Optimal. Leaf size=287 \[ \frac {14 b^2 \left (7 a^2+b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{45 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}-\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {44 a b^3 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d} \]
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Rubi [A] time = 0.41, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3842, 4076, 4047, 3768, 3771, 2641, 4046, 2639} \[ \frac {14 b^2 \left (7 a^2+b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{45 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 \left (54 a^2 b^2+15 a^4+7 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 \left (54 a^2 b^2+15 a^4+7 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {44 a b^3 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3842
Rule 4046
Rule 4047
Rule 4076
Rubi steps
\begin {align*} \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx &=\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2}{9} \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (\frac {3}{2} a \left (3 a^2+b^2\right )+\frac {1}{2} b \left (27 a^2+7 b^2\right ) \sec (c+d x)+11 a b^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {4}{63} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {21}{4} a^2 \left (3 a^2+b^2\right )+9 a b \left (7 a^2+5 b^2\right ) \sec (c+d x)+\frac {49}{4} b^2 \left (7 a^2+b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {4}{63} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {21}{4} a^2 \left (3 a^2+b^2\right )+\frac {49}{4} b^2 \left (7 a^2+b^2\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{7} \left (4 a b \left (7 a^2+5 b^2\right )\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{21} \left (4 a b \left (7 a^2+5 b^2\right )\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (15 a^4+54 a^2 b^2+7 b^4\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{15} \left (-15 a^4-54 a^2 b^2-7 b^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (4 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {8 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{15} \left (\left (-15 a^4-54 a^2 b^2-7 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 2.43, size = 256, normalized size = 0.89 \[ -\frac {2 (a+b \sec (c+d x))^4 \left (-315 a^4 \sin (c+d x)-420 a^3 b \tan (c+d x)-1134 a^2 b^2 \sin (c+d x)-60 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-378 a^2 b^2 \tan (c+d x) \sec (c+d x)+21 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )-300 a b^3 \tan (c+d x)-180 a b^3 \tan (c+d x) \sec ^2(c+d x)-147 b^4 \sin (c+d x)-35 b^4 \tan (c+d x) \sec ^3(c+d x)-49 b^4 \tan (c+d x) \sec (c+d x)\right )}{315 d \sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+b)^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} \sec \left (d x + c\right )^{5} + 4 \, a b^{3} \sec \left (d x + c\right )^{4} + 6 \, a^{2} b^{2} \sec \left (d x + c\right )^{3} + 4 \, a^{3} b \sec \left (d x + c\right )^{2} + a^{4} \sec \left (d x + c\right )\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 15.93, size = 1174, normalized size = 4.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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