Optimal. Leaf size=139 \[ -\frac {2 \cos (a+b x)}{d \sqrt {c+d x}}-\frac {2 \sqrt {b} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sqrt {b} \sqrt {2 \pi } \text {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{d^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {2 \sqrt {2 \pi } \sqrt {b} \sin \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sqrt {2 \pi } \sqrt {b} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \cos (a+b x)}{d \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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)^{3/2}} \, dx &=-\frac {2 \cos (a+b x)}{d \sqrt {c+d x}}-\frac {(2 b) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{d}\\ &=-\frac {2 \cos (a+b x)}{d \sqrt {c+d x}}-\frac {\left (2 b \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{d}-\frac {\left (2 b \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{d}\\ &=-\frac {2 \cos (a+b x)}{d \sqrt {c+d x}}-\frac {\left (4 b \cos \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 )}{d^2}-\frac {\left (4 b \sin \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 )}{d^2}\\ &=-\frac {2 \cos (a+b x)}{d \sqrt {c+d x}}-\frac {2 \sqrt {b} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sqrt {b} \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{d^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.22, size = 147, normalized size = 1.06 \begin {gather*} \frac {e^{-i a} \left (e^{2 i a-\frac {i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \text {Gamma}\left (\frac {1}{2},-\frac {i b (c+d x)}{d}\right )+e^{-i b x} \left (-1-e^{2 i (a+b x)}+e^{\frac {i b (c+d x)}{d}} \sqrt {\frac {i b (c+d x)}{d}} \text {Gamma}\left (\frac {1}{2},\frac {i b (c+d x)}{d}\right )\right )\right )}{d \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 140, normalized size = 1.01
method | result | size |
derivativedivides | \(\frac {-\frac {2 \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -b c}{d}\right )}{\sqrt {d x +c}}-\frac {2 b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -b c}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -b c}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}}{d}\) | \(140\) |
default | \(\frac {-\frac {2 \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -b c}{d}\right )}{\sqrt {d x +c}}-\frac {2 b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -b c}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -b c}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}}{d}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains complex when optimal does not.
time = 0.62, size = 129, normalized size = 0.93 \begin {gather*} \frac {{\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{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 {1}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \sqrt {\frac {{\left (d x + c\right )} b}{d}}}{4 \, \sqrt {d x + c} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 144, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} {\left (\pi d x + \pi c\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + \sqrt {2} {\left (\pi d x + \pi c\right )} \sqrt {\frac {b}{\pi d}} \operatorname {C}\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} \cos \left (b x + a\right )\right )}}{d^{2} x + c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________