Optimal. Leaf size=168 \[ -\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}-\frac {4 b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {4 b^{3/2} \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{3 d^{5/2}}+\frac {4 b \sin (a+b x)}{3 d^2 \sqrt {c+d x}} \]
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Rubi [A]
time = 0.17, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3378, 3387,
3386, 3432, 3385, 3433} \begin {gather*} -\frac {4 \sqrt {2 \pi } b^{3/2} \cos \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {4 \sqrt {2 \pi } b^{3/2} \sin \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {4 b \sin (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rubi steps
\begin {align*} \int \frac {\cos (a+b x)}{(c+d x)^{5/2}} \, dx &=-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}-\frac {(2 b) \int \frac {\sin (a+b x)}{(c+d x)^{3/2}} \, dx}{3 d}\\ &=-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}+\frac {4 b \sin (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {\left (4 b^2\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{3 d^2}\\ &=-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}+\frac {4 b \sin (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {\left (4 b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{3 d^2}+\frac {\left (4 b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{3 d^2}\\ &=-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}+\frac {4 b \sin (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {\left (8 b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{3 d^3}+\frac {\left (8 b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{3 d^3}\\ &=-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}-\frac {4 b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {4 b^{3/2} \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{3 d^{5/2}}+\frac {4 b \sin (a+b x)}{3 d^2 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.21, size = 190, normalized size = 1.13 \begin {gather*} \frac {e^{-i a} \left (-2 i e^{2 i a-\frac {i b c}{d}} \left (e^{\frac {i b (c+d x)}{d}} (-i d+2 b (c+d x))-2 i d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},-\frac {i b (c+d x)}{d}\right )\right )+e^{-i b x} \left (-2 d+4 i b (c+d x)-4 d e^{\frac {i b (c+d x)}{d}} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},\frac {i b (c+d x)}{d}\right )\right )\right )}{6 d^2 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 180, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {-\frac {2 \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -b c}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -b c}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -b c}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}}{d}\) | \(180\) |
default | \(\frac {-\frac {2 \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -b c}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -b c}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -b c}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}}{d}\) | \(180\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.63, size = 129, normalized size = 0.77 \begin {gather*} -\frac {{\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}}}{4 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 208, normalized size = 1.24 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 2 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + \sqrt {d x + c} {\left (d \cos \left (b x + a\right ) - 2 \, {\left (b d x + b c\right )} \sin \left (b x + a\right )\right )}\right )}}{3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\cos {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\cos \left (a+b\,x\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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