Optimal. Leaf size=209 \[ \frac {2 b^2 \left (29 a^2+3 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {8 a b \left (3 a^2+b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 \left (5 a^4-30 a^2 b^2-3 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {28 a b^3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d} \]
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Rubi [A] time = 0.35, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3842, 4076, 4047, 3771, 2641, 4046, 2639} \[ \frac {2 b^2 \left (29 a^2+3 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {8 a b \left (3 a^2+b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 \left (-30 a^2 b^2+5 a^4-3 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {28 a b^3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3771
Rule 3842
Rule 4046
Rule 4047
Rule 4076
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^4}{\sqrt {\sec (c+d x)}} \, dx &=\frac {2 b^2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {2}{5} \int \frac {(a+b \sec (c+d x)) \left (\frac {1}{2} a \left (5 a^2-b^2\right )+\frac {3}{2} b \left (5 a^2+b^2\right ) \sec (c+d x)+7 a b^2 \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {28 a b^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 b^2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {4}{15} \int \frac {\frac {3}{4} a^2 \left (5 a^2-b^2\right )+5 a b \left (3 a^2+b^2\right ) \sec (c+d x)+\frac {3}{4} b^2 \left (29 a^2+3 b^2\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {28 a b^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 b^2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {4}{15} \int \frac {\frac {3}{4} a^2 \left (5 a^2-b^2\right )+\frac {3}{4} b^2 \left (29 a^2+3 b^2\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (4 a b \left (3 a^2+b^2\right )\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {2 b^2 \left (29 a^2+3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {28 a b^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 b^2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{5} \left (5 a^4-30 a^2 b^2-3 b^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (4 a b \left (3 a^2+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 (3 a^2+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)}}{3 d}+\frac {2 b^2 \left (29 a^2+3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {28 a b^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 b^2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{5} \left (\left (5 a^4-30 a^2 b^2-3 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (5 a^4-30 a^2 b^2-3 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)}}{5 d}+\frac {8 a b \left (3 a^2+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)}}{3 d}+\frac {2 b^2 \left (29 a^2+3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {28 a b^3 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 b^2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 2.38, size = 146, normalized size = 0.70 \[ \frac {\sec ^{\frac {5}{2}}(c+d x) \left (b \left (80 a \left (3 a^2+b^2\right ) \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 b \sin (c+d x) \left (9 \left (10 a^2+b^2\right ) \cos (2 (c+d x))+15 \left (6 a^2+b^2\right )+40 a b \cos (c+d x)\right )\right )+12 \left (5 a^4-30 a^2 b^2-3 b^4\right ) \cos ^{\frac {5}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{30 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{4} \sec \left (d x + c\right )^{4} + 4 \, a b^{3} \sec \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \sec \left (d x + c\right )^{2} + 4 \, a^{3} b \sec \left (d x + c\right ) + a^{4}}{\sqrt {\sec \left (d x + c\right )}}, x\right ) \]
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 \[ \int \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 10.30, size = 907, normalized size = 4.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
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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