &=-\frac{\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^2 d}+\frac{\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\int \frac{\frac{3}{16} \left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right )+\frac{1}{8} a b \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sec (c+d x)+\frac{1}{16} b \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a^2}\\ &=-\frac{\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^2 d}+\frac{\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\int \frac{\frac{3}{16} \left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right )+\left (\frac{1}{8} a b \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right )-\frac{1}{16} b \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a^2}+\frac{\left (b \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right )\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{384 a^2}\\ &=-\frac{(a-b) \sqrt{a+b} \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{192 a^2 b d}-\frac{\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^2 d}+\frac{\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx}{128 a^2}-\frac{\left (b \left (9 A b^3-6 a b^2 (A+4 B)-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{384 a^2}\\ &=-\frac{(a-b) \sqrt{a+b} \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{192 a^2 b d}-\frac{\sqrt{a+b} \left (9 A b^3-6 a b^2 (A+4 B)-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{192 a^2 d}-\frac{\sqrt{a+b} \left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{64 a^3 d}-\frac{\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a^2 d}+\frac{\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac{(3 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 16.9781, size = 761, normalized size = 1.17 \[ \frac{\cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{\sin (2 (c+d x)) \left (48 a^2 A+48 a^2 C+56 a b B+3 A b^2\right )}{96 a}+\frac{1}{48} (8 a B+9 A b) \sin (c+d x)+\frac{1}{48} (8 a B+9 A b) \sin (3 (c+d x))+\frac{1}{16} a A \sin (4 (c+d x))\right )}{d (a \cos (c+d x)+b) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}-\frac{\cos ^5(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (b (a+b) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (12 a^2 b (7 A+4 (B+3 C))+8 a^3 (9 A+16 B+12 C)-6 a b^2 (3 A+4 B)+9 A b^3\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right )+3 \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+96 a^3 b B-8 a b^3 B+3 A b^4\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} \left ((a-b) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right )+2 a \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right )\right )-a \tan \left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (12 a^2 b (13 A+20 C)+128 a^3 B+24 a b^2 B-9 A b^3\right ) \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2} (a \cos (c+d x)+b)-a (a+b) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (12 a^2 b (13 A+20 C)+128 a^3 B+24 a b^2 B-9 A b^3\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right )\right )}{96 a^3 d \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2} (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.738, size = 5474, normalized size = 8.4 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{4}\,{d x} \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 ({\left (C b \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} +{\left (C a + B b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )^{4} +{\left (B a + A b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )\right )} \sqrt{b \sec \left (d x + c\right ) + a}, 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{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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