Optimal. Leaf size=652 \[ \frac{\sqrt{a+b} \cot (c+d x) \left (4 a^2 b (71 A+52 B+108 C)+8 a^3 (9 A+16 B+12 C)+2 a b^2 (59 A+132 B+192 C)+15 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{192 a d}+\frac{\sin (c+d x) \left (4 a^2 b (71 A+108 C)+128 a^3 B+264 a b^2 B+15 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{32 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (4 a^2 b (71 A+108 C)+128 a^3 B+264 a b^2 B+15 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a b d}+\frac{\sqrt{a+b} \cot (c+d x) \left (-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)-160 a^3 b B-40 a b^3 B+5 A b^4\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{64 a^2 d}+\frac{(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{24 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d} \]
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Rubi [A] time = 2.01955, antiderivative size = 652, 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} \[ \frac{\sin (c+d x) \left (4 a^2 b (71 A+108 C)+128 a^3 B+264 a b^2 B+15 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{32 d}+\frac{\sqrt{a+b} \cot (c+d x) \left (4 a^2 b (71 A+52 B+108 C)+8 a^3 (9 A+16 B+12 C)+2 a b^2 (59 A+132 B+192 C)+15 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (4 a^2 b (71 A+108 C)+128 a^3 B+264 a b^2 B+15 A b^3\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a b d}+\frac{\sqrt{a+b} \cot (c+d x) \left (-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)-160 a^3 b B-40 a b^3 B+5 A b^4\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{64 a^2 d}+\frac{(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{24 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d} \]
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))^{5/2} \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))^{5/2} \sin (c+d x)}{4 d}+\frac{1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac{1}{2} (5 A b+8 a B)+(3 a A+4 b B+4 a C) \sec (c+d x)+\frac{1}{2} b (A+8 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac{1}{12} \int \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left (\frac{3}{4} \left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right )+\frac{1}{2} \left (16 a^2 B+24 b^2 B+a b (31 A+48 C)\right ) \sec (c+d x)+\frac{1}{4} b (11 A b+8 a B+48 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac{(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac{1}{24} \int \frac{\cos (c+d x) \left (\frac{1}{8} \left (15 A b^3+128 a^3 B+264 a b^2 B+8 a^2 \left (\frac{71 A b}{2}+54 b C\right )\right )+\frac{1}{4} \left (152 a^2 b B+96 b^3 B+12 a^3 (3 A+4 C)+a b^2 (161 A+288 C)\right ) \sec (c+d x)+\frac{1}{8} b \left (104 a b B+12 a^2 (3 A+4 C)+b^2 (59 A+192 C)\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac{\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac{(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}-\frac{\int \frac{\frac{3}{16} \left (5 A b^4-160 a^3 b B-40 a b^3 B-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right )-\frac{1}{8} a b \left (104 a b B+12 a^2 (3 A+4 C)+b^2 (59 A+192 C)\right ) \sec (c+d x)+\frac{1}{16} b \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a}\\ &=\frac{\left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac{\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac{(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}-\frac{\int \frac{\frac{3}{16} \left (5 A b^4-160 a^3 b B-40 a b^3 B-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right )+\left (-\frac{1}{16} b \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right )-\frac{1}{8} a b \left (104 a b B+12 a^2 (3 A+4 C)+b^2 (59 A+192 C)\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a}-\frac{\left (b \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right )\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{384 a}\\ &=\frac{(a-b) \sqrt{a+b} \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 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 b d}+\frac{\left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac{\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac{(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}-\frac{\left (5 A b^4-160 a^3 b B-40 a b^3 B-120 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}+\frac{\left (b \left (15 A b^3+8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{384 a}\\ &=\frac{(a-b) \sqrt{a+b} \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 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 b d}+\frac{\sqrt{a+b} \left (15 A b^3+8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 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 d}+\frac{\sqrt{a+b} \left (5 A b^4-160 a^3 b B-40 a b^3 B-120 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^2 d}+\frac{\left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac{\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac{(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [B] time = 26.1058, size = 5681, normalized size = 8.71 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.706, size = 5850, normalized size = 9. \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{5}{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^{2} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right )^{4} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, 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{5}{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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