Optimal. Leaf size=187 \[ \frac {122 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {32 a^4 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {32 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d}+\frac {152 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d} \]
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Rubi [A] time = 0.26, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3238, 3791, 3769, 3771, 2639, 2641} \[ \frac {122 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {32 a^4 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {32 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d}+\frac {152 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3238
Rule 3769
Rule 3771
Rule 3791
Rubi steps
\begin {align*} \int \frac {(a+a \cos (c+d x))^4}{\sqrt {\sec (c+d x)}} \, dx &=\int \frac {(a+a \sec (c+d x))^4}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\int \left (\frac {a^4}{\sec ^{\frac {9}{2}}(c+d x)}+\frac {4 a^4}{\sec ^{\frac {7}{2}}(c+d x)}+\frac {6 a^4}{\sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^4}{\sec ^{\frac {3}{2}}(c+d x)}+\frac {a^4}{\sqrt {\sec (c+d x)}}\right ) \, dx\\ &=a^4 \int \frac {1}{\sec ^{\frac {9}{2}}(c+d x)} \, dx+a^4 \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\left (4 a^4\right ) \int \frac {1}{\sec ^{\frac {7}{2}}(c+d x)} \, dx+\left (4 a^4\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx+\left (6 a^4\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {12 a^4 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{9} \left (7 a^4\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx+\frac {1}{3} \left (4 a^4\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{7} \left (20 a^4\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{5} \left (18 a^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\left (a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 a^4 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{d}+\frac {2 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {122 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {32 a^4 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {1}{15} \left (7 a^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (20 a^4\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{3} \left (4 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{5} \left (18 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {46 a^4 \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^4 \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 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {122 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {32 a^4 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {1}{15} \left (7 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (20 a^4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {152 a^4 \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 {32 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{7 d}+\frac {2 a^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {122 a^4 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {32 a^4 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 2.12, size = 156, normalized size = 0.83 \[ \frac {a^4 \left (\frac {51072 i \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )}{\sqrt {1+e^{2 i (c+d x)}}}-11520 i \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right ) \sec (c+d x)+12240 \sin (c+d x)+3556 \sin (2 (c+d x))+720 \sin (3 (c+d x))+70 \sin (4 (c+d x))-25536 i\right )}{2520 d \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} \cos \left (d x + c\right )^{4} + 4 \, a^{4} \cos \left (d x + c\right )^{3} + 6 \, a^{4} \cos \left (d x + c\right )^{2} + 4 \, a^{4} \cos \left (d x + c\right ) + a^{4}}{\sqrt {\sec \left (d x + c\right )}}, x\right ) \]
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 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 260, normalized size = 1.39 \[ -\frac {8 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{4} \left (280 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+34 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+72 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-485 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+180 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-399 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+219 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\cos \left (c+d\,x\right )\right )}^4}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{4} \left (\int \frac {4 \cos {\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {6 \cos ^{2}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {4 \cos ^{3}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {\cos ^{4}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {1}{\sqrt {\sec {\left (c + d x \right )}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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