Optimal. Leaf size=187 \[ \frac {68 a^3 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 a^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {44 a^3 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {44 a^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {68 a^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 17, 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} \[ \frac {68 a^3 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 a^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {44 a^3 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {44 a^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {68 a^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 2641
Rule 3238
Rule 3769
Rule 3771
Rule 3791
Rubi steps
\begin {align*} \int \frac {(a+a \cos (c+d x))^3}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\int \frac {(a+a \sec (c+d x))^3}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\int \left (\frac {a^3}{\sec ^{\frac {9}{2}}(c+d x)}+\frac {3 a^3}{\sec ^{\frac {7}{2}}(c+d x)}+\frac {3 a^3}{\sec ^{\frac {5}{2}}(c+d x)}+\frac {a^3}{\sec ^{\frac {3}{2}}(c+d x)}\right ) \, dx\\ &=a^3 \int \frac {1}{\sec ^{\frac {9}{2}}(c+d x)} \, dx+a^3 \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx+\left (3 a^3\right ) \int \frac {1}{\sec ^{\frac {7}{2}}(c+d x)} \, dx+\left (3 a^3\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {6 a^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {6 a^3 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a^3 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{3} a^3 \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{9} \left (7 a^3\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx+\frac {1}{5} \left (9 a^3\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{7} \left (15 a^3\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {6 a^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {68 a^3 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {44 a^3 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {1}{15} \left (7 a^3\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{7} \left (5 a^3\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{3} \left (a^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{5} \left (9 a^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {18 a^3 \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 {2 a^3 \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^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {6 a^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {68 a^3 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {44 a^3 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {1}{15} \left (7 a^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{7} \left (5 a^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {68 a^3 \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 {44 a^3 \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 a^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {6 a^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {68 a^3 \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {44 a^3 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.29, size = 156, normalized size = 0.83 \[ \frac {a^3 \left (\frac {22848 i \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )}{\sqrt {1+e^{2 i (c+d x)}}}-5280 i \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right ) \sec (c+d x)+5820 \sin (c+d x)+2044 \sin (2 (c+d x))+540 \sin (3 (c+d x))+70 \sin (4 (c+d x))-11424 i\right )}{2520 d \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.60, size = 260, normalized size = 1.39 \[ -\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{3} \left (560 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-600 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+212 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+66 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-430 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+165 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-357 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+192 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int \frac {3 \cos {\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 \cos ^{2}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {\cos ^{3}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {1}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________